Mathematics

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3,038 domain-specific abstractions whose origin domain is Mathematics. They span 1,501 subdomains — sort by that column to group them, or click any subdomain to filter to it.

Abstraction	Subdomain	Description
(a, b)-decomposition	Graph TheoryGraph Decomposition	An (a, b)-decomposition assigns a graph's edges to a specified number a of forest parts and one residual part of maximum degree at most b.
(B, N) Pair	Group Theory	A BN-pair is an ambient group equipped with generating subgroups whose quotient-normalizer data form a Coxeter system and control Bruhat cells, parabolic subgroups, and building geometry.
0/1-polytope	Polyhedral Combinatorics	A convex polytope whose vertices are selected binary vectors from a finite-dimensional hypercube.
2-Category	Category TheoryHigher Category Theory	A strict higher category with objects, arrows between objects, and arrows between parallel arrows, composed by compatible vertical and horizontal laws.
2-group	Higher Category Theory	A monoidal groupoid in which every object has a weak inverse, categorifying the notion of a group.
2-Yoneda lemma	Higher Category Theory	A bicategorical generalization of Yoneda identifying pseudonatural transformations from a representable pseudofunctor to F with the category F assigns to the representing object.
3-Category	Higher Category Theory	A higher category with objects and composable morphisms in dimensions one through three, under specified coherence laws.
3-transposition group	Finite Group Theory	A group generated by a conjugacy class of involutions such that the product of any two generators has order at most three.
3D rotation group	Geometry and Mechanics	The Lie group SO(3) of orientation-preserving linear isometries of three-dimensional Euclidean space, represented by orthogonal matrices of determinant one.
3D4	Finite Group Theory	A twisted family of groups of Lie type obtained from type D4 by combining its order-three triality automorphism with a cubic field automorphism.
A-paracompact Space	General Topology	A topological space in which every open cover has a locally finite refinement, without requiring that the refining family itself be open.
AB5 category	Category TheoryAbelian Categories	An abelian category with arbitrary coproducts in which filtered colimits of exact sequences remain exact; adding a generator yields a Grothendieck category.
ABACABA pattern	Combinatorics on Words and Recursive Patterns	A recursively generated word obtained by placing a new central symbol between two copies of the preceding word, producing lengths one less than powers of two.
Abel Transform	Integral GeometryAxisymmetric Tomography	The symmetry-reduced line-integral transform that maps a radial field to its parallel projection and, under stated conditions, reconstructs the local radial field from that projection.
Abel's Limit Theorem	Real AnalysisPower Series	Equating a convergent coefficient series with the radial boundary limit of its associated power series.
Abelian Lie group	Lie Theory	A smooth Lie group whose multiplication is commutative, combining a finite-dimensional manifold with an abelian group structure and smooth operations.
Absolute continuity	Real Analysis	A strengthened continuity property that makes total function variation over sufficiently short disjoint intervals arbitrarily small and restores integration of the derivative.
Absolute convergence	Mathematical Analysis	Convergence of a series or improper integral after replacing every summand or integrand value by its absolute magnitude.
Absolute value	Elementary AlgebraReal Analysis	In mathematics, the absolute value or modulus of a real number x , is the (non-negative) magnitude measured without regard to its sign.
Absolute value (algebra)	Algebra	A nonnegative multiplicative or submultiplicative magnitude function on a field or integral domain that separates zero and satisfies the triangle inequality.
Absolutely convex set	Functional AnalysisTopological Vector Spaces	In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk.
Absorbing boundary condition	Numerical Wave Analysis	An artificial computational boundary rule designed to let outgoing waves exit a truncated domain with minimal reflection.
Absorbing element	Algebra	An element that returns itself whenever combined with any element under a specified binary operation.
Absorbing set	Functional AnalysisTopological Vector Spaces	A subset of a vector space whose scalar dilations eventually contain every vector, forming a basic neighborhood and boundedness concept in topological vector spaces.
Abstract additive Schwarz method	Numerical AnalysisDomain Decomposition Methods	In mathematics, the abstract additive Schwarz method, named after Hermann Schwarz, is an abstract version of the additive Schwarz method for boundary value problems on partial differential equations, formulated only in terms of linear algebra without reference to domains, subdomains, etc.
Abstract index notation	Differential Geometry	A basis-free tensor notation in which index letters name argument slots and variance rather than numerical components.
Abstract polytope	Discrete GeometryCombinatorics	In mathematics, an abstract polytope is an algebraic partially ordered set which captures certain combinatorial properties of a traditional polytope without specifying purely geometric properties such as the position of vertices.
Accelerated failure time model	Survival AnalysisEvent Time Regression	Model covariates as multiplying an event-time scale—equivalently shifting log survival time—so coefficients are interpreted through time ratios rather than the constant hazard ratios of proportional-hazards regression.
Acceptable ring	Commutative Algebra	In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein rings.
Accessibility Relation	Mathematical LogicModal Logic	A binary relation on the worlds or states of a Kripke frame that fixes which alternatives a modal operator quantifies over from each evaluation point.
Accessible category	Category Theory	A category that, for some regular cardinal λ, has λ-filtered colimits and a set of λ-presentable objects from which every object is obtainable as a λ-filtered colimit, making its large object class controllable by bounded presentability data.
Accessible quasi-category	Higher Category Theory	An infinity-category equivalent to the closure of a small infinity-category under kappa-filtered colimits for some regular cardinal kappa.
Achilles Number	Elementary Number TheoryRecreational Mathematics	A positive integer that is powerful because every prime exponent is at least two, yet is not a perfect power because those exponents have greatest common divisor one.
Action algebra	Algebraic Logic	An algebraic-logic structure combining Kleene algebra's composition and iteration with residuated semilattice implication operations.
Action groupoid	Category Theory	The groupoid whose objects are points acted on by a group and whose arrows record group elements carrying one point to another.
Acute and obtuse triangles	Euclidean Geometry	A Euclidean triangle classification based on whether all angles are below a right angle or exactly one angle exceeds it.
AD+	Set Theory	Woodin’s strengthening of the Axiom of Determinacy that combines dependent choice for reals, ordinal determinacy below Theta, and infinity-Borel definability for every set of reals.
Adams Spectral Sequence	Algebraic TopologyStable Homotopy TheoryHomological Algebra	A prime-local spectral sequence whose early terms are Ext groups over a stable cohomology-operations algebra and whose convergent limit organizes information about stable homotopy groups or maps.
Addition principle	Combinatorics	The counting rule that mutually exclusive alternatives have a total number of possibilities equal to the sum of their individual counts.
Additive function	Multiplicative Number Theory	An arithmetic function satisfying f(ab)=f(a)+f(b) whenever positive integers a and b are coprime.
Additive group	Group TheoryAbstract Algebra	A group whose binary operation is interpreted and written as addition, especially the underlying usually abelian group obtained from a ring, field, vector space, or other multi-operation structure by forgetting its other operations.
Additive inverse	Abstract Algebra	For an element in an additive algebraic structure, another element whose sum with it is the additive identity.
Additively indecomposable ordinal	Ordinal Arithmetic and Set Theory	A nonzero ordinal alpha that cannot be reached or exceeded by adding two smaller ordinals, equivalently an ordinal of the form omega raised to an ordinal power.
Adequate subcategory	Category Theory	In category theory, a branch of mathematics, an adequate subcategory of a category X is an analog of a dense subspace in topology for presheaves: namely, a subcategory i : A \hookrightarrow X such that the restriction of the Yoneda embedding X \hookrightarrow \mathbf{P}(X) along i is still fully faithful.
Adherent point	General Topology	A point every neighborhood of which intersects a selected subset, equivalently a member of that subset's closure.
Adjunction (field theory)	Field TheoryAbstract Algebra	Adjunction in field theory forms K(S), the smallest subfield of an ambient extension containing a base field K together with every element of a specified set S.
Admissible set	Set Theory	A transitive set whose membership structure satisfies Kripke–Platek set theory.
Affiliated operator	Operator AlgebrasFunctional Analysis	A closed and densely defined operator A is said to be affiliated with M if A commutes with every unitary operator U in the commutant of M.
Affine hull	Linear AlgebraConvex Geometry	In mathematics, the affine hull or affine span of a set S in Euclidean space \mathbb{R}^n is the smallest affine set containing S , or equivalently, the intersection of all affine sets containing S .
Affine logic	Substructural LogicMathematical LogicProof Theory	Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction.
Affine Process	Stochastic Processes	A Markov process whose conditional transition characteristic function is exponential-affine in its initial state.
Age (Model Theory)	—	The isomorphism-closed class of finitely generated structures that embed into a fixed model, recording exactly its finite or finitely generated local patterns.
Ahlswede–Daykin inequality	Combinatorics	A four-functions inequality on a finite distributive lattice that lifts a pointwise join–meet product bound to corresponding sums over subsets.
Airy function	Special Functions	In mathematics, the Airy function (or Airy function of the first kind) \mathbf{Ai(\boldsymbol x)} is a special function named after the British astronomer George Biddell Airy.
Airy Process	Probability Theory	A family of stationary stochastic edge-limit processes whose Fredholm-determinant finite-dimensional laws describe spatial KPZ and random-matrix fluctuations under characteristic initial geometries.
Aitken's delta-squared process	Numerical Analysis	A nonlinear sequence transformation that accelerates approximately linear convergence by extrapolating from three consecutive terms and canceling the leading error mode.
Aleph number	Set Theory	A member of the transfinite sequence of well-ordered infinite cardinalities, indexed by ordinals with aleph-null as the size of the natural numbers.
Alexander Duality	Algebraic Topology	Convert reduced homology of the complement of a qualifying compact subspace of a sphere into reduced cohomology of the subspace with the exact degree reversal q ↦ n−q−1.
Algebra bundle	Differential GeometryFiber Bundles	In mathematics, an algebra bundle is a fiber bundle whose fibers are algebras and local trivializations respect the algebra structure.
Algebra of random variables	Probability Theory	The symbolic calculus for forming functions of random variables and deriving the resulting distributions, moments and dependence-sensitive identities.
Algebra over a Ring	—	An R-module equipped with an R-bilinear internal multiplication, with associativity, unity, and commutativity imposed only at the explicitly declared convention tier.
Algebraic cobordism	Algebraic Geometry	Provide the universal oriented cohomology theory for smooth algebraic varieties, encoding projective pushforwards and first Chern classes through the universal one-dimensional commutative formal group law.
Algebraic curve	Algebraic Geometry	A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables.
Algebraic Cycle	Algebraic GeometryIntersection Theory	A dimension-graded formal integer sum of integral closed subvarieties that can be compared by geometric equivalence and assembled into Chow groups.
Algebraic Extension	Field Theory	A field extension in which every element of the larger field satisfies a nonzero polynomial with coefficients in the embedded base field.
Algebraic Matroid	Matroid TheoryCombinatorics	In mathematics, an algebraic matroid is a matroid, a combinatorial structure, that expresses an abstraction of the relation of algebraic independence.
Algebraic normal form	Boolean AlgebraFinite Fields	The unique multilinear polynomial over GF(2) representing a Boolean function as an XOR of square-free AND monomials and an optional constant.
Algebraic number field	Algebraic Number TheoryNumber Fields	A finite-degree field extension of the rational numbers, carrying arithmetic through its ring of integers, embeddings, ideals, norms, traces, units, and completions.
Algebraic Operation	AlgebraUniversal Algebra	An algebraic operation is a typed finitary mapping that takes one or more elements or structured algebraic objects as operands and returns an algebraic result under declared domain, codomain, arity, closure, and governing identities or compatibility conditions.
Algebraic set	Algebraic Geometry	A set of points defined as the simultaneous zero locus of a family of polynomial equations, without an irreducibility requirement.
Algebraic space	Algebraic GeometryGeneralized Geometric Objects	A sheaf on the étale site admitting a representable étale surjection from a scheme, generalizing schemes by allowing étale-local rather than Zariski-local affine charts.
Algebraic stack	Algebraic Geometry and Moduli	A stack with algebraic representability and smooth-cover conditions that generalizes schemes and algebraic spaces while retaining automorphism data in moduli problems.
Algebraic Structure	Abstract AlgebraUniversal Algebra	One or more carrier sets equipped with typed operations, distinguished elements, and laws that define an algebraic kind and its structure-preserving mappings.
Algebraic Surface	Algebraic GeometryBirational Geometry	A dimension-two algebraic variety over a specified field, defined by polynomial data and studied with explicit affine/projective, singularity, and birational conventions.
Algebraic Variety	Algebraic Geometry	A geometric object locally described by polynomial equations over a declared field and glued by regular maps, with coordinate algebra, Zariski topology, dimension, morphisms, and singularities interpreted under an explicit convention.
Algebraically closed field	Field Theory and Algebraic Geometry	A field in which every nonconstant univariate polynomial with coefficients in that field has a root, equivalently splits completely into linear factors.
Aliquot sum	Number Theory	In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself.
All one polynomial	Finite Field Polynomials	A polynomial whose coefficients from degree zero through its leading degree are all one, equivalently (xᵐ⁺¹−1)/(x−1).
Almost complex manifold	Differential GeometryAlmost Complex Structures	A smooth even-dimensional manifold equipped with a smoothly varying tangent-bundle endomorphism J whose square is minus the identity, providing pointwise complex linear structure without necessarily admitting complex coordinates.
Almost Flat Manifold	Riemannian Geometry	A connected closed smooth manifold that admits metrics with absolute sectional-curvature bound times diameter squared below every positive threshold.
Alphasyllabic numeral system	History of Numeration	A numeral notation that assigns numeric values to consonant–vowel syllable signs of an alphasyllabic script and composes them to encode numbers.
Alternating group	Group Theory	The subgroup A_n of the symmetric group S_n consisting of all even permutations of n objects, equivalently the kernel of the sign homomorphism, with order n!/2 for n at least two.
Alternating-Direction Implicit Method	Numerical AnalysisOperator Splitting Methods	A numerical method that alternates implicit solves across separated spatial directions or operators.
Ambient space (mathematics)	GeometryEmbeddings and Context	The surrounding mathematical space in which an object is embedded or considered, whose geometry and topology determine extrinsic relations not fixed by the object in isolation.
Amenable number	Recreational Number Theory	A positive integer n admitting a multiset of exactly n integers whose sum and product are both n.
Amicable triple	Number TheoryRecreational Mathematics	Three distinct natural numbers whose aliquot sum for each equals the sum of the other two, equivalently sharing a divisor sum equal to the triple’s total.
Amnestic Functor	Category TheoryCategorical Topology	A functor for which any source isomorphism mapped to a target identity is necessarily already an identity morphism.
Amoeba order	Set Theoretic Cardinal Invariants	An order on cardinal characteristics that compares how strongly families of measure-small sets can cover or absorb other small sets under a declared null-ideal convention.
Ample line bundle	Algebraic GeometryLine Bundle Positivity	A line bundle whose sufficiently high tensor power gives an embedding of a projective variety into projective space, expressing algebro-geometric positivity.
AM–GM Inequality	Inequalities	For any nonempty finite list of nonnegative real numbers, its geometric mean does not exceed its arithmetic mean, with equality exactly when all entries are equal.
Analytic Combinatorics	CombinatoricsAsymptotic Analysis	Walter Hayman's 1956 paper "A Generalisation of Stirling's Formula" is considered one of the earliest examples of the saddle-point method.
Analytic Function	Real and Complex Analysis	A real or complex function that, around every point of its open domain, equals a convergent power series in the relevant coordinates.
Analytic semigroup	Operator Semigroups and Pde	Extend a strongly continuous operator semigroup holomorphically into a complex-time sector, linking sectorial generators to parabolic regularization.
Andreotti–Norguet Formula	Several Complex VariablesComplex Analysis	The Andreotti–Norguet formula, first introduced by , is a higher–dimensional analogue of Cauchy integral formula for expressing the derivatives of a holomorphic function.
Anti-Diagonal Matrix	Linear AlgebraMatrix Theory	In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal).
Anticommutative property	AlgebraOperation Symmetries	A binary-operation property in which exchanging the two arguments yields the additive inverse of the original result.
Antiderivative	Calculus	A function whose derivative equals a given function on a declared domain, with all solutions differing by constants on each connected component under standard assumptions.
Antihomomorphism	Abstract AlgebraNoncommutative Algebra	A map between operation-bearing structures that preserves a binary product in reversed order: the image of a product is the product of the images in the opposite order.
Antilinear Map	Complex Linear Algebra	An additive map between complex vector spaces that transports scalar multiplication through complex conjugation, so A(lambda v) = conjugate(lambda) A(v) rather than lambda A(v).
Antimatroid	CombinatoricsConvex Geometries and Greedoids	A union-closed accessible set system modeling knowledge or construction states in which feasible elements can be added one at a time and, once available, remain available until chosen.
Antiunitary operator	Functional Analysis and Quantum Symmetry	A bijective conjugate-linear map between complex Hilbert spaces that conjugates their inner products and therefore preserves norms and transition probabilities.
Apartness relation	Constructive Mathematics	A constructive positive notion of distinction, typically irreflexive, symmetric and cotransitive, that is stronger than merely denying equality.
Aperiodic graph	Graph Theory	In the mathematical area of graph theory, a directed graph is said to be aperiodic if there is no integer k > 1 that divides the length of every cycle of the graph.
Aperiodic Semigroup	Semigroup Theory	A semigroup in which each element's positive powers eventually stop changing under one more multiplication by that element.
Apollonian Gasket	Fractal Geometry and Circle Packing	An Apollonian gasket is the fractal residual structure of an infinite circle packing formed by recursively filling every curvilinear triangular gap among three mutually tangent circles with its uniquely tangent incircle.
Apollonian network	Graph Theory	A recursively generated planar graph formed by repeatedly inserting a vertex into a triangular face and joining it to that face’s three vertices.
Arakelian's Theorem	Complex AnalysisHolomorphic Approximation	Arakelian's theorem makes connectedness and local connectedness of a one-point complement equivalent to universal uniform holomorphic approximation on a closed plane-domain set.
Arakelov theory	Arithmetic Geometry	An arithmetic geometry that augments schemes over the integers with analytic data at infinite places.
Arboricity	Graph TheoryGraph Decomposition	Measure a graph's edge density by the fewest acyclic forest layers needed to partition all its edges.
Arf ring	Commutative Algebra	A one-dimensional semilocal Cohen–Macaulay ring satisfying the Arf closure condition that controls integrally closed ideals and multiplicity sequences.
Arithmetic function	Number Theory	A function defined on positive integers, usually with complex values, that encodes a number-theoretic property of each integer.
Arithmetic Group	Arithmetic GroupsAlgebraic Groups	A group defined relative to integral points of a number-field algebraic group through finite-index commensurability.
Arithmetic number	Number Theory	A positive integer whose positive divisors have an integer arithmetic mean.
Arithmetic operation	Arithmetic	A rule-governed operation on numbers or number-like objects—such as addition, subtraction, multiplication, division, powers, roots, or logarithms—defined by its operands, domain, result, and closure conditions.
Arithmetic Progression	Elementary Algebra	Generate an ordered numeric sequence by repeatedly adding one fixed common difference, making every term affine in its discrete index.
Arithmetic progression topologies	Number Theoretic Topology	Topologies on the integers generated by selected arithmetic progressions as a basis, linking divisibility and congruence structure to topological properties.
Arithmetic surface	Arithmetic Geometry	A regular or suitably controlled two-dimensional scheme fibered over the spectrum of a Dedekind domain, with generic fiber an algebraic curve.
Arithmetic–Geometric Mean	Elliptic IntegralsNumerical Analysis	The arithmetic–geometric mean is the common limit of coupled arithmetic-mean and geometric-mean iterations on two positive numbers.
Arnold diffusion	Dynamical SystemsHamiltonian Perturbation Theory	Long-term transport of action variables along resonant pathways in a slightly perturbed integrable Hamiltonian system, enabled where surviving invariant tori do not separate the relevant phase space.
Aronszajn line	Set Theory	A linear order of cardinality aleph-one containing neither an omega-one or reverse-omega-one suborder nor an uncountable real-type suborder.
Aronszajn tree	Set Theory	A tree of height ω₁ whose levels and branches are all countable, generalized to κ-trees with levels and branches smaller than κ.
Arrangement of hyperplanes	Geometry and CombinatoricsHyperplane Arrangements	Study a finite family of affine, linear, or projective hyperplanes through its intersection poset, complement, regions, and combinatorial-topological invariants.
Arrangement of Pseudolines	Discrete Geometry	A finite family of line-like curves crosses pairwise exactly once, preserving line-arrangement combinatorics without requiring straight-line realization.
Arrowhead matrix	Matrix Theory	A square matrix whose only potentially nonzero entries lie on the main diagonal and one selected row and matching column.
Artin conductor	Algebraic Number Theory	The Artin conductor assigns a ramification-sensitive number or ideal to a Galois-group character.
Artinian Ideal	Commutative Algebra	An ideal of a polynomial ring whose quotient is Artinian, equivalently zero-dimensional over a field.
Artin–Tate lemma	Commutative Algebra	If A is commutative Noetherian, C is a finite-type A-algebra and finite as a module over an intermediate A-subalgebra B, then B is finite type over A.
Associated bundle	Fiber Bundles and Gauge Geometry	A fiber bundle obtained from a principal G-bundle and a G-space by quotienting their product under the diagonal group action.
Associated graded ring	Commutative Algebra	The graded ring formed from successive quotients of powers in an ideal filtration, preserving leading-order information while discarding higher filtration terms.
Association Scheme	Algebraic Combinatorics	An association scheme partitions a Cartesian square into binary relations satisfying identity, transpose, and uniform intersection-number conditions.
Associative algebra	Abstract Algebra	An algebra over a commutative ring whose internal multiplication satisfies associativity.
Associativity Isomorphism	Category TheoryMonoidal Categories	A natural family of isomorphisms rebracketing a categorical tensor product, constrained by the pentagon coherence identity.
Assouad–Nagata Dimension	Metric GeometryDimension Theory	A metric dimension defined by uniformly bounded, uniformly low-multiplicity covers at every scale, with cover diameter growing at most linearly with scale.
Asymmetric graph	Graph Theory	A graph whose automorphism group is trivial, so no nonidentity permutation of vertices preserves adjacency.
Asymmetric simple exclusion process	Stochastic Processes	A continuous-time interacting-particle model on a lattice where biased nearest-neighbor jumps occur only into vacant sites.
Asymptote	Analytic GeometryCalculus	A straight line approached arbitrarily closely by an unbounded branch of a plane curve in a specified limiting direction.
Asymptotic analysis	Mathematical AnalysisLimiting Approximations	The study of approximations and expansions that become accurate as a variable approaches a specified limit, emphasizing dominant scales and controlled remainder behavior.
Atom (measure theory)	Measure TheoryAtomic and Nonatomic Measures	Identify a measurable set of positive measure whose measurable subsets have either zero measure or the atom’s full measure.
Atom (Order Theory)	Order Theory	In the mathematical field of order theory, an element a of a partially ordered set with least element 0 is an atom if 0 0 has an atom a below it, that is, there is some a such that b ≥ a :> 0.
Atomic Sentence	Formal Logic	A closed formula formed by an atomic base clause, not by a connective or quantifier, in a specified formal logic.
Aubin–Lions lemma	Functional Analysis	A compactness result for time-dependent functions combining spatial compact embedding with control of a time derivative in a weaker space.
Auslander–Reiten theory	Representations of Artin Algebras and Additive Categories	A representation-theoretic framework organizing indecomposable modules and morphisms through almost-split sequences and Auslander–Reiten quivers.
Automatic Differentiation	Numerical ComputingDerivative Computation	A family of program-evaluation and transformation techniques that decomposes an executed numerical computation into differentiable primitives and composes their local derivative rules to obtain derivatives accurate to working precision, chiefly through forward Jacobian–vector or reverse vector–Jacobian accumulation.
Automatic Group	Geometric Group Theory	A finitely generated group with a regular covering language of representative words and synchronous finite-state recognition of multiplication by generators.
Automorphic number	Recreational MathematicsNumber Theory	A base-b natural number whose numeral reappears as the final k digits of its square, equivalently an idempotent n²≡n modulo b^k for k equal to its digit length.
Automorphism Group	Abstract AlgebraAutomorphism Groups	The group obtained by collecting every structure-preserving self-isomorphism of a fixed mathematical object and using composition as the group operation.
Auxiliary function	Number TheoryTranscendence Methods	A deliberately constructed function with engineered zeros, growth or arithmetic properties that converts a target claim—especially in transcendence theory—into an estimate or contradiction.
Average Order of an Arithmetic Function	Arithmetic Function AsymptoticsSummatory Methods	Describe an arithmetic function's aggregate growth with a simpler function whose initial-interval sums are asymptotically equivalent.
Axiom of countable choice	Weak Choice Principles	Assert that every countable family of nonempty sets admits a choice function, retaining a strictly weaker set-theoretic commitment than full Choice and a different strength from Dependent Choice.
Axiom of Dependent Choice	Set TheoryWeak Choice Principles	Assert an infinite coherent sequence whenever every element of a nonempty set has a permitted successor.
Axiom of determinacy	Set Theory	A set-theoretic axiom asserting that every infinite two-player perfect-information game on natural numbers has a winning strategy for one player.
Axiom of infinity	Mathematical LogicAxiomatic Set Theory	Assert in Zermelo–Fraenkel set theory that an inductive set exists—one containing the empty set and closed under the successor x mapped to x union singleton x—thereby supplying a set from which omega and the natural-number sequence can be isolated.
Axiom of Regularity	Set TheoryMathematical Logic	The ZF foundation axiom requiring every nonempty set to contain a member disjoint from it, enforcing well-founded membership and ruling out self-containing or endlessly descending membership structures under stated background axioms.
Axiom schema	Mathematical Logic	A metalanguage template with substitution conditions that generates a potentially infinite family of object-language axioms.
Aztec Diamond	Enumerative Combinatorics	Use an order-indexed diamond-shaped square-lattice region whose domino exact covers support a power-of-two enumeration, reversible shuffling, path representations, and an arctic-circle limit shape.
A∞-operad	Algebraic TopologyHomotopy Theory	In mathematics, an A ∞ -operad is a type of operad used in algebraic topology and homotopy theory to describe algebraic structures where the property of associativity is loosened.
B, C, K, W system	Combinatory Logic	A basis for combinatory logic using composition, permutation, constant and duplication combinators as primitives.
Ba space	Measure Theory and Functional Analysis	The Banach space of bounded finitely additive signed measures on an algebra of sets, equipped with the total-variation norm.
Back-and-Forth Method	Model TheoryCountable Structures	A countable-structure isomorphism method that alternately extends finite partial isomorphisms to cover the next source and target elements, then unions the chain.
Background field method	Quantum Field TheoryEffective Action Methods	A quantum-field-theory method that splits a field into a prescribed background and a fluctuating quantum part so effective actions can be computed while preserving useful covariance or gauge symmetry.
Baer group	Group Theory	A group in which every cyclic subgroup is subnormal.
Baer norm	Group Theory	The characteristic subgroup formed by intersecting the normalizers of every subgroup of a group.
Bailey–Borwein–Plouffe Formula	Computational Number Theory	A base-16 rational series for π whose radix alignment lets modular exponentiation recover hexadecimal or binary digits at a distant position without first generating the intervening expansion.
Bailout Embedding	Dynamical Systems	Embed a dynamical system in a larger phase space whose transverse dynamics repel trajectories from unwanted unstable regions and contract them back onto selected stable invariant motion.
Baire function	Descriptive Set TheoryBaire Classes	A real-valued function obtainable from continuous functions by countable transfinite iteration of pointwise sequential limits, classified by the least countable ordinal stage required.
Baire one star function	Real Analysis	A real function whose restriction is continuous on some nonempty relatively open piece of every perfect set.
Balanced hypergraph	Hypergraph Theory	A hypergraph with no strong odd cycle, equivalently one whose incidence matrix is balanced and supports bipartite-like integrality properties.
Balanced set	Topological Vector Spaces	A subset of a real or complex vector space closed under multiplication by every scalar of absolute value at most one.
Balancing domain decomposition method	Numerical Linear Algebra and Domain Decomposition	A nonoverlapping domain-decomposition preconditioner that combines independent subdomain solves with a coarse correction assembled from local nullspaces to solve symmetric positive-definite finite-element systems.
Balayage	Classical and Probabilistic Potential Theory	An operator that sweeps an interior measure onto a domain boundary while preserving its Newtonian potential outside the closed domain.
Balinski's theorem	Polyhedral Combinatorics	In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional convex polyhedra and higher-dimensional convex polytopes.
Ball Tree	Computational GeometryMetric Space Indexing	Index points in a metric space with a hierarchy of enclosing balls so triangle-inequality lower bounds prune whole subtrees during exact nearest-neighbor and geometric search.
Banach Algebra	Functional Analysis	A Banach algebra is called unital if it has an identity element for the multiplication whose norm is 1, and commutative if its multiplication is commutative.
Banach bundle	Global Analysis and Infinite Dimensional Geometry	A fiber bundle whose fibers are Banach spaces and whose local trivializations and transition maps preserve compatible continuous linear structure.
Banach Space	Functional Analysis	Complete a normed real or complex vector space so every norm-Cauchy approximation sequence converges to an element of the same space, making limits compatible with linear combination and continuous-operator analysis.
Banach–Alaoglu Theorem	Functional Analysis	The normed-space theorem making every closed dual ball compact for pointwise-on-the-predual, or weak-*, convergence.
Banach–Mazur compactum	Functional Analysis	The compact metric space of isometry classes of fixed-dimensional normed spaces under logarithmic Banach–Mazur distance.
Bar complex	Homological Algebra	A canonical chain complex built from iterated tensor products to resolve an algebra, group or related object.
Baranyai's theorem	Combinatorial Design	The theorem that a complete uniform hypergraph can be decomposed into perfect matchings whenever the edge size divides the number of vertices.
Barron space	Approximation TheoryNeural Network Function Spaces	A function space characterized by integral representations or spectral moment bounds that control approximation by two-layer neural networks with dimension-favorable error rates.
Bartlett's method	Time Series Analysis	A power-spectrum estimator that averages periodograms from non-overlapping equal-length segments to reduce variance at the cost of frequency resolution.
Bartlett's theorem	Applied Probability	A Poisson-arrival occupancy theorem for independently moving individuals in a system's regions.
Barycentric-sum problem	Combinatorial Number Theory	The barycentric-sum problem asks for the minimum sequence length that guarantees a subsequence containing a term equal to its modular average.
Base Rate	Probability and StatisticsBayesian Inference	The prevalence or prior probability of a class before case-specific evidence is incorporated.
Baskakov operator	Functional AnalysisApproximation Theory	In functional analysis, a branch of mathematics, the Baskakov operators are generalizations of Bernstein polynomials, Szász–Mirakyan operators, and Lupas operators.
Bates Distribution	Probability Theory	The probability distribution of the arithmetic mean of n independent identically distributed uniform random variables, a scaled Irwin–Hall sum.
Bauer Maximum Principle	Convex Analysis	A convex upper-semicontinuous function on a nonempty compact convex set attains its maximum at an extreme point.
Baumslag–Gersten Group	Combinatorial Group Theory	The two-generator one-relator group whose conjugation tower couples an ascending HNN structure to unusually large filling complexity, non-residual finiteness, and a sharply non-elementary yet tractable word problem.
Baum–Connes Conjecture	Operator AlgebrasNoncommutative Geometry	An assembly-map conjecture relating a group's geometric equivariant K-homology to the analytic K-theory of its reduced group C-star algebra.
Baxter Permutation	CombinatoricsPermutation Pattern Theory	A finite permutation avoiding the adjacency-sensitive patterns 2-41-3 and 3-14-2, forming an enumerated family with recursive, planar, tree, rectangulation, and algebraic representations.
Bayes Factor	Statistics	The ratio of the marginal likelihoods of the data under two specific models, isolating the data's weight of evidence between them and updating prior model odds to posterior odds multiplicatively — with an automatic Occam penalty on flexible models.
BCK algebra	Universal Algebra and Substructural Logic	An algebra with a binary implication-like operation and a distinguished zero satisfying the BCI identities plus an axiom that enforces the BCK weakening condition.
Beck's monadicity theorem	Category Theory	A categorical criterion determining when a functor is equivalent to the forgetful functor from algebras for the monad induced by its adjunction.
Beck–Chevalley Condition	Category TheoryCategorical Logic	The invertibility of the canonical comparison between base change and an adjoint operation around a categorical square.
Behnke–Stein Open Riemann Surface Theorem	Complex Analysis	Every connected noncompact Riemann surface is Stein, with global holomorphic separation and convexity.
Behrend function	Algebraic Geometry and Moduli	An intrinsic integer-valued constructible weight on a complex scheme whose Euler integral can recover suitable virtual counts.
Belief Aggregation	Probability TheoryProbabilistic Opinion Pooling	Pool several probability assignments over a common event space into one collective distribution under an explicit rule.
Berge–Zhukovskii Equilibrium	Noncooperative Game TheoryAlternative Equilibrium Concepts	A normal-form game profile at which, holding any one player's own strategy fixed, no joint change by all the other players can raise that player's payoff—the mutual-support counterpart to Nash's own-deviation stability.
Bernoulli number	Number Theory and AnalysisSpecial Number Sequences and Generating Functions	Define a rational-number sequence by the exponential generating function for x divided by exp(x) minus one, yielding coefficients that govern power sums, Euler-Maclaurin correction, and zeta-value identities.
Bernstein inequalities (probability theory)	Probability Theory	Bernstein inequalities are exponential concentration bounds that control the deviation of sums of independent bounded random variables using both variance and a bound on individual magnitude.
Bernstein polynomial	Approximation Theory and Computer Graphics	A polynomial represented in the Bernstein basis, whose nonnegative partition-of-unity weights support stable approximation, shape preservation, and Bézier geometry.
Bernstein's theorem (approximation theory)	Approximation Theory	In approximation theory, Bernstein's theorem is a converse to Jackson's theorem.
Bernstein's Theorem (Polynomials)	Polynomial InequalitiesApproximation Theory	Bernstein's polynomial theorem sharply bounds a complex polynomial's unit-circle derivative norm by its degree times its value norm.
Bernstein–Zelevinsky classification	Representation Theory	In mathematics, the Bernstein–Zelevinsky classification, introduced by and , classifies the irreducible complex smooth representations of a general linear group over a local field in terms of cuspidal representations.
Bertrand paradox (probability)	Probability Foundations	A geometric-probability paradox in which different seemingly natural random-chord constructions produce different answers, revealing that randomness requires a specified measure.
Besov space	Functional Analysis	A scale of function spaces measuring smoothness through integrability and multiscale difference or frequency-decay parameters, generalizing Sobolev and Hölder spaces.
Bessel Function	Special FunctionsOrdinary Differential Equations	A parameterized special-function family solving Bessel's equation and furnishing radial modes for cylindrical separation problems.
Bessel–Clifford function	Special FunctionsBessel and Hypergeometric Functions	Use an entire reciprocal-gamma-normalized power series whose derivative shift, differential equation, and square-root substitutions organize ordinary and modified Bessel functions.
Beth number	Set Theory	A transfinite cardinal sequence beginning at countable infinity and repeatedly applying power set at successors and supremum at limit ordinals.
BF Algebra	Universal Algebra	A type-(2,0) algebra with a distinguished zero and a binary operation satisfying self-cancellation, right-zero identity, and zero-mediated reversal.
Bhargava factorial	Number Theory and Integer Valued Polynomials	A factorial sequence attached to an arbitrary subset of the integers through p-orderings, generalizing n-factorial while preserving divisibility and integer-valued-polynomial properties.
Bialgebra	Abstract AlgebraAlgebra Coalgebra Compatibility	A vector space carrying compatible unital associative algebra and counital coassociative coalgebra structures, so multiplication and unit are coalgebra maps equivalently comultiplication and counit are algebra maps.
Biased Graph	Graph Theory	A graph with selected balanced cycles satisfying the theta rule: no theta subgraph has exactly two balanced cycles.
Biclique-free graph	Graph Theory	A graph excluding some fixed complete bipartite graph as a subgraph.
Bicommutant	Operator AlgebrasAbstract Algebra	In algebra, the bicommutant of a subset S of a semigroup (such as an algebra or a group) is the commutant of the commutant of that subset.
Bicomplex Number	Abstract AlgebraHypercomplex Number Systems	Extend complex arithmetic with a second commuting imaginary unit, producing a four-real-dimensional commutative algebra whose idempotent decomposition reveals two coupled complex components and zero divisors.
Biconnected Component	Graph Theory	A maximal connected edge-bearing block of an undirected graph with no internal articulation vertex, counting a bridge as a one-edge block under an explicit convention.
Bicyclic semigroup	Structure Theory of Semigroups	Use the universal monoid generated by two elements whose product in one order is the identity while the reverse product is not, yielding a canonical countable inverse-semigroup test object.
Bidiagonal matrix	Linear AlgebraStructured Matrices	A banded matrix whose potentially nonzero entries lie only on the main diagonal and one adjacent superdiagonal or subdiagonal, yielding simple determinants, eigenvalues, products, and efficient structured computations.
Bijective numeration	Numeral Systems	A numeral system giving every nonnegative integer exactly one finite digit string, with no zero digit or leading-zero ambiguity in its positional form.
Bijective proof	Enumerative CombinatoricsCombinatorics	A proof of equal cardinality by an explicit invertible map pairing every object in one combinatorial class with exactly one object in another.
Bimodule	Abstract AlgebraModule Theory	An abelian group with a left action by one ring and a right action by another that commute, so applying the two actions in either compatible order gives the same result.
Binary relation	Set Theory and Discrete Mathematics	Represent which ordered pairs from two declared sets stand in a relation by selecting a subset of their Cartesian product, enabling converse, composition, closure, and relational properties.
Binary space partitioning	Computational Geometry	A recursive geometric partition that divides a space by hyperplanes into two half-spaces and stores the resulting hierarchy in a binary tree.
Binomial (polynomial)	AlgebraPolynomials and Toric Geometry	A polynomial consisting of exactly two nonzero monomial terms, whose sparse form supports binomial expansions, toric ideals and algebraic varieties governed by exponent differences.
Binomial number	Number Theory	An integer generated from a homogeneous two-term power form, principally xⁿ+yⁿ or the normalized difference (xⁿ−yⁿ)/(x−y), under stated integer conditions.
Binomial transform	Combinatorics	An invertible triangular sequence transform that combines source terms with signed or unsigned binomial coefficients under a declared convention.
Biordered set	Semigroup TheoryIdempotent Structures	An abstract set of idempotent-like elements equipped with compatible left and right quasiorders and partial basic products that axiomatize the idempotent structure of a semigroup.
Biquadratic field	Algebraic Number Theory	A degree-four Galois extension of the rational numbers with Klein-four Galois group.
Biregular graph	Graph Theory	A bipartite graph in which all vertices on each side have one common degree, with the two sides allowed different degrees.
Birkhoff Factorization	Loop Groups	A loop-group decomposition expressing an invertible Laurent-polynomial or suitable loop-valued matrix as a product of positive-power, diagonal monomial, and negative-power factors, generalizing finite-dimensional LU decomposition.
Birman–Schwinger Principle	Spectral TheoryOperator Theory	Turn a perturbed-operator eigenvalue question into a fixed-threshold eigenvalue question for a resolvent-sandwiched auxiliary operator, preserving eigenspace and counting information under stated hypotheses.
Bisection Method	Numerical Analysis	Locate a zero of a continuous real function by preserving an opposite-sign interval and repeatedly replacing it with the sign-changing half, obtaining a deterministic enclosure whose width halves at every step.
Bismut Connection	Differential GeometryHermitian Geometry	The unique connection on a Hermitian manifold that preserves both the metric and complex structure while having totally skew-symmetric torsion, reducing to Levi–Civita in the Kähler case.
BIT predicate	Mathematical LogicArithmetic Coding	A binary relation that returns the zero-based j-th binary digit of a nonnegative integer i, enabling finite-set membership tests by bit position.
Bivariegated graph	Graph Theory	An even-order graph whose vertices split into equal parts so every vertex has exactly one neighbor across the split.
Björling problem	Minimal Surface Geometry	The analytic minimal-surface problem of constructing a minimal surface through a prescribed real-analytic space curve with a prescribed compatible normal field.
BK-space	Functional Analysis	A Banach sequence space in which every coordinate projection is continuous.
Blackwell Informativeness Order	Statistical Decision Theory	Compare experiments by whether one can be garbled into another, equivalently by universal Bayes decision value under standard finite assumptions.
Blackwell–Girshick Equation	Probability Theory	A two-term identity separating the variance of an independent-count random sum into mark-size and count-uncertainty contributions.
Blaschke Product	Complex AnalysisHardy Spaces	A finite or convergent infinite product of disk-automorphism factors that realizes prescribed zeros while forming an inner analytic function.
Block Graph	Graph Theory	is a type of undirected graph in which every biconnected component (block) is a clique.
Block LU decomposition	Numerical Linear Algebra	Factorization of a partitioned matrix into lower and upper block-triangular factors, governed by Schur complements.
Block walking	CombinatoricsEnumerative Combinatorics	Block walking is a combinatorial proof technique that traverses structured blocks of a configuration according to local transition rules so a global counting, ordering, or existence result follows from the walk.
Blocking set	Finite GeometryBlocking Sets in Incidence Structures	Choose points in an incidence structure so every designated line or block meets the set, usually excluding a whole contained line to remove trivial solutions.
Bochner integral	Functional Analysis	The Banach-space-valued extension of the Lebesgue integral defined by norm limits of integrals of simple functions.
Bochner's theorem (orthogonal polynomials)	Orthogonal Polynomials	A classification theorem identifying the classical orthogonal-polynomial sequences that are eigenfunctions of a second-order differential operator with polynomial coefficients.
Bochner–Martinelli formula	Several Complex Variables	An integral representation for continuously differentiable functions on domains in several complex variables using the Bochner–Martinelli kernel.
Bockstein homomorphism	Coefficient Connecting Homomorphisms	Construct the degree-shifting connecting homomorphism in homology or cohomology induced by a short exact sequence of coefficient groups, detecting whether a class lifts through the middle coefficient object.
Bockstein Spectral Sequence	Algebraic TopologySpectral Sequences	Use successive mod-p homology pages and higher Bockstein differentials to distinguish p-power torsion from free homology.
Bogdanov–Takens bifurcation	Bifurcation TheoryDynamical Systems	A generic codimension-two equilibrium bifurcation with a double zero eigenvalue whose unfolding organizes nearby saddle-node, Hopf, and homoclinic bifurcation curves.
Bohemian matrices	Experimental Linear Algebra and Matrix Theory	A family of matrices whose entries are restricted to a fixed finite discrete population, often bounded-height integers, sometimes with additional Toeplitz, Hessenberg or other structure.
Bombieri norm	Polynomial Geometry	A unitarily invariant weighted coefficient norm on homogeneous polynomials that makes distinct monomials orthogonal with factorial-ratio squared norms.
Bonnet Theorem	Differential GeometrySurface Theory	The fundamental theorem of surface theory reconstructs a surface immersion from compatible first and second fundamental forms, uniquely up to rigid motion.
Boole's rule	Numerical Analysis	A closed Newton–Cotes quadrature rule using five equally spaced samples to approximate an integral by a weighted quartic interpolant.
Boolean algebra	Algebraic Logic	An algebraic structure with conjunction, disjunction and complementation satisfying laws that model two-valued logic and set operations.
Borel Functional Calculus	Operator Theory	Assign a Borel-measurable scalar function to an operator by integrating it against the operator's spectral measure, extending continuous and polynomial evaluation while preserving algebraic and spectral relations.
Borel measure	Measure Theory	A measure defined on the Borel sigma-algebra generated by the open subsets of a topological space.
Borel regular measure	Measure Theory	In mathematics, an outer measure μ on n-dimensional Euclidean space R n is called a Borel regular measure if the following two conditions hold.
Borel right process	Markov Processes and Potential Theory	A strong Markov process on a suitable Borel state space with right-continuous paths and a transition semigroup satisfying the regularity needed for potential theory.
Borel Set	Measure Theory	A subset generated from a topological space’s open sets by complement and countable union, equivalently an element of the topology-generated Borel σ-algebra.
Borel–de Siebenthal Theory	Lie Theory	Classify connected maximal-rank subgroups of a compact connected Lie group by retaining a maximal torus and reading admissible full-rank root subsystems from its extended Dynkin diagram.
Borel–Weil–Bott theorem	Representation Theory and Algebraic Geometry	A theorem realizing irreducible representations of compact or complex semisimple Lie groups as the unique nonzero cohomology of suitable line bundles on flag varieties.
Bott cannibalistic class	Algebraic Topology	A K-theory characteristic class measuring how an Adams operation acts on the Thom class of a complex vector bundle or representation.
Bound variable	Formal Logic and Variable BindingMathematical Logic	A variable occurrence governed by a quantifier, function parameter, summation, integral, or other binder within its specified scope.
Boundary Particle Method	Numerical AnalysisMeshfree Methods	In applied mathematics, the boundary particle method (BPM) is a boundary-only meshless (meshfree) collocation technique, in the sense that none of inner nodes are required in the numerical solution of nonhomogeneous partial differential equations.
Boundary Value Problem	Differential Equations	A differential equation on a domain paired with conditions on the domain's boundary, such that the interior solution is a derived consequence of the law plus the edge data rather than freely specified — the equilibrium counterpart of the marchable initial value problem.
Bounded arithmetic	Mathematical LogicProof Complexity	A family of weak arithmetic theories whose bounded quantifiers and restricted induction calibrate feasible reasoning, linking provably total functions and proofs to computational-complexity classes and propositional proof systems.
Bounded complete poset	Order TheoryCompleteness Properties	A partially ordered set in which every subset having an upper bound also has a least upper bound, expressing completeness for mutually consistent collections.
Bounded deformation	Calculus of Variations and Continuum Mechanics	Place a vector field in BD when its symmetrized distributional derivative is a finite Radon measure, retaining the infinitesimal-strain quantity required by elasticity and fracture while allowing more singular displacement behavior than bounded variation.
Bounded operator	Functional Analysis	A linear operator between normed spaces whose output norm is at most a fixed constant times the input norm.
Box Spline	Multivariate Approximation Theory	A compactly supported multivariate piecewise-polynomial function generated from a finite multiset of direction vectors, equivalently by repeated convolution of uniform segment measures or projection of a higher-dimensional box.
Boy or girl paradox	Probability Reasoning	A probability puzzle showing that conditioning on family sex composition depends on how the information was selected and reported.
Bracket algebra	Invariant Theory	A quotient algebra encoding projective invariants through bracket symbols over a signed alphabet.
Bracket polynomial	Knot Theory	A Laurent-polynomial state-sum invariant of framed unoriented link diagrams whose normalization yields the Jones polynomial for oriented links.
Branch Decomposition	Graph TheoryStructural Graph Theory	A tree arrangement of a graph's edges whose cuts expose shared-vertex interfaces, allowing the arrangement's largest interface to be measured.
Branched manifold	Differential Topology and Dynamics	A manifold-like space locally formed from finitely many smooth sheets sharing regions under compatible charts, with a well-defined tangent structure despite controlled branching.
Branching process	Probability Theory	A stochastic population process in which individuals independently generate random descendants according to a declared reproduction law.
Branching Quantifier	Mathematical Logic	A partially ordered quantifier prefix that represents independence among variable choices, allowing existential dependencies to branch rather than follow the single linear order of ordinary first-order quantification.
Brandt matrix	Arithmetic Geometry	A matrix encoding counts or weighted correspondences among ideal classes of a definite quaternion algebra, realizing Hecke operators on quaternionic modular forms.
Brauer's Induction Theorem	Finite Group Representation TheoryCharacter Theory	Every complex virtual character of a finite group is an integer combination of characters induced from linear characters of elementary subgroups.
Brauer's theorem on forms	Algebraic Number Theory	A theorem guaranteeing large linear spaces of common zeros for sufficiently many-variable homogeneous forms over fields with bounded diagonal-form obstruction.
Brauner space	Functional AnalysisLocally Convex and Stereotype Spaces	A complete compactly generated locally convex space whose compact subsets are cofinal in one countable increasing family, forming the stereotype dual class paired with Fréchet spaces.
Brownian meander	Stochastic Processes	A Brownian-motion-derived stochastic process on a fixed interval conditioned to remain nonnegative after leaving zero.
Brownian Skorokhod Embedding	Stochastic ProcessesProbability Theory	Represent a prescribed probability law as Brownian motion observed at an adapted stopping time, with admissibility conditions stated separately.
Brownian snake	ProbabilityPath Valued Processes	A Markov process taking values in finite stopped paths whose lifetime evolves like reflected Brownian motion and whose path tips encode spatial genealogies such as superprocesses.
Brun sieve	Analytic Number Theory	A combinatorial sieve that estimates integers avoiding specified prime divisibility conditions by truncating inclusion-exclusion with alternating upper and lower bounds.
Buchsbaum ring	Commutative Algebra	A Noetherian local ring in which every system of parameters is a weak sequence under a maximal-ideal colon condition.
Bundle Gerbe	Differential Geometry	A geometric gerbe presentation built from a surjective submersion, a line bundle on paired fibers, and associative multiplication over triples.
Bundle metric	Differential Geometry of Vector Bundles	A smoothly varying nondegenerate bilinear form on each fibre of a vector bundle.
Bundle of Principal Parts	Algebraic Geometry	A finite-order vector bundle of local section data over a smooth algebraic variety.
Burau representation	Braid Groups and Low Dimensional Topology	A parameter-dependent linear representation of the braid group obtained from its action on the homology of an infinite cyclic cover of a punctured disk.
Burning Ship fractal	Fractal GeometryComplex Dynamics	The escape-time set generated by iterating a quadratic complex map after replacing both real and imaginary components with their absolute values at every step.
Burnside category	Equivariant Homotopy TheoryCategory Theory	An additive category of finite G-sets whose morphisms are group-completed equivalence classes of equivariant spans composed by pullback.
Burnside's lemma	Group Actions	The orbit-counting result that the number of orbits of a finite group action equals the average number of elements fixed by a group element.
Butcher Group	Numerical AnalysisB Series and Runge Kutta Methods	The group of normalized rooted-tree coefficient maps whose product is induced by composition of B-series, turning sequential composition, formal inversion, order conditions, and structure-preserving subclasses of numerical integrators into algebraic operations.
Butson-type Hadamard matrix	Combinatorial Matrix Theory	A complex Hadamard matrix whose entries are q-th roots of unity.
C space	Functional AnalysisSequence Spaces	The Banach space of all convergent real or complex sequences equipped with the supremum norm, containing c0 as the closed subspace of sequences converging to zero.
C*-Algebra	Functional AnalysisOperator Algebras	A C*-algebra is a complex Banach algebra equipped with an involution whose norm satisfies the C*-identity, equivalently realizable up to star-isomorphism as a norm-closed adjoint-closed algebra of bounded operators on a complex Hilbert space.
C-minimal theory	Model Theory	A model-theoretic theory whose definable one-variable sets are finite Boolean combinations of cones determined by a ternary C-relation.
C-Theorem	Quantum Field TheoryRenormalization Group	Zamolodchikov's two-dimensional field-theory theorem supplies a monotone c-function along renormalization-group flow whose fixed-point value is the conformal central charge.
C0-Semigroup	Operator Semigroups	A nonnegative-time family of bounded linear operators on a Banach space that composes by time addition and is strongly continuous on every state.
Caccioppoli set	Geometric Measure TheoryFinite Perimeter Sets	A measurable set whose characteristic function has locally bounded variation, equivalently a set of locally finite perimeter in the geometric-measure-theory sense.
Cage (Graph Theory)	Graph Theory	An r-regular graph of girth g having the minimum possible number of vertices among all graphs with that degree and girth.
Cake number	Combinatorial Geometry	In mathematics, the cake number, denoted by C n , is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly n planes.
Calabi–Yau manifold	Complex Geometry	A compact Kähler complex manifold with trivial canonical bundle, admitting a Ricci-flat Kähler metric.
Caliber (mathematics)	Topological Cardinal Properties	Classify a cardinal as a caliber of a topological space when every equally large family of nonempty open sets contains an equally large subfamily sharing one common point, with caliber-star and precaliber variants kept distinct.
Calkin algebra	Operator Algebras	The C*-algebra obtained by quotienting all bounded operators on an infinite-dimensional separable Hilbert space by the ideal of compact operators.
Calkin correspondence	Operator TheoryFunctional Analysis	In mathematics, the Calkin correspondence, named after mathematician John Williams Calkin, is a bijective correspondence between two-sided ideals of bounded linear operators of a separable infinite-dimensional Hilbert space and Calkin sequence spaces (also called rearrangement invariant sequence spaces).
Calkin–Wilf tree	Number Theory	A binary tree that enumerates every positive rational number exactly once in lowest terms.
Cancellation property	AlgebraSemigroup Properties	An algebraic property allowing a common left or right factor to be removed from an equality, even when no inverse element is available.
Cannon–Thurston map	Geometric Group TheoryHyperbolic Geometry	A Cannon–Thurston map is the continuous boundary map induced, when it exists, by extending a specified inclusion of hyperbolic spaces or groups to their compactifications.
Canonical ring	Birational Geometry	The graded ring formed from global sections of all nonnegative tensor powers of a variety's canonical bundle or canonical divisor.
Canonical Sheaf	Algebraic Geometry	In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega , which is the n th exterior power of the cotangent bundle \Omega on V .
Cantor set	Topology	The compact perfect nowhere-dense subset obtained by repeatedly deleting open middle thirds from a closed interval.
Cantor tree surface	Geometric TopologyClassification of Infinite Type Surfaces	Form the unique orientable boundaryless surface of genus zero whose end space is a Cantor set and whose every end is planar, equivalently a sphere with a Cantor set removed.
Capacitated Arc Routing Problem	Combinatorial OptimizationVehicle Routing	Find minimum-cost depot-returning vehicle tours that service demand-bearing network links while each tour stays within vehicle capacity.
Capacity of a set	Potential TheoryCapacity Theory	Assign a potential-theoretic size to a set through an extremal energy or admissible-function problem, detecting thinness and polar sets beyond additive volume.
Capelli's identity	Representation TheoryNoncommutative Determinant Identities	Correct the determinant identity det(AB)=det(A)det(B) for matrices of noncommuting multiplication and differentiation operators by adding an ordered diagonal shift, yielding a central invariant in gl_n representation theory.
Carleman linearization	Dynamical SystemsLinearization Methods	A lifting method that represents a finite-dimensional nonlinear dynamical system as an infinite-dimensional linear system over monomials, then truncates it for approximation.
Carleman's equation	Logarithmic Kernel Integral Equations	Recover an unknown density on a finite interval from a first-kind integral equation with logarithmic kernel, retaining the endpoint weight and the exceptional interval-length solvability condition.
Carlyle Circle	Coordinate GeometryGeometric Equation Solving	A coefficient-defined circle whose intersections with the horizontal axis geometrically realize the real roots of a normalized quadratic equation.
Carmichael number	Computational Number Theory	A composite integer that satisfies the Fermat congruence for every integer base, so it systematically imitates prime behavior under the basic Fermat test.
Carré du champ operator	Markov Semigroups and Geometric Analysis	A symmetric bilinear operator derived from a Markov generator that measures its failure to satisfy the Leibniz derivation rule and plays the role of a squared gradient.
Cartan matrix	Lie Theory	A (symmetrizable) generalized Cartan matrix is a square matrix A = (a_{ij}) with integer entries such that.
Cartesian closed category	Category Theory	A category with a terminal object, binary products and exponential objects representing morphisms out of products.
Cartesian monoidal category	Category Theory	In mathematics, specifically in the field known as category theory, a monoidal category where the monoidal ("tensor") product is the categorical product is called a cartesian monoidal category.
Cartesian product of graphs	Graph TheoryGraph Products and Factorization	Construct a graph on ordered vertex pairs in which an edge changes exactly one coordinate along an edge of its corresponding factor while holding the other coordinate fixed.
Casimir effect	Quantum Field TheoryFluctuation Induced Forces	A force or pressure on macroscopic boundaries arising from changes in quantum-field fluctuations and mode structure imposed by geometry, materials and boundary conditions.
Castelnuovo–Mumford Regularity	Algebraic GeometryProjective Sheaf Cohomology and Graded Resolutions	Locate the least projective twist whose diagonal higher-cohomology vanishings persist, yielding one integer bound on global generation, Hilbert-function stabilization, and graded syzygy degrees.
Casting out nines	Arithmetic VerificationElementary Number Theory	Casting out nines is any of three arithmetical procedures.
CAT(k) space	Metric Geometry	A geodesic metric space whose triangles are no thicker than comparison triangles in the constant-curvature model space of curvature k, within the prescribed perimeter range.
Catalan number	Enumerative Combinatorics	The integer sequence C_n=(1/(n+1)) binomial(2n,n) that counts many recursively nested structures such as balanced parentheses, polygon triangulations and binary trees.
Categorical Lift	Category TheoryLifting Properties	A morphism through an object over a target that restores a prescribed commutative triangle or fills a commutative-square lifting problem.
Categorical quotient	Algebraic GeometryGeometric Invariant Theory	A universal invariant morphism from an object with group action through which every other invariant morphism factors uniquely.
Categorical set theory	Foundations of Mathematics	Set-theoretic foundations formulated or analyzed through category-theoretic objects, morphisms, and axioms such as ETCS.
Categorical Trace	Category TheorySymmetric Monoidal Categories	Close an endomorphism of a dualizable object in a symmetric monoidal category into an endomorphism of its unit.
Category of compactly generated weak Hausdorff spaces	Algebraic Topology	A convenient category of spaces whose topology is detected by compact Hausdorff probes and whose compact images are closed.
Category of Manifolds	Differential GeometryCategory Theory	The category whose objects are manifolds of a declared C^p class and whose morphisms are C^p maps, with variants fixing model spaces, dimension, boundary, or smoothness conventions.
Category of Markov kernels	Categorical Probability	A category whose objects are measurable spaces and whose morphisms are Markov kernels, composed by integrating one conditional probability kernel through another.
Category of metric spaces	Category Theory	The category whose objects are metric spaces and whose morphisms are nonexpansive maps.
Category of Modules	Module TheoryCategory Theory	For a fixed unital ring and action side, the category whose objects are all its modules and whose arrows are all its linear maps.
Category of relations	Category Theory	The category Rel whose objects are sets and whose morphisms are binary relations composed by existential relational composition.
Category of representations	Representation Theory	A category whose objects are representations of a fixed algebraic structure and whose morphisms are equivariant maps.
Category of sets	Category TheoryFoundational Categories	The category Set whose objects are sets and morphisms are total functions, with function composition, serving as a foundational categorical universe in which products, coproducts, limits, exponentials, and element-based intuitions are realized.
Category of Small Categories	Category Theory	In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories.
Category theory	Foundations and Structural Mathematics	A mathematical framework studying objects through composable morphisms, identities, functors, natural transformations, and universal properties.
Catholic Semigroup	Semigroup Theory	A semigroup whose inverse-set map separates elements: no two distinct elements have exactly the same set of semigroup inverses.
Cauchy matrix	Linear AlgebraStructured Matrices	A structured matrix with entries 1/(x_i−y_j) for distinct parameter sequences with nonzero cross-differences, possessing explicit determinant, inverse, displacement rank, and totally nonsingular submatrix formulas.
Cauchy surface	Lorentzian GeometryGlobal Causality and Initial Value Surfaces	Select an achronal hypersurface that every inextendible timelike curve crosses exactly once, thereby carrying complete global causal initial data and characterizing globally hyperbolic spacetime.
Cauchy's integral formula	Complex Analysis	A boundary integral that reconstructs every value and derivative of a holomorphic function inside a contour.
Cauchy–Schwarz inequality	Inner Product Geometry	The inequality bounding the absolute inner product of two vectors by the product of their norms, with equality exactly when the vectors are linearly dependent under the usual hypotheses.
Cayley Graph	Group Theory	A graph with group elements as vertices and edges given uniformly by multiplication by a chosen generating set.
Cayley Transform	Matrix Analysis and Lie Groups	A fractional-linear transformation that maps suitable skew-adjoint or Lie-algebra elements to orthogonal/unitary group elements—and, in a scalar form, maps a line or half-plane to a circle—while excluding points where the denominator is singular.
Cellular algebra	Graham Lehrer Cellular Algebras	Equip an associative algebra with a poset-indexed cellular basis and involution whose multiplication is triangular modulo lower cells, enabling standard cell modules and representation-theoretic filtrations.
Cellular homology	Algebraic Topology	A homology theory for CW complexes computed from a chain complex with one generator per cell and boundary maps determined by attaching maps.
Cellular space	Algebraic TopologyCw Complexes	A cellular space is a compact Hausdorff space that has the structure of a CW complex.
Center (group theory)	Group Theory	The subgroup of elements that commute with every element of a group.
Centered Polygonal Number	Figurate Numbers	A figurate number formed from one central point and successive k-sided dot rings, with one-based count C_k(n)=1+k n(n−1)/2.
Centerpoint (Geometry)	Discrete Geometry	Choose a point that every containing closed halfspace shares with at least ⌈n/(d+1)⌉ of n points in d dimensions.
Central angle	Euclidean Geometry	An angle with vertex at a circle's center and sides along radii to two circumference points, measuring a chosen subtended arc.
Central simple algebra	Ring and Brauer Theory	A finite-dimensional associative algebra over a field that has no nontrivial two-sided ideals and whose center is exactly the base field.
Central Subgroup	Group Theory	A subgroup contained in the center of an ambient group, equivalently one whose elements commute with every element of that group.
Chain complex	Homological Algebra	A graded sequence of modules or abelian groups connected by boundary homomorphisms whose consecutive composition is zero.
Chainstore paradox	Game Theory	The conflict between backward-induction accommodation and reputation-based deterrence in a finitely repeated entry game with sequential challengers.
Champernowne constant	Number Theory	The real number formed by concatenating the positive integers in order in a chosen base, with the base-ten version proved normal.
Change of fiber	Algebraic Topology	The homotopy-equivalence map between fibers of a fibration induced by transporting along a path in the base space.
Change of Rings	Commutative AlgebraModule Theory	In algebra, change of rings refers to one of several related ways for changing the coefficient ring of a module.
Channel surface	Differential Geometry	In geometry and topology, a channel surface or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve, its directrix.
Chaotic scattering	Dynamical Systems	Scattering dynamics in which arbitrarily small changes in incoming conditions produce fractal changes in exit channel, angle or delay time.
Character Theory	Representation Theory	The study of group representations through trace-valued class functions whose orthogonality and arithmetic encode irreducible decomposition and group structure.
Character variety	Moduli TheoryAlgebraic Geometry	In the mathematics of moduli theory, given an algebraic, reductive, Lie group G and a finitely generated group \pi , the G -character variety of \pi is a space of equivalence classes of group homomorphisms from \pi to G .
Characteristic (algebra)	Ring Theory	The characteristic of a ring is the least positive integer n for which n copies of the multiplicative identity sum to zero, or zero when no such integer exists.
Characteristic function (probability theory)	Probability Theory	The Fourier–Stieltjes transform of a probability law, whose values uniquely determine the distribution.
Characteristic polynomial of a graph	Spectral Graph Theory	In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix.
Characterization (mathematics)	Mathematical PracticeMathematical Logic	In mathematics, a characterization of an object is a set of conditions that, while possibly different from the definition of the object, is logically equivalent to it.
Cheap Talk	Game TheorySignaling and Communication	Costless, non-binding communication whose credibility rests entirely on incentive alignment: as the gap between sender and receiver interests widens, the information a costless message can carry shrinks through ever-coarser partitions down to meaningless babble.
Chebyshev's Inequality	Probability Theory	A distribution-free bound on deviation from a mean using finite quadratic dispersion and a positive threshold.
Chern–Weil homomorphism	Differential Geometry	The map sending invariant polynomials on a Lie algebra to de Rham cohomology classes represented by curvature forms of principal-bundle connections.
Chordal bipartite graph	Graph Theory	A bipartite graph in which every cycle of length at least six has a chord, equivalently a bipartite graph with no induced cycle longer than four.
Chow–Liu Tree	Information TheoryGraphical Models	Approximate a discrete joint distribution by the tree-factorized model whose edges maximize total pairwise mutual information.
Chow–Rashevsky theorem	Sub Riemannian Geometry	A sub-Riemannian accessibility theorem stating that any two points of a connected manifold can be joined by a horizontal path when the allowed distribution is bracket generating.
Christoffel–Darboux formula	Orthogonal PolynomialsKernel Identities	An identity collapsing a finite weighted sum of products of orthogonal polynomials into a quotient involving only two consecutive polynomial degrees.
Chromatic number	Graph TheoryGraph Coloring	The smallest number of colors needed to color a graph is called its chromatic number, and is often denoted .
Chromatic polynomial	Algebraic Graph Theory	A graph polynomial P(G,k) whose value at each nonnegative integer k counts the proper vertex colorings of graph G using k labeled colors.
Chromatic symmetric function	Algebraic Graph Theory	A symmetric-function graph invariant formed as the weight-generating function of all proper vertex colorings by positive integers.
Chu space	Category Theory	A three-part relational structure of points, states, and values whose duality and morphisms generalize topological and linear spaces.
Circulant matrix	Linear AlgebraStructured Matrices and Fourier Analysis	A square matrix generated by cyclically shifting one row, equivalently with entries depending only on index difference modulo n, and diagonalized by the discrete Fourier transform.
Circular algebraic curve	Algebraic Geometry	A real plane algebraic curve whose highest-degree homogeneous part is divisible by x squared plus y squared, equivalently passing through both circular points at infinity.
Circular arc	Euclidean Geometry	A connected portion of a circle's circumference between two endpoints, specified by its center, radius, orientation, and central-angle span.
Cissoid	Plane Geometry	A plane curve constructed from two base curves and a pole by placing, on each ray through the pole, a point whose directed radius combines the two ray intersections.
Class number formula	Algebraic Number Theory	A number-theoretic identity relating a Dedekind zeta function’s special behavior to a field’s class number, regulator, roots of unity, embeddings, and discriminant.
Classical group	Group TheoryLinear and Lie Groups	A member of the principal matrix-group families associated with finite-dimensional vector spaces and nondegenerate bilinear, quadratic, Hermitian or symplectic forms.
Classification of Discontinuities	—	A real-analysis taxonomy that diagnoses failure of continuity by the existence, finiteness, equality, and function-value agreement of one-sided limits.
Classification of Fatou components	Complex Dynamics	The dynamical classification of periodic stable regions of rational maps into attracting, parabolic, Siegel, Herman and related component types.
Classification theorem	Mathematical Methodology	A theorem enumerating every object of a declared mathematical type up to a stated equivalence, without omission or redundant equivalence classes.
Classifying space	Algebraic TopologyBundle Classification	A space BG representing principal G-bundles up to homotopy, obtained from a contractible free G-space EG and characterized by pullback classification.
Classifying space for O(n)	—	The classifying space BO(n) represents rank-n real vector bundles: homotopy classes of maps into BO(n) classify such bundles over suitable base spaces.
Classifying space for SO(n)	Algebraic TopologyFiber Bundles	In mathematics, the classifying space \operatorname{BSO}(n) for the special orthogonal group \operatorname{SO}(n) is the base space of the universal \operatorname{SO}(n) principal bundle \operatorname{ESO}(n)\rightarrow\operatorname{BSO}(n) .
Classifying space for SU(n)	Algebraic TopologyFiber Bundles	Classifying space for SU(n) is a recurring identity in mathematics, logic, and statistics defined by this frozen evidence: In mathematics, the classifying space \operatorname{BSU}(n) for the special unitary group \operatorname{SU}(n) is the base space of the universal \operatorname{SU}(n) principal bundle \operatorname{ESU}(n)\rightarrow\operatorname{BSU}(n) .
Claw-free graph	Graph TheoryForbidden Induced Subgraphs	A graph containing no induced subgraph isomorphic to the four-vertex star K1,3.
Clifford Module	Clifford Algebra	A vector space or module equipped with an action of a Clifford algebra, so vectors act as operators satisfying the quadratic form's square and anticommutation relations.
Clifford theory	Representation TheoryNormal Subgroup Restriction	Representation-theoretic results describing how irreducible representations of a group restrict to a normal subgroup and how subgroup constituents extend or induce back to the group.
Clique (Graph Theory)	Graph Theory	A vertex subset of an undirected graph in which every two distinct vertices are adjacent, equivalently an induced complete subgraph.
Clique graph	Graph TheoryIntersection Graph Operators	Transform an undirected graph into the intersection graph of its maximal cliques, making each maximal clique a vertex and joining two when their underlying vertex sets intersect.
Clique-sum	Graph TheoryGraph Decomposition and Minors	Combine graphs by choosing equally sized complete subgraphs, identifying their vertices through a bijection, and optionally deleting interface-clique edges under an explicit convention.
Clique-width	Graph Theory	The minimum number of labels needed to construct a graph by vertex creation, disjoint union, complete joins between labels, and relabeling.
Clone (Universal Algebra)	Universal Algebra	Collect finitary operations on one carrier so that every projection is present and arbitrary finitary substitution of member operations produces another member.
Clopen Set	General Topology	Identify a subset that is both open and closed in the same topology, so it and its complement form a topological separation when both are nonempty.
Closed Immersion	Algebraic Geometry	A scheme morphism that identifies its source with a closed subscheme through locally surjective maps from ambient functions.
Closed Linear Operator	Functional AnalysisOperator Theory	A partially defined linear operator whose graph is a closed subset of the product of its domain's ambient space and codomain.
Closed monoidal category	Category Theory	A monoidal category in which tensoring by any object has a right adjoint represented by an internal hom object.
Closed Preordered Set	Set Theory and Order Theory	A preorder is closed to a stated chain length when every descending chain shorter than that length has a common lower bound, so transfinite strengthening can continue through limit stages without leaving the order.
Closed Set	Topology Analysis	Certify with one structural bit that no legitimate process inside a set — taking a limit (topology) or applying an operation (algebra) — can carry you outside it, discharging every boundary check in a proof at once.
CLRg property	Fixed Point TheoryMetric Spaces	A common-limit-in-the-range condition for self-maps f and g: some sequence x_k has both f(x_k) and g(x_k) converge to the same point g(x), placing the shared limit in the range of g without requiring that range to be closed.
Club principle	Set Theory	A set-theoretic guessing principle asserting a sequence of cofinal subsets that is fully contained in every unbounded set at some indexed stage.
Cluster algebra	Algebra and Combinatorics	A commutative algebra generated from overlapping algebraically independent clusters by iterated birational mutations governed by exchange matrices or quivers.
Coarea formula	Geometric Measure TheoryMultivariable Calculus	For smooth functions the formula is a result in multivariate calculus which follows from a change of variables.
Cochran's Theorem	Mathematical StatisticsNormal Quadratic Forms	A rank criterion that identifies when a complete quadratic-form partition of centered spherical Gaussian variation yields mutually independent chi-square components.
Cocurvature	Differential Geometry	The vector-valued two-form that detects nonintegrability of the image distribution of a tangent-bundle projection.
Codensity monad	Category Theory	The monad given by the right Kan extension of a functor along itself when that extension exists.
Coefficient	Symbolic Algebra	A multiplicative factor attached to a term in an algebraic expression, series, equation or linear combination, determining that term's scale under a stated basis or representation.
Coequalizer	Category Theory	A universal quotient-like object that makes two parallel morphisms equal and factors every other morphism that equalizes them uniquely.
Cograph	Graph TheoryHereditary Graph Classes	A graph generated from single vertices by repeated disjoint union and complementation, equivalently a graph with no induced four-vertex path and a recursive cotree decomposition.
Cohen ring	Commutative Algebra	A field or complete discrete valuation ring of mixed characteristic whose maximal ideal is generated by the residue characteristic, used to lift residue fields in local algebra.
Cohen–Macaulay Ring	Commutative Algebra	A commutative Noetherian ring whose localizations all have depth equal to Krull dimension.
Coherency (homotopy theory)	Homotopy TheoryHigher Category Theory	In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism".
Coherent category	Category TheoryCategorical Logic and Topos Theory	Equip a regular category with finite unions of subobjects that remain stable under pullback, providing categorical semantics for finite-limit, existential, and finite-disjunctive reasoning.
Coherent sheaf	Algebraic Geometry	A sheaf of modules locally having a finite presentation whose relations are themselves finitely generated, providing a stable algebraic model of geometric data.
Cohomological dimension	Group CohomologyHomological Algebra	Cohomological dimension denotes invariant of a group within group cohomology.
Cohomology of a Stack	Algebraic GeometryHomological Algebra	Cohomology of a stack extends sheaf-cohomological constructions to algebraic stacks by choosing an appropriate site and deriving global sections, including equivariant interpretations for quotient stacks.
Cohomology Ring	Algebraic TopologyHomological Algebra	The graded direct sum of a space's cohomology groups with coefficients in a ring, equipped with the degree-adding cup product and its graded-commutative multiplication.
Coin Problem	Number Theory	Find the largest nonnegative integer not representable as a nonnegative integer combination of given coprime positive denominations—the Frobenius number of their numerical semigroup.
Cointerpretability	Mathematical LogicRelations Between Formal Theories	Compare formal theories by a logic-preserving translation in the reverse language direction that reflects the translated theory's theorems, a dual of interpretability tied to Σ₁-conservativity for suitable arithmetical theories.
Cokernel	Category Theory and Homological Algebra	The universal quotient of a morphism's codomain that makes the morphism vanish, realized for linear maps as codomain modulo image.
Colin de Verdière graph invariant	Spectral Graph Theory	A minor-monotone graph parameter defined by the maximum corank of a constrained symmetric matrix with one negative eigenvalue and the Strong Arnold property.
Collapsing manifold	Riemannian Geometry	A Riemannian manifold or sequence whose metric geometry approaches a lower-dimensional limit while specified curvature or diameter controls are retained.
Collar neighbourhood	Differential Topology	A neighborhood of a manifold's boundary that is identified with the product of that boundary and a half-open interval while fixing the boundary at interval coordinate zero.
Collectionwise Normal Space	General Topology	Separate every discrete indexed family of closed sets at once by matching pairwise-disjoint open neighborhoods, extending ordinary two-set normality to arbitrary family size.
Collectively exhaustive events	Probability TheoryEvent Algebra and Sample Spaces	Require a declared family of events to cover the entire sample space, so every possible outcome belongs to at least one member without requiring the members to be disjoint.
Collineation	Projective Geometry	A bijection of projective spaces that preserves collinearity.
Colombeau algebra	Generalized Function Theory	A differential algebra of generalized functions that embeds distributions while permitting nonlinear multiplication and retaining compatibility with smooth-function products.
Combinatorial number system	Enumerative Combinatorics	A bijective representation of nonnegative integers as fixed-size combinations through sums of binomial coefficients, enabling direct ranking and unranking without listing earlier combinations.
Combinatory Logic	Mathematical LogicApplicative Calculi	A family of applicative formal systems in which fixed combinators and application express functional abstraction and computation without binding variables in the object language.
Common logarithm	Elementary MathematicsLogarithms	The base-ten logarithm, the inverse of raising ten to a power and historically central to decimal calculation tables, scientific notation and orders of magnitude.
Commutative magma	Abstract Algebra	A set with a closed binary operation satisfying commutativity but not necessarily associativity, identity or inverses.
Commutative ring	Commutative AlgebraRing Structures	A ring whose multiplication is commutative, providing the algebraic setting in which ideals, localization, spectra and polynomial geometry acquire their standard symmetric forms.
Commutator	Algebra	An algebraic expression that measures failure of two elements or operators to commute, such as aba⁻¹b⁻¹ in a group or ab−ba in a ring.
Compact closed category	Category TheoryMonoidal Duality	A symmetric monoidal category in which every object has a dual with unit and counit morphisms satisfying the snake identities.
Compact element	Domain TheoryOrder Theory	Compact element denotes those elements of a partially ordered set that cannot be subsumed by a supremum of any directed set that does not already contain them in domain theory.
Compact embedding	Functional Analysis and Topology	An embedding whose inclusion map is compact, so bounded sequences in the source possess subsequences converging in the target; in topology, related notation can instead mean compact containment.
Compact Operator	—	A bounded linear operator that sends bounded sets to relatively compact sets, giving infinite-dimensional problems finite-dimensional-like approximation and spectral behavior.
Compact Quasi-Newton Representation	Numerical OptimizationQuasi Newton Methods	A block low-rank form of accumulated quasi-Newton Hessian or inverse-Hessian updates, built from secant-pair history to replace dense matrix operations with tall factors and a small system.
Compact semigroup	Semigroup Theory	A semigroup whose solution sets for arbitrary systems of word equations are already determined by some finite subsystem, under the algebraic compactness convention.
Compactly supported homology	Homology TheoryAlgebraic Topology	In mathematics, a homology theory in algebraic topology is compactly supported if, in every degree n, the relative homology group H n (X, A) of every pair of spaces.
Compactness	Topology and Analysis	Certify that a space is finite-like — every open cover has a finite subcover, equivalently every sequence a convergent subsequence — so a whole package of existence theorems fires automatically on it and fails without it.
Comparability Graph	Graph Theory	Join exactly the comparable pairs of a partial order in an undirected graph, equivalently requiring that its edges admit a transitive orientation.
Complement (group theory)	Group TheoryAbstract Algebra	A subgroup that factors a group with another subgroup and intersects it only at the identity.
Complement (set theory)	Set Theory	The set of elements in a declared universe that are not members of a selected set, or the elements of one set left after removing another.
Complement graph	Graph Theory	In the mathematical field of graph theory, the complement or inverse of a graph is a graph on the same vertices such that two distinct vertices are adjacent (connected) in if and only if they are not adjacent in .
Complementary event	Probability Theory	For an event in a sample space, the event containing exactly the outcomes in the sample space that are not in the original event.
Complete Bipartite Graph	Graph Theory	A bipartite graph K_{m,n} with no within-part edges and every possible edge between its two vertex parts present.
Complete category	Category TheoryLimits and Colimits	A category possessing a limit for every diagram indexed by a small category, equivalently all small products and equalizers under standard size conventions.
Complete field	Analysis and Number TheoryComplete Valued Fields	Equip a field with a compatible absolute value or metric and require every Cauchy sequence to converge within the field, making limits available without leaving its algebraic carrier.
Complete Heyting algebra	Order Theory and Point Free Topology	Combine arbitrary joins and meets with Heyting implication, equivalently requiring finite meets to distribute over arbitrary joins, to form the algebraic objects called frames.
Complete intersection	Algebraic Geometry	A scheme or variety whose defining ideal is locally generated by exactly its codimension number of elements, giving the expected minimal equation count.
Complete lattice	Order and Lattice Theory	A partially ordered set in which every subset, including the empty set, has both a supremum and an infimum.
Complete measure	Measure TheoryMeasure Space Completion	A measure space in which every subset of every measurable null set is itself measurable and has measure zero.
Complete sequence	Additive Number Theory	A sequence of natural numbers whose distinct finite subset sums represent every positive integer.
Complete variety	Algebraic Geometry	An algebraic variety whose projection after product with any variety is a closed map, the algebro-geometric analogue of compactness.
Complete-linkage clustering	Data AnalysisHierarchical Clustering	Build an agglomerative hierarchy by defining intercluster distance as the farthest cross-cluster pair and repeatedly merging the pair with the smallest such maximum.
Completely distributive lattice	Order TheoryComplete Lattices	A complete lattice in which arbitrary meets distribute over arbitrary joins according to the choice-function identity, equivalently satisfying the self-dual complete distributivity law.
Completely metrizable space	Topology	A topological space whose topology is induced by at least one complete metric, whether or not every compatible metric is complete.
Completely regular space	General Topology	In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.
Completely Uniformizable Space	General TopologyUniform Spaces	A topological space whose topology is induced by at least one complete uniformity, also called Dieudonné complete under conventions that may additionally require Hausdorffness.
Completion of a ring	Commutative AlgebraAdic Topology and Completion	Replace a ring by the inverse limit of its quotients by successive powers of an ideal, producing an ideal-adically complete ring together with the canonical map from the original ring.
Complex Affine Space	Affine GeometryAlgebraic Geometry	An affine space modeled on a complex vector space: points admit complex-vector translations and differences but no intrinsically distinguished origin.
Complex conjugate	Complex AnalysisComplex Arithmetic and Symmetry	Reflect a complex number across the real axis by reversing the sign of its imaginary component, producing an involutive field automorphism that preserves real scalars, sums, products, and modulus.
Complex conjugate representation	Representation Theory	For a complex representation, the representation on the conjugate vector space obtained by conjugating scalar structure and the representing matrices or linear maps, with its relation to the dual depending on unitarity or pseudounitarity.
Complex differential form	Differential Geometry	A differential form whose coefficients take complex values, often decomposed into holomorphic and antiholomorphic bidegrees.
Complex Hadamard matrix	Matrix Theory	A square complex matrix whose entries all have unit modulus and whose rows are mutually orthogonal.
Complex hyperbolic space	Differential GeometryComplex Hyperbolic and Symmetric Spaces	Realize the simply connected complete Kähler manifold of complex dimension n and constant negative holomorphic sectional curvature, equivalently the rank-one Hermitian symmetric space acted on transitively by PU(n,1).
Complex Lie group	Lie TheoryComplex Geometry	In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^{-1} is holomorphic.
Complex normal distribution	Probability and Signal ProcessingComplex Random Vectors	A distribution for complex random vectors whose stacked real and imaginary parts are jointly Gaussian, characterized by mean, covariance, and relation (pseudo-covariance) matrices, with circular proper Gaussian as a special case.
Complex number	Complex AnalysisAlgebra	A field element a+bi extending the reals by i²=−1, equivalently a plane point with field multiplication that encodes rotation and scaling.
Complex polytope	Complex Geometry	A regular incidence geometry modeled in complex unitary space that generalizes real regular polytopes through complex reflections and phase-valued incidence.
Complex random vector	Probability Theory	A random element of a finite-dimensional complex vector space, equivalently a jointly distributed collection of complex-valued random variables.
Complex representation	Representation Theory	A group or Lie-algebra representation on a complex vector space; in a narrower usage, one whose complex-conjugate representation is inequivalent to it, distinguishing it from real and pseudoreal types.
Complex-base system	Number RepresentationComplex Radix Numeration	Represent real or complex numbers positionally using a nonreal radix and a finite digit alphabet, allowing a single unsigned expansion to encode multiple dimensions when admissibility, convergence, and uniqueness conditions hold.
Complexification (Lie group)	Lie Theory	The universal map from a real Lie group into a complex Lie group through which every continuous homomorphism from the real group to a complex Lie group factors uniquely as a holomorphic homomorphism.
Component (graph theory)	Graph Theory	A maximal connected subgraph of an undirected graph; the graph's components uniquely partition its vertex set.
Composite number	Elementary Number Theory	A positive integer greater than one that can be expressed as a product of two smaller positive integers.
Composition Algebra	Nonassociative AlgebraQuadratic Forms	A unital algebra with a nondegenerate quadratic form whose value is multiplicative under the algebra product.
Compound Poisson process	Stochastic Processes	A jump process formed by summing independent random jump sizes at event times of a Poisson counting process.
Computability logic	Mathematical LogicComputability Theory	Computability logic (CoL) is a research program and mathematical framework for redeveloping logic as a systematic formal theory of computability, as opposed to classical logic, which is a formal theory of truth.
Computable real function	Computable Analysis	A real-valued function for which arbitrarily accurate output approximations can be generated effectively from arbitrarily accurate representations of its real inputs.
Computer-Assisted Proof	Computer Assisted Mathematics	Establish a mathematical theorem with a proof whose evidential chain materially depends on machine-generated search, exhaustive case checking, rigorous computation, or formal proof checking.
Concrete category	Category TheoryConcretization	A category equipped with a faithful functor to Set, allowing its objects to be regarded as sets with structure and its morphisms as distinguishable structure-preserving functions.
Condensed Detachment	—	An inference rule that unifies an implication’s antecedent with a minor premise and detaches the most general resulting consequent.
Condition Number	Numerical Analysis	Quantify the worst local amplification of admissible relative input perturbations into relative output changes for a specified mathematical problem, point, and choice of norms.
Conditional convergence	Mathematical Analysis	Convergence of a series or improper integral despite failure of convergence after taking absolute values.
Conditioned Disjunction	Propositional Logic	A ternary Boolean connective whose middle argument selects the first branch when true and the third branch when false: [p,q,r] = (q ∧ p) ∨ (¬q ∧ r).
Conductance (graph theory)	Spectral Graph Theory and Markov Chains	A normalized bottleneck measure comparing the edge flow leaving a vertex set with the smaller stationary volume of that set and its complement.
Conductor (ring theory)	Commutative Algebra and Orders	The largest ideal shared by a commutative ring and an extension ring, equal to the annihilator of the quotient; when the extension is the normalization, it measures the smaller ring's failure to be integrally closed.
Cone (algebraic geometry)	Algebraic Geometry	A relative affine scheme obtained as the spectrum of a graded quasi-coherent algebra, carrying the scaling action induced by its grading and admitting an associated projective cone.
Cone (category theory)	Category Theory	In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor.
Cone (topology)	Algebraic Topology	In topology, especially algebraic topology, the cone of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point.
Conference graph	Graph TheoryAlgebraic Graph Theory	In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, and It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares.
Conference Matrix	Combinatorial Matrix TheoryDesign Theory	A square zero-diagonal ±1 matrix whose columns are mutually orthogonal with squared norm n−1, equivalently satisfying CᵀC=(n−1)I.
Conflict-free coloring	Graph and Hypergraph Coloring	A hypergraph vertex coloring in which every hyperedge contains at least one vertex whose color occurs exactly once within that edge.
Congruence ideal	Arithmetic Geometry	An ideal measuring congruences between an eigencomponent and the complementary part of an arithmetic algebra, commonly obtained from the image of an annihilator under a quotient character.
Congruence Subgroup	Modular Forms	A congruence subgroup contains the kernel of an integral matrix group's reduction modulo some level.
Conic Optimization	Convex Optimization	Conic optimization is a subfield of convex optimization that studies problems consisting of minimizing a convex function over the intersection of an affine subspace and a convex cone.
Conical combination	Convex GeometryLinear Algebra	A finite nonnegative linear combination of vectors; the collection of all such combinations is their conical hull, the minimal convex cone generated from the origin.
Conical surface	Ruled and Developable Surfaces	Generate a two-napped ruled surface as the union of complete straight lines through one fixed apex and points of a directrix, preserving the apex singularity and distinguishing the general object from a solid cone or circular special case.
Conjugacy class	Group TheoryGroup Actions	An equivalence class of group elements related by inner automorphisms, containing all elements of the form gag⁻¹ for a fixed a and varying g.
Conjugacy class sum	Representation Theory and Group Algebras	The sum in a group algebra of all basis elements belonging to one conjugacy class of a finite group.
Conjugacy problem	Combinatorial Group TheoryAlgorithmic Group Theory	Decide whether two words in a finitely generated or finitely presented group represent conjugate elements, with solvability depending on the group class and presentation rather than group axioms alone.
Conjugate Gradient Method	—	A Krylov-subspace solver for symmetric positive-definite linear systems that builds mutually A-conjugate directions while minimizing the associated quadratic.
Conjugate index	Banach Space Geometry	For a Banach space, the largest exponent for which its dual is guaranteed to have the corresponding finite cotype, expressed through Hölder-conjugate type and cotype behavior under the governing convention.
Connected Dominating Set	Graph TheoryDomination Theory	A vertex subset of an undirected graph that dominates every excluded vertex while inducing a connected subgraph on its selected vertices.
Connected relation	Relation Theory	A binary relation that compares every pair of distinct elements in at least one direction.
Connected ring	Commutative AlgebraRing Decomposition	A commutative ring with no idempotents other than zero and one, equivalently one whose prime spectrum is connected in the Zariski topology.
Constant Term	Polynomials Formal Series and Algebraic Combinatorics	The coefficient of the multiplicative-identity monomial in a polynomial, series, or Laurent expression—the component independent of every declared variable and recoverable by evaluation at zero only when negative powers are absent.
Constant-recursive sequence	Discrete Mathematics	A sequence satisfying a fixed finite-order linear recurrence with constant coefficients.
Constrained optimization	Mathematical Optimization	In mathematical optimization, constrained optimization (in some contexts called constraint optimization) is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables.
Constructible sheaf	Algebraic and Microlocal Geometry	A sheaf that becomes locally constant with finite-type stalks on each piece of a finite stratification of its underlying space.
Constructible topology	Algebraic GeometrySpectral Spaces	The compact Hausdorff totally disconnected refinement of the Zariski topology on a scheme or spectrum, generated by making quasi-compact open sets and their complements open.
Construction of the Real Numbers	Real AnalysisFoundations of Mathematics	A consequence of the axioms is that this structure is unique up to an isomorphism, and thus, the real numbers can be used and manipulated, without referring to the method of construction.
Constructive Logic	Mathematical LogicConstructive Reasoning	A family of proof systems in which establishing a claim requires the construction appropriate to its logical form, rather than unrestricted reliance on classical excluded middle or double-negation elimination.
Contact graph	Geometric Graph Theory	Represent a family of geometric objects by one vertex per object and an edge exactly when two objects satisfy a declared boundary-contact relation without prohibited interior overlap or crossing, making the permitted notion of touching part of the graph class.
Container (type theory)	Type Theory	A shape-and-positions representation of strictly positive collection-like type constructors in dependent type theory.
Continued Fraction	Number TheoryDiophantine Approximation	A finite or infinite recursively nested fraction defined by sequences of partial numerators and denominators.
Continuity Set	Measure TheoryProbability Theory	A Borel set whose boundary has measure zero for a specified measure.
Continuous function	Mathematical Analysis and Topology	A function that preserves arbitrarily local closeness: inverse images of open sets are open, equivalently limits can pass through the function under suitable structures.
Continuous function (ordinal theory)	Set TheoryOrdinal Functions	An ordinal-indexed sequence whose value at every limit index equals the supremum of its earlier values, usually considered together with monotonicity in transfinite constructions.
Continuous Group Action	Topological Transformation Groups	An action of a topological group on a topological space whose joint evaluation map is continuous, organizing the space into compatible orbits, stabilizers, fixed-point sets, and an orbit quotient.
Continuous Uniform Distribution	ProbabilityContinuous Distributions	The bounded continuous probability law whose density is constant, assigning probability in direct proportion to interval length.
Continuous-time Markov chain	Stochastic Processes	A stochastic process with the Markov property on a discrete state space whose state changes occur in continuous time according to exponential holding rates and transition intensities.
Continuous-Time Random Walk	Stochastic ProcessesAnomalous Transport	A jump-process model that interleaves random state displacements with random waiting times, turning an event-indexed walk into a trajectory indexed by continuous calendar time.
Continuous-time stochastic process	Probability and Stochastic Processes	A collection of random variables indexed by a continuous parameter set, usually a real time interval, without implying that its sample paths are continuous.
Contou-Carrère symbol	Algebraic K TheoryArithmetic Geometry	A multiplicative Steinberg symbol on pairs of invertible Laurent series over an Artinian ring, valued in the ring's units and defined through winding exponents, leading coefficients, and positive-negative coefficient pairings.
Contour integration	Complex Analysis	Integration of a complex function along an oriented path in the complex plane, with deformation and residues enabling evaluation.
Contraction Morphism	Algebraic Geometry	Map a projective variety onto another without a nontrivial finite Stein factor, so the fibers remain connected.
Convergence Group	Geometric Group TheoryBoundary Dynamics	A group action on a compact space whose induced action on distinct triples is properly discontinuous, equivalently exhibiting subsequential collapse away from a repelling point in the metrizable case.
Conversion between quaternions and Euler angles	Rotation RepresentationsQuaternion Algebra	A convention-bound transformation between a normalized unit quaternion and an ordered Euler-angle parameterization representing the same three-dimensional rotation.
Convex bipartite graph	Graph Theory	In the mathematical field of graph theory, a convex bipartite graph is a bipartite graph with specific properties.
Convex body	Convex Geometry	A compact convex subset of finite-dimensional Euclidean space with nonempty ambient interior under the standard convention.
Convex conjugate	Convex Analysis	The supremum transform mapping an extended-real function on a vector space to the greatest affine lower-bound gap over its dual space.
Convex hull	Convex Geometry	The smallest convex set containing a given set, equivalently all finite convex combinations of its points.
Conway criterion	Tessellation TheoryMonohedral Tiling Criteria	A sufficient boundary-symmetry test guaranteeing that a topological-disk prototile can tile the plane by translations and half-turns.
Coordinate Singularity	Differential GeometryGeometry of Manifolds	A coordinate chart breaks down at a locus where the underlying geometry remains locally regular in another valid description.
Cop number	Graph Theory	The minimum number of pursuers required to guarantee capture of an evader in the standard cops-and-robber game on a graph.
Cop-win graph	Graph Pursuit Games	A graph on which one pursuer has a strategy that guarantees capture of one evader in the alternating vertex-movement game.
Coprime integers	Number Theory	In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1.
Coproduct	Category Theory	A categorical colimit receiving one morphism from each object and universal among all such cocones.
Corank	Linear AlgebraRank and Nullity	A rank-deficiency quantity, commonly the codomain dimension minus rank of a linear map or matrix, equivalently the dimension of its cokernel in finite-dimensional linear algebra.
Core Model	Inner Model TheorySet Theory	A fine-structural, iterable inner model built under stated upper bounds on large cardinals to approximate the universe of sets canonically and support covering, comparison, and consistency-strength analysis.
Core-compact space	General Topology and Domain Theory	A topological space whose lattice of open sets is a continuous poset, equivalently an exponentiable object in the category of topological spaces.
Corner-point grid	Computational Geometry	A three-dimensional hexahedral grid whose cells are defined by ordered pillars and independently positioned corner points along those pillars.
Corona product	Graph Theory	A graph operation joining each vertex of one graph to every vertex of its own attached copy of a second graph.
Correlation Dimension	Chaos TheoryStrange Attractor Characterization	A fractal-dimension statistic given by the small-radius scaling exponent of the probability or normalized count that two sampled points lie within distance epsilon.
Correlation function (quantum field theory)	Quantum Field Theory	A vacuum or state expectation value of an ordered product of quantum field operators at specified spacetime points.
Correlation integral	Nonlinear Time Series and Dynamical Systems	The probability, estimated from state pairs, that two independently sampled points on a trajectory or invariant measure lie within a specified distance.
Cosheaf	Category TheorySheaf Theory	In topology, a branch of mathematics, a cosheaf is a dual notion to that of a sheaf that is useful in studying Borel-Moore homology.
Cosmos (category theory)	Enriched Category Theory	A complete and cocomplete symmetric closed monoidal category chosen as the base of enrichment for categories, functors, natural transformations, limits and tensors.
Cosocle	Group Theory	In group theory, a cosocle of a group G, denoted by Cosoc(G), is the intersection of all maximal normal subgroups of G.
Costas array	Combinatorial Design	A permutation array whose displacement vector between every pair of dots is unique.
Cotangent sheaf	Algebraic Geometry	The sheaf of relative Kahler differentials that universally represents derivations for a morphism of schemes or ringed spaces.
Cotorsion group	Abelian Group Theory	An abelian group M for which every extension by a torsion-free abelian group splits, equivalently Ext(F,M)=0 for every torsion-free F.
Countably Generated Module	Module Theory	A module spanned by an at-most-countable set using finite linear combinations over its ring.
Countably quasi-barrelled space	Topological Vector Spaces	A topological vector space in which every strongly bounded dual subset that is a countable union of equicontinuous sets is itself equicontinuous.
Counting measure	Measure Theory	The counting measure can be defined on any measurable space (that is, any set X along with a sigma-algebra) but is mostly used on countable sets.
Courant bracket	Generalized Geometry	An antisymmetric differential-geometric bracket on sections of a tangent-plus-cotangent bundle that extends the Lie bracket and has an exact-form Jacobiator.
Cousin problems	Several Complex VariablesComplex Analytic Geometry	Ask whether compatible local meromorphic data on a complex manifold glue to a global meromorphic function, with additive and multiplicative versions carrying distinct cohomological obstructions.
Covariance Matrix	Multivariate Probability	The square array of every pairwise covariance among a random vector's components, representing their joint second-order variation.
Covariant (Invariant Theory)	Invariant Theory	A polynomial map between group representations that transforms equivariantly, carrying the symmetry action on its input into the corresponding action on its output.
Covariant derivative	Differential GeometryConnections	A connection-defined derivative on vector or tensor fields that corrects ordinary differentiation so results transform consistently across changing bases on a manifold.
Covariant transformation	Tensor Calculus	A component-transformation rule in which lower-index tensor components change with the inverse-coordinate or dual-basis matrix so the underlying geometric object and contractions remain invariant.
Cover (topology)	Topology	A family of subsets whose union contains a specified set or space, with open covers restricting the members to open subsets.
Covering design	Finite Covering Designs	Choose fixed-size blocks from a finite point set so every required smaller subset lies in at least one block, and minimize the number of blocks through the covering number under explicit parameter and multiplicity conventions.
Covering number	Metric Geometry	The minimum number of radius-r balls required to cover a specified subset of a metric or pseudometric space.
Covering Set	Number Theory	Certify that every term of a modularly periodic integer sequence has a divisor from one finite prime set by covering every index class with a periodic divisibility congruence.
Cox process	Point Processes	A point process that is conditionally Poisson given a random intensity measure, thereby representing clustered or environment-driven event rates.
Coxeter Element	Group TheoryCoxeter Groups	A product containing each simple reflection of a Coxeter system exactly once, organizing conjugacy, order, spectrum, and reflection geometry.
Cramér's Theorem (Large Deviations)	Probability TheoryLarge Deviations	An i.i.d.-sample-mean large-deviation principle whose exponential rate is the convex conjugate of the one-observation log moment-generating function.
Creative and Productive Sets	Computability Theory	Formalize effective diagonal escape: a productive set computably supplies an element outside any enumerated subset of itself, while a creative set is enumerable and has a productive complement.
Credal Set	Imprecise Probability	Represent imprecise probabilistic belief by a set of admissible probability measures, deriving lower and upper expectations as envelopes while keeping convexity, closure, conditioning, and independence choices explicit.
Cremona Group	Birational GeometryGroups of Birational Transformations	The group of birational self-transformations of projective n-space over a fixed field, equivalently the field automorphisms of a purely transcendental function field.
Crinkled Arc	Functional Analysis	A continuous Hilbert-space curve whose chords over nonoverlapping parameter intervals are orthogonal, yielding a nowhere-tangent path with a unique normalized form up to reparameterization and unitary equivalence.
Crofton formula	Integral Geometry	An integral-geometric identity recovering a curve’s length from the invariant-measure average number of intersections with lines.
Cross Section (Geometry)	Geometry	Expose the geometry of a body by intersecting it with a plane or, in higher dimension, a hyperplane, retaining the induced lower-dimensional figure together with the cutter's position and orientation.
Cross-Covariance Matrix	Probability Theory	The rectangular matrix of pairwise second central moments between the components of two random vectors, preserving direction, units, and linear transformation structure.
Cross-listed Classification	Research Classification and Indexing	Assign a bridging work several classifications at once — one focal primary plus any number of secondaries — so it stays findable from every domain it spans, without forcing a single arbitrary home or minting a hybrid category.
Crossed Product Algebra	—	An algebra built from an algebra and a group action so adjoined group operators implement the action by conjugation, with a specified analytic completion where required.
Crossing number (graph theory)	Topological Graph Theory	The minimum number of edge intersections over all plane drawings of a graph under a specified crossing convention.
Crout matrix decomposition	Numerical Linear Algebra	An LU factorization convention placing arbitrary diagonal entries in the lower triangular factor and unit diagonal entries in the upper factor, with pivoting when needed.
Cube (algebra)	Elementary AlgebraPowers and Roots	The third power x³ of a number or algebraic expression, obtained by multiplying three equal factors and inverted on suitable domains by the cube-root operation.
Cubic Form	Algebra	A homogeneous degree-three formal polynomial in declared variables over a specified coefficient domain.
Cubic Fourfold	Algebraic Geometry	An irreducible four-dimensional projective hypersurface defined by a homogeneous cubic equation in P⁵.
Cubic function	Algebra	A polynomial function of degree exactly three with nonzero leading coefficient.
Cubic Graph	Graph Theory	A graph in which every vertex has degree exactly three, creating a sparse regular class with distinctive matching, coloring, symmetry, and Hamiltonicity theory.
Cubic Hermite Spline	Numerical AnalysisSpline Interpolation	A piecewise cubic interpolant specified by knot values and first derivatives, continuous through the first derivative.
Cubical Set	Algebraic Topology	Represent a combinatorial or homotopical object as a contravariant set-valued functor on a declared cube category, with functorial face, degeneracy, and optional richer cube operations.
Cullen number	Number TheoryInteger Sequences	An integer of the form C_n = n·2^n + 1, forming a named exponential sequence whose rare prime terms are Cullen primes.
Cumulant	Probability Theory	Encode a probability law by the coefficients of the logarithm of its generating function, so independent sums become coefficientwise addition and joint cumulants isolate connected dependence.
Cumulative distribution function	Probability TheoryDistribution Representations	For a real-valued random variable X, the function F(x)=P(X≤x), a nondecreasing right-continuous map whose limits are zero and one and which uniquely determines the distribution.
Cumulative Hierarchy	Axiomatic Set Theory	Build a set universe in ordinal stages by retaining earlier contents, adding all available subsets at successors, and uniting stages at limits.
Cunningham function	Special Functions	A special-function family expressed through the confluent hypergeometric U function and used in higher-order density expansions and diffusion equations.
Cunningham number	Number TheoryExponential Integer Sequences	An integer of the form b^n−1 or b^n+1 with integer base b that is not itself a perfect power, organizing prominent exponential families for factorization and primality study.
Curl (mathematics)	Vector Calculus	The vector differential operator measuring the local infinitesimal circulation and rotation axis of a three-dimensional vector field.
Curve	Geometry	A one-dimensional path represented, in its broad topological form, as the continuous image of an interval in a space.
Curvelet Transform	—	A directional multiscale transform with parabolically scaled, increasingly elongated fine-scale elements that sparsely represent smooth curves and curved singularities.
Curvilinear coordinates	Differential Geometry and Mathematical Physics	A locally invertible coordinate system on Euclidean space or a manifold whose coordinate curves or surfaces may be curved, with geometry represented through basis variation, metric coefficients and a Jacobian.
Cusp Form	Modular FormsArithmetic Geometry	A modular form of specified group and weight is cuspidal when its local expansion has zero constant term at every cusp.
Cut Rule	Proof TheorySequent Calculus	A sequent-calculus inference rule that composes two derivations through a matched intermediate formula, discharging it from the result.
Cut-Elimination Theorem	Proof Theory	For a specified proof calculus, every derivation using an intermediate cut formula can be replaced by a cut-free derivation of the same end judgment, when the calculus satisfies the theorem's rule conditions.
CW complex	Algebraic Topology	A topological space constructed inductively by attaching open cells of increasing dimension under closure-finiteness and weak-topology conditions.
Cycle Graph (Algebra)	Group TheoryVisual Group Theory	An undirected diagram representing finite-group elements as vertices and cyclic subgroups as identity-sharing polygons.
Cyclic Algebra	Algebra	A central simple algebra built from a cyclic Galois extension K/F, a generator sigma, and a scalar a, with a twisting element u satisfying u^n=a and uk=sigma(k)u.
Cyclic Category	Category TheoryAlgebraic Topology	The category of finite cyclically ordered sets and degree-one monotone maps, represented by periodic integer lifts modulo target-period translation.
Cyclic group	Group Theory	A group generated by repeated integer powers of one element, so every member lies on a single algebraic cycle or infinite progression.
Cyclic homology	Homological AlgebraNoncommutative Geometry	A homology theory imposing cyclic symmetry on Hochschild chains to provide de Rham-like invariants for associative, including noncommutative, algebras.
Cyclic module	Ring TheoryModule Theory	In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element.
Cyclic Number	Number Theory	Encode a repeating unit fraction as a digit block whose consecutive nonzero multiples appear as cyclic rotations of that block in a fixed base.
Cyclic number (group theory)	Finite Group Theory and Number Theory	A positive integer n such that every group of order n is cyclic, equivalently n is coprime to Euler's totient phi(n).
Cyclic quadrilateral	Euclidean Geometry	In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertices all lie on a single circle, making the sides chords of the circle.
Cyclic surgery theorem	Low Dimensional Topology	A three-manifold theorem bounding the distance between two Dehn fillings of an irreducible boundary-torus manifold when both filled manifolds have cyclic fundamental group.
Cyclically Ordered Group	Ordered Group Theory	Equip a group with a ternary cyclic order that is preserved by multiplication on both sides, making the algebraic translations act as orientation-preserving symmetries of a circle-like order.
Cyclomatic number	Graph Theory	The dimension of an undirected graph's cycle space, equal to the minimum number of edges whose removal makes it acyclic and computed as edges minus vertices plus connected components.
Cyclotomic polynomial	Algebra	The monic irreducible integer polynomial whose roots are exactly the primitive nth roots of unity.
Cylinder set	Product Spaces	A subset of a Cartesian product determined by restrictions on only finitely many coordinates, forming basic sets for product topology and generators for cylinder sigma-algebras.
Cylinder Set Measure	Infinite Dimensional Analysis	A consistent family of finite-dimensional distributions represented on the cylinder algebra of an infinite-dimensional linear space, often only finitely additive until an extension or radonification criterion produces a genuine countably additive measure.
Cylindrical Algebraic Decomposition	Real Algebraic Geometry	Partition real coordinate space into finitely many connected semialgebraic cells that are cylindrically compatible under projection and sign-invariant for a declared polynomial family.
Cylindrification	Computability Theory	Extend a numbering by pairing every original index with an ignored auxiliary coordinate, creating infinitely many computably organized names for each numbered object and converting ordinary reducibility comparisons into one-one form.
Càdlàg Function	Real AnalysisStochastic Processes	A time-indexed function whose value agrees with its right limit and whose pre-time left limit exists, allowing jumps with a defined before-and-after state.
Céa's lemma	Numerical AnalysisFinite Element Method	Céa's lemma bounds the error of a Galerkin finite-element approximation by a continuity-to-coercivity constant times the best approximation error available in the chosen subspace.
D-interval hypergraph	Graph TheoryHypergraphs	In graph theory, a -interval hypergraph is a kind of a hypergraph constructed using intervals of real lines.
Dagger Compact Category	Category TheoryMonoidal Duality	A compact closed category with an involutive adjoint compatible with its tensor structure and duality cups and caps.
Daniell Integral	—	A function-first integration construction that extends a positive monotone-continuous linear functional from an elementary function lattice and derives measure afterward.
Danzer set	Discrete Geometry	A set of points in Euclidean space that intersects every convex body of unit volume, studied through the unresolved question of whether bounded-density examples exist.
Darboux vector	Differential Geometry	The instantaneous angular-velocity vector of a moving orthonormal frame along a space curve, combining curvature and torsion.
Davenport–Schinzel sequence	Combinatorics	In combinatorics, a Davenport–Schinzel sequence is a sequence of symbols in which the number of times any two symbols may appear in alternation is limited.
Davies–Bouldin index	Cluster AnalysisInternal Validation	An internal cluster-validity score averaging, over clusters, the worst ratio of combined within-cluster scatter to between-centroid separation, with lower values indicating better compactness-separation tradeoff.
Dawson–Gärtner theorem	Probability Theory	A projective-limit theorem lifting compatible finite-dimensional large-deviation principles to the inverse-limit space.
Day Convolution	Category Theory	Lift an indexing category's tensor to a product of functors by left Kan extending their value-wise tensor along it.
Ddbar lemma	Complex and Kahler Geometry	A complex-geometric result stating, under its standard hypotheses, that a differential form closed under both ∂ and ∂̄ and exact for d is also ∂∂̄-exact.
De Bruijn Torus	Combinatorics on WordsCombinatorial Design	A two-dimensional periodic symbol array in which every possible fixed-size window appears exactly once per period, so each window identifies one position.
De Morgan's Laws	Mathematical LogicBoolean Algebra	In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference.
Decimal	Radix Notation and Elementary Arithmetic	A radix-ten numeral system and notation in which place positions encode powers of ten for integer and fractional quantities.
Decimal representation	Number Representation	A positional base-ten digit expansion of a nonnegative real number, with finite integer part and finite or infinite fractional part.
Decomposable measure	Measure Theory	A measure space partitionable into measurable pieces of finite measure so every measurable set is assembled compatibly from its intersections with those pieces.
Dedekind cut	Order Theory and Foundations	Represent a boundary in a linear order by a downward-closed lower part with no greatest element, constructing order completion and the real numbers from rational cuts.
Dedekind eta function	Modular Forms	A holomorphic function on the upper half-plane defined by q^(1/24) times the infinite product of (1−q^n), with a weight-one-half modular transformation law.
Dedekind psi function	Number Theory	The multiplicative arithmetic function ψ(n)=n times the product of (1+1/p) over the distinct prime divisors of n.
Dedekind zeta function	Algebraic Number TheoryZeta and L Functions	Encode the nonzero ideals of a number field in a Dirichlet series and Euler product whose analytic behavior carries arithmetic information about the field.
Deductive closure	Mathematical LogicConsequence Operators	The smallest superset of a set of formulas that contains every formula derivable from it under a specified consequence relation or proof system.
Defective matrix	Linear Algebra	A square matrix lacking a full basis of eigenvectors and therefore not diagonalizable over the stated field.
Deficiency (graph theory)	Graph TheoryMatching Theory	A matching-theoretic shortfall measure, commonly the maximum over vertex subsets of the number of selected vertices minus the size of their neighborhood.
Deficient number	Number TheoryDivisor Sum Classification	Classify a positive integer as deficient when the sum of its positive proper divisors is smaller than the integer itself, equivalently when its divisor sum is less than twice the integer.
Definable set	Model Theory	A subset or relation in a mathematical structure consisting exactly of tuples satisfying a first-order formula, with or without named parameters.
Definite quadratic form	Linear Algebra	A real quadratic form that is strictly positive on every nonzero vector or strictly negative on every nonzero vector.
Degeneracy (graph theory)	Graph Theory	The least k such that every nonempty subgraph has a vertex of degree at most k, equivalently the maximum core number in a graph.
Degeneration (algebraic geometry)	Algebraic Geometry	A family of algebraic varieties or schemes whose general fibers specialize to a distinguished, often more singular, fiber, with flatness controlling which invariants are preserved.
Degree (graph theory)	Graph TheoryVertex Invariants	The number of edge ends incident to a vertex, with loops counted twice in an undirected multigraph and distinct in-degree and out-degree counts for directed graphs.
Degree diameter problem	Extremal Graph Order Under Degree and Diameter	Maximize the number of vertices in a finite graph subject to simultaneous maximum-degree and diameter bounds, comparing constructions with the breadth-first Moore upper bound.
Degree of a field extension	Field Theory	The vector-space dimension of an extension field over its base field.
Degree of an algebraic variety	Algebraic Geometry	The multiplicity-counted number of intersections between an embedded algebraic variety and a general complementary-dimensional linear subspace.
Dehn surgery	Geometric TopologyThree Manifold Constructions	A 3-manifold construction that removes tubular neighborhoods of link components and glues solid tori back along specified boundary slopes.
Del	Vector Calculus	The vector differential operator whose combinations with scalar or vector fields denote gradient, divergence and curl.
Delaporte Distribution	Probability and StatisticsOverdispersed Count Distributions	A discrete count distribution formed as the convolution of independent Poisson and negative-binomial components, equivalently a Poisson count with a mean containing fixed and gamma-random parts.
Deligne–Lusztig theory	Representation Theory and Algebraic Geometry	A geometric construction of representations of finite groups of Lie type from compactly supported l-adic cohomology of varieties associated with reductive groups, Frobenius maps, and maximal tori.
Delone Set	Discrete Geometry	A metric-space point set with both a positive uniform-separation bound and a finite covering-radius bound, so it is nowhere arbitrarily crowded and nowhere arbitrarily sparse.
Delta Functor	Homological AlgebraCategory Theory	A graded family of additive functors whose natural connecting maps turn short exact sequences into long exact sequences.
Delta set	Algebraic TopologySemi Simplicial Objects	A semi-simplicial object consisting of sets of n-simplices with face maps satisfying simplicial identities but no required degeneracy maps, providing flexible combinatorial models for gluing and homology.
Denjoy's theorem on rotation number	Dynamical Systems and Circle Maps	A regularity theorem stating that an orientation-preserving circle diffeomorphism with irrational rotation number and derivative of bounded variation is topologically conjugate to the corresponding irrational rotation.
Dense Graph	Graph Theory	A graph whose edge population is a substantial, usually non-vanishing fraction of all possible vertex pairs, placing it in the quadratic-edge rather than sparse regime under a declared asymptotic convention.
Densely defined operator	Operator TheoryFunctional Analysis	In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function.
Depth (ring theory)	Commutative Algebra	A homological invariant measuring the length of a maximal regular sequence acting on a module, equivalently the first degree of nonvanishing Ext under standard local Noetherian hypotheses.
Derangement	Enumerative Combinatorics	A permutation with no fixed points, so every element moves away from its original position.
Derivative	Calculus Real Analysis	The limit of a difference quotient, f′(x) = lim_{h→0}[f(x+h)−f(x)]/h, that turns 'rate of change at a single point' into a defined object — simultaneously the tangent slope, the best local linear approximation, and the output's sensitivity to an infinitesimal input nudge.
Derivative of the Exponential Map	Lie Theory	The Lie exponential's differential converts an algebra perturbation into a group tangent using a commutator correction.
Derived functor	Homological Algebra	A functor obtained by resolving objects relative to an exactness-deficient functor and taking homology, systematically measuring its failure to preserve exact sequences.
Derived scheme	Derived Algebraic Geometry	A scheme-like geometric object whose structure sheaf carries homotopical or differential-graded information encoding higher intersections and deformation data.
Desargues's Theorem	Projective Geometry	A projective-incidence rule linking concurrence of corresponding triangle-vertex lines to collinearity of corresponding side intersections in a Desarguesian setting.
Descartes Number	Multiplicative Number Theory	An odd composite integer equipped with a coprime factorization D=km that satisfies the perfect-number divisor-sum equation only when the composite factor m is formally treated as a single prime-like factor.
Descent (Mathematics)	Algebraic GeometryDescent Theory	Recover a global mathematical object from compatible local objects by comparing pullbacks on fiber-product overlaps, enforcing a cocycle, and proving the resulting datum effective.
Descriptive Geometry	GeometryEngineering Graphics	Represent and solve three-dimensional spatial relations through coordinated planar projections whose view directions are deliberately changed to expose true lengths, shapes, incidences, and distances.
Determinacy	Descriptive Set Theory	Classify a specified perfect-information win-or-lose game by whether one player has a strategy that defeats every possible counterplay.
Determinant	Linear Algebra	Map a square matrix or finite-dimensional endomorphism to the unique normalized alternating multilinear scalar that tracks invertibility, oriented volume scaling, and composition multiplicatively.
Determinantal variety	Algebraic GeometryCommutative Algebra	The radical ideal defining the determinantal variety is generated by the (r + 1) × (r + 1) minors of the matrix (Bruns-Vetter, Theorem 2.10).
Development (topology)	General Topology	A countable sequence of open covers whose stars at each point form a neighborhood base, characterizing developable spaces.
Deviation of a local ring	Commutative Algebra	A sequence of nonnegative homological invariants counting generators in an acyclic closure and measuring successive departures of a local ring from regularity and complete-intersection structure.
DF-space	Functional AnalysisLocally Convex Topological Vector Spaces	Characterize a locally convex topological vector space by countable quasi-barrelledness together with a fundamental sequence of bounded sets, capturing the structural class modeled by strong duals of metrizable spaces.
Diaconescu's theorem	Constructive Set Theory	The constructive-logic result that a sufficiently strong axiom of choice entails the law of excluded middle.
Diagonal form	Algebra	A homogeneous polynomial containing only pure powers of individual variables and no mixed monomials.
Diagonal functor	Category Theory	The functor sending each object and morphism to a constant tuple or constant diagram, whose adjoints characterize categorical products, coproducts, limits and colimits.
Diagonal Matrix	Linear Algebra	A square matrix with every off-diagonal entry equal to zero, representing independent coordinate scaling in the chosen basis.
Diagonal Morphism	Category Theory	For an object with a categorical self-product, select the unique map into that product whose composites with both projections are the identity.
Diagonal subgroup	Group Theory	The subgroup of a direct power G^n consisting of tuples whose every coordinate is the same group element.
Diagram (category theory)	Category TheoryDiagrams Limits and Colimits	A functor from an index category into a target category, encoding a shaped family of objects together with all indexed morphisms and composition relations for limits, colimits, and universal constructions.
Diagram (mathematical logic)	Model Theory	The set of first-order sentences with named parameters that are true in a structure, with atomic and elementary variants preserving different amounts of information.
Diameter (graph theory)	Graph TheoryGraph Distance and Invariants	Measure a connected graph by the maximum shortest-path distance over all pairs of vertices.
Diameter (group theory)	Finite Group Theory	The largest Cayley-graph distance required to reach elements of a finite group under a specified generating-set convention, sometimes maximized over all generating sets.
Diamond Operation	Higher Category Theory	Combine two simplicial sets by gluing the endpoint faces of their product cylinder to the respective factors, producing a simplicial set over the 1-simplex that is categorically equivalent to their join.
Diamond principle	Set Theory	A set-theoretic guessing principle asserting a sequence that correctly anticipates every subset of the first uncountable ordinal on a stationary set.
Dictator Game	Game TheoryExperimental Economics	An experimental protocol where one participant unilaterally splits an endowment with a powerless recipient — by stripping away every strategic lever, it forces any positive transfer onto the non-strategic ledger, so its lattice of single-bit variants decomposes the sources of pro-social behaviour.
Diffeomorphism	Differential TopologyDifferential Geometry	A bijection between differentiable manifolds whose forward map and inverse are differentiable to the stated class, establishing equivalence of their smooth structures.
Difference of Two Squares	AlgebraFactorization	The algebraic identity a² − b² = (a − b)(a + b), converting a difference of commuting squares into conjugate factors.
Different ideal	Algebraic Number Theory	An ideal measuring the failure of the ring of integers of a number field to be self-dual under the trace pairing, inverse to the codifferent fractional ideal.
Differentiable curve	Differential Geometry	A parametrized path in a manifold or Euclidean space whose coordinate representation has the declared degree of differentiability.
Differentiable stack	Differential GeometryDifferentiable Stacks and Lie Groupoids	Represent quotient-like smooth geometry as a stack on manifolds admitting a smooth atlas, equivalently through a Lie-groupoid presentation considered up to Morita equivalence.
Differentiable vector-valued functions from Euclidean space	Functional Analysis	Differentiable maps from an open Euclidean domain into a topological vector space, with derivatives encoded by continuous multilinear maps.
Differential Calculus over Commutative Algebras	Commutative AlgebraDifferential Geometry	An algebraic calculus in which derivations, finite-order operators, differentials, and jets are defined from a commutative algebra, its modules, multiplication commutators, and universal properties.
Differential equation	Dynamical Systems	A mathematical equation relating an unknown function to its own derivatives, encoding the law that a quantity's rate of change depends on its current state — so specifying the local rule plus initial or boundary conditions fixes the entire trajectory.
Differential form	Differential Geometry	An alternating covariant tensor field that can be integrated over oriented manifolds of matching dimension.
Differential game	Game Theory	A continuous-time strategic game whose shared state evolves by differential equations controlled by multiple players with distinct objectives.
Differential Inclusion	Dynamical Systems	A continuous-time evolution law that permits a state-dependent set of instantaneous velocities rather than prescribing just one.
Differential invariant	Differential Geometry	A function of variables and derivatives that remains unchanged under the prolonged action of a transformation group.
Differential of a function	CalculusDifferential Calculus	The linear map giving the first-order change of a differentiable function at a point, written df and represented in one variable by dy=f′(x)dx.
Differential operator	Differential Equations and Geometry	An operator built from derivatives and coefficient functions that maps functions or sections to new functions or sections according to a declared finite order.
Differential Structure	Differential GeometrySmooth Manifolds	A maximal compatible atlas that determines which coordinate descriptions on a topological manifold count as differentiable.
Differentiation of trigonometric functions	Differential Calculus	The calculus rule family that maps trigonometric functions and their compositions to derivatives through their periodic identities, limit behavior and the chain rule.
Differintegral	Fractional CalculusFractional Operators	A fractional-calculus operator D^q that unifies differentiation for positive order and integration for negative order, with integer cases recovered under a specified convention.
Diffusion Process	Probability TheoryStochastic Processes	A continuous-path continuous-time Markov process governed locally by drift and covariance and globally by its transition law.
Digamma function	Special Functions	The logarithmic derivative ψ(z)=Γ′(z)/Γ(z) of the gamma function, extending shifted harmonic-number relations to complex arguments.
Digon	Polygon GeometrySpherical Geometry	A two-sided polygonal object with two vertices and two edges, degenerate under ordinary straight-sided Euclidean realization but nondegenerate in settings such as spherical geometry.
Dihedral Angle	GeometrySpatial Angles	Two intersecting planes or oriented half-planes are compared around their common line, producing an unsigned fold angle or a convention-dependent signed torsion that encodes relative orientation in geometry, molecules, chains, and polyhedra.
Dimension (vector space)	Linear AlgebraVector Spaces	The cardinality of any basis of a vector space over a specified field, well-defined because all bases have the same cardinality.
Dimension of an algebraic variety	Algebraic Geometry	The intrinsic number of independent parameters of an algebraic variety, equivalently the Krull dimension of its coordinate ring in the affine irreducible case.
Dini continuity	Real AnalysisModulus of Continuity	A continuity refinement requiring the modulus of continuity to have a finite scale-weighted integral near zero, equivalently a summable geometric-scale modulus under standard assumptions.
Diophantine Equation	Number TheoryDiophantine Analysis	A polynomial equation with integer coefficients whose admissible solutions are required to be integers.
Diophantine quintuple	Number TheoryDiophantine Tuples	A five-element set of positive integers for which the product of every two distinct elements plus one is a perfect square.
Dirac delta function	Distribution Theory	A distribution concentrated at one point whose action on a test function returns that function’s value at the point.
Dirac Structure	Differential GeometryGeometric Mechanics	A maximally isotropic relation between vectors and covectors whose geometric form additionally requires integrability and whose port form encodes power-conserving interconnection.
Direct image with compact support	Sheaf TheorySix Functor Formalism	Send a sheaf along a continuous map while retaining only local sections whose support is proper over the target open set, yielding the functor conventionally written f-shriek.
Direct Limit	Category TheoryAlgebra	A directed colimit that assembles compatible forward stages into an object universal among cocones from those stages.
Direct Product	Abstract Algebra	A product of algebraic structures formed from all factor tuples with coordinatewise operations and structure-preserving projections.
Direct product of groups	Group Theory	In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted .
Direct sum of groups	Abstract AlgebraGroup Constructions	A group assembled from mutually commuting normal subgroups with trivial intersections so every element decomposes uniquely into component elements, with finite support in infinite families.
Direct Sum of Topological Groups	Topological AlgebraTopological Groups	Decompose a topological group into subgroup factors whose multiplication map is simultaneously a group isomorphism and a homeomorphism, preserving both algebraic and topological structure.
Directed algebraic topology	Directed TopologyAlgebraic Topology	In mathematics, directed algebraic topology is a refinement of algebraic topology for directed spaces, topological spaces and their combinatorial counterparts equipped with some notion of direction.
Direction (geometry)	Geometry	An equivalence class of parallel oriented lines, rays, or nonzero vectors that retains orientation while discarding position and usually magnitude.
Directional Derivative	Multivariable Analysis and Differential Geometry	The instantaneous rate of change of a function at a point when the input moves along one specified vector, computed as a one-dimensional path limit and, under differentiability, by applying the derivative to that vector.
Dirichlet algebra	Uniform Algebras and Operator Theory	A closed unital subalgebra of continuous functions on a compact Hausdorff space whose real parts are uniformly dense in all continuous real-valued functions.
Dirichlet convolution	Multiplicative Number Theory	A binary operation on arithmetic functions defined by summing f(d)g(n/d) over the positive divisors d of n.
Dirichlet density	Analytic Number Theory	An analytic density of a set of primes defined by its weighted prime Dirichlet series as the exponent approaches one from above.
Dirichlet Eigenvalue	Spectral TheoryElliptic Boundary Value Problems	An eigenvalue of the negative Laplacian on a domain is selected by a nonzero mode that vanishes on the domain boundary.
Dirichlet Eta Function	Analytic Number Theory	Extend the alternating reciprocal-power series into an entire complex function tied to zeta by η(s)=(1−2^(1−s))ζ(s), gaining convergence on Re(s)>0 and a characteristic extra zero lattice.
Dirichlet Kernel	Harmonic Analysis	The finite symmetric Fourier-mode selector whose periodic convolution produces an ordinary Fourier partial sum.
Dirichlet series	Analytic Number Theory	A complex series whose nth term is a coefficient multiplied by n raised to a complex negative exponent, serving as a multiplicative generating function in analytic number theory.
Discontinuous linear map	Functional Analysis	A linear transformation between topological vector spaces that fails the continuity or boundedness condition imposed by their topologies.
Discrepancy theory	Combinatorics	The study of how evenly discrete points, signs, or colors can approximate a desired continuous or balanced distribution over a family of test sets.
Discrete Chebyshev Transform	Numerical AnalysisApproximation Theory	Convert values on a Chebyshev grid to finite polynomial-basis coefficients and back under a specified nodal convention.
Discrete Hartley Transform	Digital Signal Processing	A real, finite cas-kernel transform encodes a real sequence in N real coefficients and uses the same kernel for inversion up to scale.
Discrete measure	Measure Theory	A measure concentrated on an at most countable set, representable as a countable weighted sum of point masses under the stated measurable-space convention.
Discrete space	General Topology	A topological space in which every subset is open, equivalently every point is isolated and the topology is the full power set.
Discrete-time Markov chain	Probability Theory	A stochastic sequence whose next-state distribution depends on the current state and transition step but not on the earlier path once the present is known.
Discrete-time proportional hazards	Survival AnalysisDiscrete Duration Models	A grouped-duration model whose conditional event probability in each interval is linked to covariates so their effects correspond to proportional underlying hazards or a specified discrete analogue.
Disjunct matrix	Combinatorial Group Testing	A binary nonadaptive group-testing design in which every column has a row containing 1 while any chosen set of at most d other columns all contain 0.
Disjunctive normal form	Boolean Logic	A Boolean formula represented as a disjunction of conjunctions of literals—an OR of AND terms.
Disk-Covering Problem	Discrete GeometryCovering Problems	One of the covering disks is placed central and the remaining five in a symmetrical way around it.
Dispersion Function	Probability TheoryStochastic Orders	The convex location-indexed functional \(D_X(u)=\mathbb{E}|X-u|\), whose slopes recover an integrable real distribution.
Dispersion point	TopologyConnectedness and One Point Connectifications	Identify a point whose removal turns a connected topological space into a space with no nontrivial connected component, concentrating the original connectedness at one indispensable point.
Displaced Poisson Distribution	ProbabilityDiscrete Distributions	A Poisson-tail count law obtained by conditioning on at least an integer threshold and counting the excess, equivalently shifting the Poisson probability recurrence.
Dissociation number	Graph Theory and Combinatorial Optimization	The maximum number of vertices in a graph whose induced subgraph has maximum degree at most one.
Distance (graph theory)	Graph Theory	The length of a shortest path between two graph vertices, with infinity or undefined value when no admissible path connects them.
Distance Matrix	Discrete MathematicsMetric Geometry	An indexed square array whose entry records a declared pairwise distance or dissimilarity, with metric properties present only when the underlying function satisfies them.
Distribution (Differential Geometry)	Differential Geometry	A distribution on a smooth manifold assigns to each point a smoothly varying subspace of its tangent space, encoding admissible infinitesimal directions.
Distributive Law Between Monads	Category Theory	A natural transformation λ:TS⇒ST coherent with both monads' units and multiplications, allowing the ordered composite ST to inherit a monad structure.
Distributivity	Abstract Algebra	A universal compatibility law between two operations, equating action on a combined operand with the combination of separate actions, with distinct left and right forms when order matters.
Distributivity (order theory)	Order Theory	A family of order-theoretic laws governing how infima and suprema interact, from finite lattice distributivity to complete and infinite distributive variants.
Divide-and-Conquer Eigenvalue Algorithm	Numerical Linear Algebra	A symmetric tridiagonal eigensolver recursively splits the matrix with a rank-one tear, then merges child spectra through a stabilized secular update.
Division (mathematics)	Arithmetic and AlgebraBasic Operations	Recover a quotient q from dividend a and nonzero divisor b by solving bq=a, with exact, remainder, rational, field and algorithmic meanings determined by the ambient number system.
Division Algebra	Abstract AlgebraAlgebraic Structures	A nonzero algebra over a field in which left and right multiplication by every nonzero element are bijective, permitting unique division on both sides under the declared associativity convention.
Divisor (Algebraic Geometry)	Algebraic Geometry	An algebraic-geometric divisor records codimension-one zero-and-pole data as a Weil cycle or compatible local Cartier equations, with class and line-bundle relations set by the space.
Divisor Function	Number TheoryArithmetic Functions	The multiplicative arithmetic-function family \(\sigma_z(n)=\sum_{d\mid n}d^z\), including divisor count and divisor sum as distinguished cases.
Divisor summatory function	Analytic Number Theory	In number theory, the divisor summatory function is a function that is a sum over the divisor function.
Dixmier Trace	Operator AlgebrasNoncommutative GeometryFunctional Analysis	A singular trace on the weak trace-class ideal obtained by applying a generalized limit to logarithmically normalized partial sums of an operator's ordered eigenvalues or singular values.
Dixon's identity	CombinatoricsHypergeometric Series	A family of binomial and hypergeometric summation identities associated with A. C. Dixon, including a terminating triple-binomial evaluation.
Dolbeault Cohomology	Complex Geometry	Measure complex-geometric obstructions by taking ∂̄-closed forms modulo ∂̄-exact forms in each bidegree.
Dold–Kan correspondence	Homological Algebra and Simplicial Methods	An equivalence between simplicial abelian groups and nonnegatively graded chain complexes, matching homotopy groups with homology groups and simplicial homotopy with chain homotopy.
Dollar Auction	Game TheoryAuctions and Mechanism Design	Mechanize the escalation trap with one rule change — the second-highest bidder also pays and gets nothing — so that backward induction makes each additional bid locally rational and rational players escalate past the prize's value.
Doléans–Dade Exponential	Stochastic CalculusSemimartingale Theory	Map a semimartingale driver to the unique multiplicative process solving dZ = Z_- dX, with continuous quadratic-variation and jump-product corrections that ordinary exponentiation omits.
Domain (ring theory)	Ring Theory	A nonzero ring with no nonzero left or right zero divisors.
Domain coloring	Complex Analysis Visualization	A visualization of a complex-valued function that maps each input-plane point to color properties encoding output argument, magnitude and sometimes contour structure.
Domain of holomorphy	Complex AnalysisSeveral Complex Variables	A domain in several complex variables that supports a holomorphic function unable to extend across any strictly larger connected domain, equivalently a holomorphically convex natural domain.
Domatic number	Graph TheoryDomination Partitions and Graph Invariants	Assign a graph the largest number of blocks in a vertex partition for which every block is a dominating set.
Dominant functor	Category Theory	A functor whose target objects are all retracts of objects in its image.
Dominated Strategy	Game TheoryGame Theoretic Solution Concepts	Rule out an action that yields a lower payoff than some alternative for every possible profile of opponents' actions — a belief-free test requiring no model of the opponent, licensing iterated elimination and the dominant-strategy robustness that mechanism design targets.
Donaldson–Thomas theory	Enumerative Algebraic Geometry	An enumerative theory assigning virtual counts to moduli spaces of stable sheaves, ideal sheaves, or related objects on Calabi–Yau threefolds.
Doob Decomposition Theorem	—	An adapted integrable discrete-time process splits uniquely into its initial value, a martingale of one-step surprises, and a predictable cumulative conditional drift.
Doob martingale	ProbabilityMartingales and Filtrations	Track the evolving conditional expectation E[Y|F_t] of an integrable target as a filtration reveals information, producing a martingale of progressively refined best predictions.
Doob–Dynkin lemma	Probability TheoryMeasure Theoretic Probability and Conditional Expectation	Characterize when a measurable quantity carries no information beyond a measurable map: under the stated measurable-space conditions it factors as a measurable function of that map exactly when it is measurable with respect to the map's generated sigma-algebra.
Door space	Topology	A topological space in which every subset is open, closed or both.
Dot Product	Linear AlgebraEuclidean Vector Geometry	A symmetric bilinear pairing of real coordinate vectors that sums componentwise products and thereby encodes Euclidean length, angle, orthogonality, and projection.
Double (manifold)	Differential Topology	The boundaryless manifold formed by gluing two copies of a manifold with boundary point-for-point along their entire common boundary.
Double affine braid group	Representation TheoryDouble Affine Structures	A braid-like group associated with an affine root system that adds a second affine translation structure and whose group algebra leads to double affine Hecke algebras.
Double category	Category TheoryHigher Categories	A two-dimensional categorical structure with objects, horizontal arrows, vertical arrows and squares that compose in both directions subject to an interchange law.
Double graph	Graph Products and Transformations	A graph construction that replaces every vertex with two nonadjacent copies and replaces each original edge with all four edges between the corresponding copy pairs.
Double Mersenne number	Number Theory	In mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2^p-1}-1 where p is prime.
Double tangent bundle	Differential Geometry	The tangent bundle of the total space of a manifold’s tangent bundle, carrying two compatible vector-bundle projections and a canonical flip.
Doubling space	Metric GeometryFinite Dimensional Metric Spaces	A metric space in which every radius-r ball can be covered by at most a fixed number of radius-r/2 balls, giving a scale-uniform finite doubling dimension.
Doubly stochastic matrix	Matrix TheoryStochastic and Convex Matrices	A square nonnegative matrix whose every row and column sums to one, equivalently a convex combination of permutation matrices and a point in the Birkhoff polytope.
Drazin inverse	Matrix Theory	A unique generalized inverse for a square matrix defined through its index, commutation, and power equations.
Du Bois singularity	Algebraic Geometry	Classify a reduced characteristic-zero scheme by requiring its structure sheaf to agree quasi-isomorphically with degree zero of the Du Bois complex, equivalently through a log-resolution criterion.
Dual (category theory)	Category Theory	The principle that reversing every morphism and composition order converts any categorical statement into a dual statement valid in the opposite category.
Dual curve	Projective Algebraic Geometry	The curve in the dual projective plane whose points correspond to tangent lines of a given plane curve.
Dual lattice	Geometry of Numbers and Lattice Theory	The lattice of vectors pairing integrally with every vector of a full-rank lattice under a declared inner product.
Dual matroid	Matroid TheoryCombinatorics	In matroid theory, the dual of a matroid M is another matroid M^\ast that has the same elements as M , and in which a set is independent if and only if M has a basis set disjoint from it.
Dual module	Module TheoryAbstract Algebra	In mathematics, the dual module of a left (respectively right) module M over a ring R is the set of left (respectively right) R-module homomorphisms from M to R with the pointwise right (respectively left) module structure.
Dual number	Algebra and Automatic Differentiation	An element a+b epsilon of a two-dimensional commutative algebra with nonzero nilpotent epsilon satisfying epsilon squared equals zero.
Dual polyhedron	Polyhedral Geometry	In geometry, every polyhedron is associated with a second dual structure, wherein the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.
Dual System	Functional Analysis	Pair two vector spaces by a nondegenerate bilinear form so each separates points of the other and determines weak, polar, and compatible locally convex structures.
Dual wavelet	Wavelet and Frame Theory	A wavelet family biorthogonal to a primal wavelet family so analysis coefficients and synthesis functions reconstruct signals even when the basis is not orthonormal.
Duality (order theory)	Order Theory	The order-reversing construction that replaces a partially ordered set by the same elements with every comparison reversed.
Dualizing module	Commutative AlgebraHomological Algebra	In abstract algebra, a dualizing module, also called a canonical module, is a module over a commutative ring that is analogous to the canonical bundle of a smooth variety.
Dually chordal graph	Graph Theory	A graph admitting a maximum-neighborhood ordering, equivalently one whose maximal-clique hypergraph has a join tree with the graph’s vertex set as host.
Dubins–Spanier Theorems	Fair Division TheoryMeasure Theory	A family of measure-theoretic results showing that attainable participant-by-piece valuation matrices are compact and convex under countably additive nonatomic measures, with fair-division existence corollaries.
Ducci Sequence	Discrete MathematicsInteger Dynamical Systems	An orbit of a cyclic integer tuple under repeated replacement by the absolute differences of adjacent entries, studied through its transient, zero-reaching, and periodic behavior.
Dudeney number	Recreational Number Theory	A base-dependent natural number that is a perfect cube whose digit sum equals its cube root.
DuPont Analysis	Financial Statement AnalysisProfitability Analysis	Diagnose return on equity by decomposing it into multiplicative profitability, asset-use, and leverage components, with extended forms isolating tax and financing effects.
Durfee square	Partition Theory	The largest square contained in the Ferrers or Young diagram of an integer partition.
Dvoretzky–Kiefer–Wolfowitz inequality	Probability Theory	A distribution-free exponential bound on the probability that an empirical cumulative distribution function deviates uniformly from its population distribution.
Dyadic cubes	Harmonic Analysis and Geometric Measure Theory	A nested multiscale grid of half-open cubes whose side lengths are powers of two, partitioning Euclidean space at each scale and giving every cube a unique parent and finitely many children.
Dyadic Rational	Number Theory	A rational number expressible as an integer divided by a power of two, equivalently a rational number with a terminating binary expansion.
Dying percolation conjecture	Percolation Theory	The conjecture that critical Bernoulli bond percolation on every integer lattice of dimension at least two has no infinite cluster.
Dynamical Set	Dynamical Systems	A dynamical set is a subset of a dynamical system's state space defined or characterized by the behavior of points, orbits, iterates, images, preimages, recurrence, escape, stability, or invariance under a specified transformation or group or semigroup action.
Dyson Brownian Motion	Random Matrix Theory	Evolve an ordered spectrum as coupled Brownian particles whose inverse-gap drift prevents collisions and encodes eigenvalue repulsion inherited from stochastic motion of a symmetry-class matrix.
Dévissage	Algebraic GeometryAlgebraic K Theory	Reduce a statement about algebraic objects to finite filtrations of simpler subquotients and a justified transfer rule.
E-function	Transcendental Number TheoryArithmetic Differential Equations	A Siegel E-function is an entire exponential-generating series with algebraic coefficients of controlled conjugate size and denominator growth that also satisfies a linear differential equation over the polynomials.
Eberlein–Šmulian theorem	Functional Analysis	The Banach-space result equating weak compactness with weak sequential compactness and weak limit-point compactness.
Eckmann–Hilton Duality	Homotopy Theory	Generate and test homotopy-theoretic counterpart concepts by expressing a construction categorically and reversing its arrows, with adjunctions and universal properties guiding—but never automatically validating—the transfer.
Edge Coloring	Graph TheoryCombinatorial Optimization	Assign colors to graph edges so that no two edges sharing a vertex receive the same color.
Edge Covering Number	Graph Theory	The minimum cardinality of an edge cover of a graph: the smallest set of edges incident to every vertex, defined only when isolated vertices are absent or under an explicit extension, and equal to |V| minus the maximum-matching size for finite graphs without isolated vertices.
Edge Tessellation	Geometric Tilings	A congruent polygonal tiling closed under reflection across every tile edge, so one tile and its edge reflections generate the entire tiling.
Eells–Kuiper Manifold	Differential Topology	Recognize the exceptional closed manifolds in dimensions 2, 4, 8, or 16 that admit a three-critical-point Morse function and have projective-plane-like compactification and cohomology structure.
Effaceable Functor	Homological AlgebraAbelian Categories	An additive functor for which every object embeds by a map whose induced functor morphism is zero.
Effective Polish space	Computable AnalysisEffective Descriptive Set Theory	A complete separable metric space supplied with a computable dense presentation that makes basic distance comparisons effectively decidable or enumerable.
Egorov's theorem	Measure TheoryReal Analysis	It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian mathematician and geometer, who published independent proofs respectively in 1910 and 1911.
EHP spectral sequence	Algebraic Topology	A spectral sequence derived from EHP fibrations that inductively relates unstable homotopy groups of spheres to stable homotopy after localization at a prime.
Eigenvector Centrality	Network Analysis	A network node score recursively weighted by the scores of connected nodes and selected from a leading adjacency eigenvector.
Eight-Node Quadratic Serendipity Quadrilateral (Q8)	Finite Element AnalysisSerendipity Finite Elements on Quadrilaterals	A two-dimensional C0 isoparametric finite element with four vertex and four midside degrees of freedom whose reference shape-function space contains every total-degree-two polynomial while omitting the Q9 interior node and tensor-product term.
Eilenberg–MacLane space	Algebraic TopologyHomotopy Types	A connected space K(G,n) with exactly one nontrivial homotopy group, G in degree n, serving as a representing space for cohomology.
Eilenberg–Mazur swindle	Algebraic Topology	A proof method using infinite self-similar sums or decompositions to cancel or absorb an object in a seemingly paradoxical way.
Einstein manifold	Differential Geometry	A Riemannian or pseudo-Riemannian manifold whose Ricci curvature tensor is everywhere a scalar multiple of its metric.
Eisenstein reciprocity	Algebraic Number Theory	A higher-power reciprocity law relating residue symbols in cyclotomic integer rings.
El Farol Bar Problem	Game TheoryCongestion and Minority Games	The congestion model in which each agent independently decides whether to attend a capacity-limited bar, enjoyable only if uncrowded — so any prediction rule shared by all self-destructs, and the system resolves only through heterogeneous inductive predictors rather than convergence.
Elementary Amenable Group	Group TheoryAmenability	A group obtainable from finite and abelian groups by isomorphism, subgroups, quotients, extensions, and directed unions.
Elementary function arithmetic	Proof Theory	A weak first-order arithmetic theory whose provably total functions are the elementary recursive functions, typically extending bounded arithmetic with exponentiation.
Elementary substructure	Model TheoryMathematical Logic	A substructure preserving the truth of every first-order formula with its own parameters, equivalently satisfying the Tarski–Vaught existential-witness condition.
Elementary theory of abstract categories	Category TheoryCategorical Foundations	Lawvere's first-order axiomatization of categories and functors, treating objects indirectly through identity arrows and composition rather than through set-theoretic membership.
Elementary-embedding large-cardinal schema	Set TheoryLarge Cardinals	A set-theoretic schema that classifies certain large-cardinal properties by an elementary embedding from V to a transitive class, its critical point, and an explicitly quantified target-model condition.
Elimination theory	Computational Algebraic GeometryPolynomial Systems	Remove selected variables from polynomial systems while preserving the algebraic consequences in the retained variables, linking elimination ideals, resultants, Gröbner bases, projection, and implicitization.
Ellipse	Analytic Geometry	A closed planar curve whose distances to two fixed foci sum to a constant greater than their separation.
Ellipsoid packing	Discrete GeometryPacking Problems	The geometric optimization problem of arranging congruent ellipsoids in three-dimensional space without overlap so their asymptotic occupied-volume fraction is maximized.
Elliptic Complex	Differential GeometryGlobal Analysis	An elliptic complex is a differential-operator chain whose principal-symbol sequence is exact at every nonzero cotangent vector.
Elliptic Cylindrical Coordinates	Coordinate GeometryMathematical Physics	A three-dimensional orthogonal coordinate system obtained by extending planar elliptic coordinates along a perpendicular axis, with coordinate surfaces formed by confocal elliptic prisms, hyperbolic prisms, and planes.
Elliptic Divisibility Sequence	Arithmetic of Elliptic Curves	An integer divisibility sequence generated by the nonlinear recurrence of elliptic-curve division polynomials, translating multiplication of a rational point into term divisibility, height growth, ranks of apparition, and primitive-divisor structure.
Elliptic operator	Partial Differential EquationsElliptic Operators	A differential operator whose principal symbol is invertible away from the zero covector, excluding real characteristic directions and supporting strong regularity for its solutions.
Elliptic unit	Algebraic Number TheorySpecial Units	A distinguished algebraic unit in an abelian extension of an imaginary quadratic field, constructed from special values of modular or elliptic functions and forming an Euler system.
Empirical Measure	Probability TheoryEmpirical Processes	The random atomic probability measure P_n = n^{-1} sum_i delta_Xi that assigns equal mass to realized observations and turns sample averages into integration against a measure.
Empirical process	Probability and Statistics	A stochastic process indexing the centered and scaled difference between an empirical measure and its population expectation over a class of functions or sets.
Empty Sum	AlgebraSummation Conventions	A summation with no terms, assigned the additive identity of its carrier.
En (Lie algebra)	Lie Theory	The Lie or Kac–Moody algebra family associated with the E-n branching Dynkin diagram, including exceptional finite cases and indefinite extensions.
End (graph theory)	Infinite Graph Theory	An equivalence class of one-way infinite paths that remain connected in the same unbounded component after deletion of every finite vertex set.
Energetic Space	Functional Analysis and Mathematical PhysicsCoercive Operators and Variational Problems	The Hilbert completion of a symmetric strongly positive operator's domain in the induced quadratic-form norm, continuously embedded in the ambient Hilbert space and serving as the natural weak-solution and error space.
Energy functional	Calculus of VariationsPotential Theory	The condenser capacity also has a variational characterization: C(Σ, S) is the infimum of the Dirichlet's energy functional.
Engel expansion	Number Representation	A representation of a positive real number as a sum of reciprocals of cumulative products from a unique nondecreasing integer sequence.
Engquist–Majda absorbing boundary condition	Numerical AnalysisNumerical Pdes	In numerical methods for partial differential equations, Engquist–Majda absorbing boundary conditions or Lindman–Engquist–Majda absorbing boundary conditions are a hierarchy of absorbing boundary conditions for the numerical solution of wave equations.
Enriques–Kodaira classification	Algebraic Geometry	A birational classification of compact complex surfaces by Kodaira dimension and minimal-model invariants.
Entanglement (graph measure)	Graph TheoryStructural Graph Measures	A directed-graph complexity measure equal to the least number of cops needed to capture a perpetually moving robber in the entanglement game.
Envelope (category theory)	Category Theory	A universal embedding of a category or structured object into a larger completed category satisfying a specified closure or completion property.
Envelope (mathematics)	Differential GeometryCurve Families	A locus tangentially contacted by varying members of a specified family of curves, subject to regularity and parameter-domain checks.
Epigraph	Convex AnalysisOptimization	The upward-closed set of real-height pairs above an extended-real-valued function, preserving each function value as a fiber infimum.
Epigroup	Semigroup Theory	A semigroup in which every element is group-bound: some positive power of it lies in a subgroup of the semigroup.
Equally Spaced Polynomial	Finite Field ArithmeticBinary Extension Field Representation	Form a binary polynomial whose nonzero unit coefficients occupy the arithmetic progression of exponents 0, s, 2s, through rs, yielding a substitution-structured family used when irreducible members support regular finite-field arithmetic.
Equation-Free Modeling	Applied MathematicsMultiscale Scientific Computing	A multiscale framework that uses lift–simulate–restrict queries to a fine-scale model as an on-demand coarse time-stepper, enabling macroscopic analysis without explicit closed coarse equations.
Equational logic	Mathematical LogicUniversal Algebra	First-order equational logic consists of quantifier-free terms of ordinary first-order logic, with equality as the only predicate symbol.
Equiareal map	Differential Geometry	A smooth map between surfaces that preserves the area of every region, equivalently pulling back the target area element to the source area element.
Equichordal point problem	Convex Geometry History	The plane-geometry question whether a convex body can have two distinct interior points through each of which every chord has the same point-specific length, answered negatively.
Equicontinuity	Real AnalysisFunctional Analysis	Control the output variation of every function in a family with the same input neighborhood.
Equidistributed sequence	Number TheoryUniform Distribution	Require the limiting frequency of sequence terms in every subinterval to equal that subinterval’s normalized length.
Equidistribution Theorem	Number Theory	The theorem that successive multiples of an irrational number, reduced modulo one, become uniformly distributed on the unit interval or circle.
Equilateral Dimension	Metric Geometry	Measure how large an exactly pairwise-equidistant subset a metric space can support, with the distance scale and attainment convention stated explicitly.
Equisatisfiability	Mathematical LogicSatisfiability Preserving Transformations	The logical relation in which two formulas share the same existence-of-a-model status—both satisfiable or both unsatisfiable—without requiring the same models, enabling compact satisfiability-preserving transformations such as Tseitin encoding and Skolemization.
Equitable coloring	Graph Theory	A proper vertex coloring whose color classes differ in size by at most one, combining adjacency separation with balanced allocation.
Equivalence (measure theory)	Measure TheoryAbsolute Continuity and Measure Classes	Treat two measures on one measurable space as equivalent exactly when each is absolutely continuous with respect to the other, so they have the same null sets.
Equivalence of metrics	Metric Geometry	A relation between metrics that captures equality of induced topology, uniformity, Lipschitz structure, or another declared level of geometric behavior.
Equivalent Radius	Applied GeometryShape Measurement	The radius of a reference circle or sphere that matches a noncircular or nonspherical object on one explicitly selected measure.
Equivariant bundle	Equivariant TopologyFibre Bundles	For example, an equivariant vector bundle is an equivariant bundle such that the action of G restricts to a linear isomorphism between fibres.
Equivariant cohomology	Equivariant TopologyAlgebraic Topology	Cohomology of a group's homotopy quotient, recording topology together with an action on a space.
Equivariant differential form	Equivariant Differential Geometry	A group-equivariant polynomial map from a Lie algebra to differential forms on a manifold, representing a cochain in the Cartan model of equivariant cohomology.
Erdős–Ko–Rado Theorem	Extremal CombinatoricsIntersection Theorems	For n at least 2k, a pairwise-intersecting family of k-subsets of an n-element set has at most C(n-1,k-1) members, attained by every full star.
Erdős–Woods number	Elementary and Computational Number Theory	A positive integer k for which some interval of k+1 consecutive integers has every interior member sharing a nontrivial common divisor with at least one endpoint.
Ergodic process	Stochastic ProcessesErgodic Theory and Time Series	A stochastic process for which specified long-run time averages along almost every realization equal the corresponding ensemble expectations, allowing one sufficiently long trajectory to represent the regime.
Ergodicity	Dynamical SystemsErgodic Theory	A measure-preserving dynamical property in which every invariant measurable set has measure zero or full measure, making the system statistically indecomposable.
Error analysis (mathematics)	Numerical Analysis	The study of how approximation, rounding, truncation, measurement, discretization, and conditioning create and propagate uncertainty in mathematical results.
Esscher transform	Probability TheoryExponential Change of Probability Measure	Exponentially tilt a probability law by a parameter and normalize by its moment-generating function, producing a new law that shifts event weights while preserving absolute-continuity structure on the finite domain.
Essential Extension	Module TheoryHomological Algebra	An inclusion of modules in which the included submodule meets every nonzero submodule of the ambient module.
Essential manifold	Geometric TopologySystolic Geometry	In geometry, an essential manifold is a special type of closed manifold.
Essentially surjective functor	Category Theory	A functor whose image contains an object isomorphic to every object in its codomain.
Etemadi's Inequality	Probability TheoryMaximal Inequalities	A maximal inequality bounding excursions of independent partial sums by the worst partial-sum tail probability at a smaller threshold.
Eternal dominating set	Graph Theoretic Security Games	A set of graph vertices occupied by mobile guards that can respond to every infinite sequence of vertex attacks while remaining a dominating set after each move.
Euclid number	Number Theory	An integer one greater than the product of the first n primes, with a second-kind variant one less than that primorial.
Euclidean domain	Ring Theory	An integral domain equipped with a Euclidean function that supports division with remainder of strictly smaller value and therefore the Euclidean algorithm.
Euclidean Neighborhood Retract	TopologyGeometric Topology	A topological space homeomorphic to a subset X of some Euclidean space for which an open neighborhood U of X admits a continuous retraction r:U→X fixing every point of X.
Euclidean ordered field	Ordered Algebra	An ordered field in which every nonnegative element has a square root within the field.
Euclidean random matrix	Random Matrix Theory	A random matrix whose entries are deterministic functions of randomly placed points in Euclidean space, coupling matrix statistics to spatial geometry.
Euclidean Rotation	Euclidean GeometryLinear Algebra	A proper Euclidean point map of an oriented plane or three-space that preserves distance and orientation while fixing a center or axis, with identity at angles congruent to zero modulo 2π.
Euclidean Space	Euclidean Geometry	Combine finite-dimensional real affine structure with a positive-definite inner product so displacement, distance, angle, orthogonality, projection, and rigid motion form one coherent flat geometry.
Euclidean Vector	Euclidean GeometryLinear Algebra	A free Euclidean vector is a location-independent class of directed segments with linear operations and Euclidean length and angle.
Euclid–Mullin sequence	Number Theory	A recursively defined prime sequence taking each new term as the least prime factor of one plus the product of all preceding terms.
Euler Characteristic	Algebraic Topology	A homotopy-invariant integer obtained by alternating the cell counts—or homology ranks—of a finite cell complex.
Euler diagram	Diagrammatic Logic	A set diagram representing only the inclusion, overlap and disjointness relations that actually hold among the depicted classes.
Euler Line	Triangle Geometry	The line of a non-equilateral triangle on which its circumcenter, centroid, orthocenter, and nine-point center lie in fixed affine ratios.
Euler Method	Numerical AnalysisInitial Value Methods	An explicit first-order ODE integrator that advances an initial state by adding the step size times the derivative evaluated at the current numerical state.
Euler sequence	Projective Algebraic Geometry	A canonical exact sequence of sheaves on projective space relating relative differentials to twists of the structure sheaf.
Euler's elliptic differential equation	Elliptic IntegralsDifferential Equations	A branch-qualified relation between two elliptic differentials sharing a nonsingular quartic, whose local integration yields an algebraic addition relation.
Euler's Formula	Complex Analysis	Euler's formula equates the complex exponential at a real imaginary angle with cosine and sine coordinates: exp(ix) = cos x + i sin x.
Euler's Four-Square Identity	Algebra and Number TheoryComposition of Quadratic Forms	A bilinear four-component composition law whose output represents the product of two sums of four squares as another sum of four squares.
Euler's totient function	Number Theory	The arithmetic function counting residue classes modulo a positive integer that are coprime to it.
Eulerian number	Enumerative Combinatorics	A permutation-counting number classified by an exact number of ascents or, under shifted conventions, descents.
Even-hole-free graph	Graph TheoryStructural Graph Theory	Graph that contains no induced cycle of even length ≥ 6.
Event (probability theory)	Probability Theory	A measurable subset of a sample space representing the collection of outcomes for which a probabilistic proposition holds.
Exact coloring	Graph Coloring	A proper vertex coloring in which each unordered pair of distinct colors occurs on exactly one edge.
Exact sequence	Homological Algebra	A sequence of morphisms in which the image of every map is exactly the kernel of the next, encoding that each stage contains no unexplained residue between arrival and annihilation.
Exceptional Inverse Image Functor	Sheaf TheoryAlgebraic Geometry	Exceptional Inverse Image Functor is a recurring identity in formal models and representations, mathematics, logic, and statistics defined by this frozen evidence: In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves.
Exchange matrix	Linear Algebra	The permutation matrix with ones on the antidiagonal that reverses coordinate, row, or column order.
Excisive triad	Algebraic TopologyExcision Conditions	A topological triad (X;A,B) in which X is covered by the interiors of subspaces A and B, supplying the cover condition used by excision and Mayer–Vietoris arguments.
Excursion probability	Stochastic Processes	The probability that a stochastic process crosses a specified threshold somewhere over a declared index region.
Exhaustion by compact sets	TopologyCompact Exhaustions	A nested sequence of compact subsets whose interiors successively contain earlier terms and whose union covers the whole topological space.
Existential Instantiation	Mathematical LogicNatural Deduction	In predicate logic, existential instantiation (also called existential elimination) is a rule of inference which says that, given a formula of the form (\exists x) \phi(x) , one may infer \phi(c) for a new constant symbol c.
Existential Quantification	Mathematical Logic	Bind a variable to assert that at least one object in a logical domain satisfies a specified formula.
Exotic R4	Smooth Four Manifold Topology	A smooth four-manifold homeomorphic but not diffeomorphic to ordinary Euclidean four-space, exposing the unique dimension-four split between topological and smooth equivalence.
Expander graph	Graph Theory	A sparse finite graph in which every not-too-large vertex set has a proportionally large boundary, equivalently exhibiting strong combinatorial or spectral expansion.
Expansive Homeomorphism	Topological Dynamical Systems	A homeomorphism of a compact metric space for which every distinct point pair is separated by at least one forward or backward iterate beyond one fixed positive expansivity scale.
Experiment (Probability Theory)	Probability Theory	A mathematical model of a repeatable trial whose possible outcomes form a sample space, measurable events form a sigma-algebra, and probabilities are assigned by a measure.
Exponential Integrator	—	A time-integration method that propagates a selected linear part through its exponential and approximates the remaining variation-of-constants contribution.
Exponential map (Lie theory)	Lie Theory	The canonical smooth map sending a Lie-algebra element to the time-one point of its one-parameter subgroup in the Lie group.
Exponential polynomial	Exponential PolynomialsAnalysis	In mathematics, exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function.
Exponential Sheaf Sequence	Complex GeometrySheaf Cohomology	In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry.
Exponential Stability	Dynamical SystemsStability Theory	An equilibrium's trajectory deviations obey a common exponentially decaying bound relative to their initial size over a specified domain.
Exponentially Modified Gaussian Distribution	Probability TheoryStatistical Distributions	The distribution of an independent Gaussian value plus a positive exponential value, yielding a precise right-skewed convolution family.
Exportation (logic)	Propositional Logic	Convert an implication from a conjunction, (P ∧ Q) → R, into nested implications, P → (Q → R), under formal conjunction and implication rules.
Extension (simplicial set)	Algebraic TopologySimplicial Homotopy Theory	The Ex endofunctor on simplicial sets, right adjoint to subdivision, that replaces a simplicial set by maps from subdivided simplices and iteratively improves horn-filling behavior.
Extensive category	Category Theory	A category with finite coproducts that are disjoint and stable enough that objects over a coproduct decompose equivalently into objects over its summands.
Exterior Algebra	Multilinear Algebra	Turn alternating multilinear combinations of a module into ordinary linear algebra by quotienting its tensor algebra so every repeated degree-one factor vanishes, producing a graded wedge product with a universal mapping property.
Exterior derivative	Differential GeometryDifferential Forms	The metric-independent differential operator that sends each differential k-form to a (k+1)-form, obeys a graded product rule, and squares to zero.
External Ray	Complex Dynamics	A constant-angle curve in an exterior conformal coordinate that approaches a Julia-set or connectedness-locus boundary from infinity and may land at a boundary point.
Extraneous and Missing Solutions	AlgebraEquation Solving	Opposite solution-set errors in equation solving: a non-equivalent step admits candidates absent from the original equation or discards genuine solutions.
Extremally disconnected space	General Topology	A topological space in which the closure of every open set is open.
E∞-operad	Higher Algebra	An operad whose spaces of n-ary operations are contractible with suitably free symmetric-group actions, encoding multiplication associative and commutative up to coherent higher homotopies.
F-crystal	Arithmetic Geometry	A finite free Witt-vector module equipped with an injective Frobenius-semilinear endomorphism, encoding crystalline-cohomological Frobenius structure.
F-space	Functional Analysis	A real or complex vector space equipped with a complete translation-invariant metric whose addition and scalar multiplication are continuous.
Faber polynomials	Complex Analysis	Polynomials canonically associated with a normalized Laurent series or conformal map, defined by canceling the principal part of its powers.
Factor Complexity Function	Combinatorics on WordsFactor Complexity	The length-indexed count of distinct contiguous factors in a finite or infinite word—equivalently the density function of its factor language—whose growth separates periodicity, low-complexity aperiodicity, and entropy-bearing pattern diversity.
Factor-critical graph	Graph TheoryMatching Theory	An odd-order graph in which deleting any single vertex leaves a graph with a perfect matching, equivalently every vertex can be the unmatched vertex of a near-perfect matching.
Factorial moment	Probability TheoryMoments of Discrete Distributions	Take the expectation of a random variable’s falling factorial to expose counting structure and probability-generating-function derivatives.
Factorial Number System	Combinatorial Enumeration	A mixed-radix positional representation with factorial place values and digit bound 0 through i at the i-factorial place, giving every nonnegative integer a unique expansion and a natural rank/unrank bridge to permutations.
Factorization of polynomials	Computational and Commutative Algebra	The decomposition of a polynomial over a specified coefficient domain into a unit and irreducible polynomial factors, unique only under appropriate factorization properties and normalization.
Factorization system	Category Theory	A pair of morphism classes in a category through which every morphism factors, with a unique lifting property characterizing the two classes against one another.
False position method	Numerical AnalysisRoot Finding	In mathematics, the regula falsi, method of false position, or false position method is a family of algorithms used to solve linear equations and smooth nonlinear equations for a single unknown value.
Fano factor	Stochastic Processes	A count-dispersion ratio equal to variance divided by mean, with one marking a Poisson baseline.
Farey sequence	Elementary Number Theory	The increasing sequence of reduced rational numbers between zero and one whose denominators do not exceed a chosen order.
Farsightedness (game theory)	Game Theory	A strategic solution assumption under which players evaluate deviations by anticipating subsequent moves and terminal outcomes rather than only immediate payoffs.
Fat object (geometry)	Computational GeometryShape Regularity	A geometric object whose extent is comparable in every direction under a declared fatness criterion, excluding arbitrarily thin or needle-like shapes and enabling stronger algorithmic bounds.
Faulhaber's formula	Enumerative Number Theory	A Bernoulli-number polynomial formula for the sum of equal nonnegative integer powers over the first n positive integers.
Feasible Region	Optimization	Turn an optimization problem's list of constraints into a single geometric set — the intersection of everything they permit — whose shape (empty, convex, bounded, its boundary) governs whether a solution exists, where it sits, and which methods will find it.
Fermat number	Number TheorySpecial Integer Sequences	Generate the integer sequence F_n = 2^(2^n) + 1, whose product recurrence makes distinct terms pairwise coprime and whose rare prime members connect to constructible polygons.
Fermat's Little Theorem	Elementary Number TheoryModular Arithmetic	For prime p, every integer a satisfies a^p ≡ a modulo p; nonzero residues satisfy a^(p−1) ≡ 1.
FETI-DP	Numerical AnalysisIterative Substructuring Methods	A dual–primal nonoverlapping domain-decomposition solver that assembles selected interface degrees of freedom globally while enforcing continuity of the remaining duplicated interface variables with Lagrange multipliers.
Fiber Bundle	Geometry and TopologyDifferential Topology	A fiber bundle is a mathematical structure consisting of a total space mapped onto a base space so that each base point has an associated fiber and the structure is locally equivalent to a product of an open base neighborhood with a typical fiber through compatible trivializations.
Fiber Product of Schemes	Algebraic Geometry	The universal scheme of pairs of maps compatible over a common base, realized affinely by a tensor product and serving as the engine of base change, scheme-theoretic fibers, and intersections.
Fibrations of graphs	Graph TheoryGraph Morphisms	Directed-graph homomorphisms with a unique lifting property for incoming edges, making each base edge lift uniquely to every vertex over its target.
Fibred Category	Category Theory	Organize categories of objects varying over a base category so every base morphism admits a universal cartesian pullback, making reindexing coherent up to canonical isomorphism.
Fictitious domain method	Numerical Analysis and Partial Differential Equations	A numerical PDE method that embeds an irregular physical domain in a simpler computational domain and enforces the original boundary or interface conditions without a boundary-conforming background mesh.
Field (Algebraic)	Abstract Algebra	Guarantee that you can always add, subtract, multiply, and divide by anything non-zero by demanding one axiom package — two commutative-group operations bound by distributivity — which certifies the whole apparatus of linear algebra in a single membership check.
Field Extension	Field TheoryAbstract Algebra	A field extension is a pair of fields L/K together with an embedding identifying K as a subfield of L, so that L becomes a vector space and algebra over K and can be studied by degree, generators, algebraicity, separability, normality, and automorphisms.
Field of fractions	Commutative Algebra	The smallest field containing an integral domain, constructed from equivalence classes of ratios of its elements with nonzero denominators.
Filling radius	Riemannian GeometrySystolic Geometry	In Riemannian geometry, the filling radius of a Riemannian manifold X is a metric invariant of X.
Filtered algebra	AlgebraFiltered Structures	An algebra equipped with a nested exhaustive sequence of subspaces whose multiplication sends filtration levels p and q into level p+q.
Filtered category	Category Theory	A nonempty category in which every finite diagram admits a compatible cocone, equivalently objects and parallel arrows can be jointly advanced and equalized.
Filtering problem (stochastic processes)	Stochastic Processes	The sequential inference problem of estimating a hidden stochastic state from noisy partial observations available up to the present time.
Filters in topology	General Topology	A set-family formalism that characterizes convergence, continuity, closure, compactness, and limits in arbitrary topological spaces without relying on sequences.
Filtration (algebra)	Abstract AlgebraHomological Algebra	An ordered family of subobjects of an algebraic structure whose stages are nested with the index order, often with operations respecting degree—for example `F_i F_j ⊆ F_{i+j}`—so complexity or information accumulates by level.
Filtration (Probability Theory)	Probability TheoryStochastic Processes	An ordered family of nested event σ-algebras on one probability space that represents the information available at successive indices.
Fine topology (potential theory)	Potential Theory	The coarsest topology on a potential-theoretic domain that makes every subharmonic function—equivalently every superharmonic function—continuous, refining Euclidean topology where ordinary continuity is too coarse.
Finite algebra	Commutative Algebra	An R-algebra that is finitely generated as an R-module, a stronger finiteness condition than finite generation as an algebra.
Finite difference	Numerical Analysis	The difference between function values at finitely separated arguments, used as a discrete operator and as an approximation to derivatives.
Finite Difference Coefficient	Numerical Analysis	A stencil weight determined by derivative order, evaluation point, sample-node offsets, and polynomial exactness so their weighted samples approximate the target derivative with a declared truncation order.
Finite Difference Method	—	A numerical method samples a differential equation on a discrete grid, replaces derivatives with finite-difference stencils, closes the resulting algebraic system, and analyzes truncation error, stability, and convergence under refinement.
Finite Element Method	—	A numerical method poses a boundary-value problem in weak form, chooses piecewise finite-dimensional trial and test spaces over a mesh, assembles local element contributions, and solves the resulting global algebraic system with controlled approximation error.
Finite extensions of local fields	Algebraic Number TheoryLocal Fields	The above implies that there is an equivalence of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K.
Finite morphism	Algebraic GeometryScheme Morphisms	A morphism of schemes that is affine and whose induced coordinate-ring algebra is finite as a module, generalizing maps with algebraically finite fibers.
Finite set	Set Theory and Combinatorics	A set equipotent with the natural numbers below some n, equivalently one whose elements can be completely counted and assigned a natural-number cardinality.
Finite subdivision rule	Combinatorial Topology and Conformal Dynamics	A finite recursive prescription replacing each tile type by a patterned subdivision to generate successively finer cell structures.
Finite-Valued Logic	—	A logic is characterized by a finite logical matrix: finitely many semantic values, designated values defining consequence, and truth functions interpreting its connectives.
Finiteness Properties of Groups	Group CohomologyGeometric Group Theory	A hierarchy measuring whether an often infinite group admits finite low-dimensional classifying-space skeleta or finite projective-resolution data, including F_n, FP_n, F∞, and F.
FinSet	Category Theory	The category whose objects are finite sets and whose morphisms are all functions between them, equivalent to the small skeleton FinOrd of finite ordinals and supporting finite products, coproducts, and exponentials.
Firing-Squad Synchronization Problem	Cellular AutomataFinite State Synchronization Protocols	Find one finite local rule that, from a single endpoint trigger, makes every cell in an arbitrarily long finite cellular-automaton line enter a firing state simultaneously for the first time.
First Fundamental Form	Differential Geometry	First Fundamental Form is a recurring differential geometry identity in which the ambient inner product induces a quadratic metric on a surface tangent space for lengths, angles, areas, and curvature calculations.
First-countable space	General Topology	A topological space in which every point has a countable neighborhood basis.
First-Order Arithmetic	Mathematical LogicTheories of Arithmetic	Formal first-order theories of natural-number arithmetic defined by a signature, axioms, models, and a specified induction scheme.
Five lemma	Homological AlgebraExact Sequence Diagram Lemmas	Infer that the middle vertical morphism in a commutative five-object diagram with exact rows is an isomorphism when the two neighboring outer maps meet the required isomorphism, epimorphism, and monomorphism conditions.
Five-term exact sequence	Homological Algebra	The low-degree exact sequence extracted from a first-quadrant spectral sequence, linking edge terms, an early differential and the first two groups of the abutment.
Fixed points of isometry groups in Euclidean space	Euclidean Geometry	The common-fixed-point set of a Euclidean isometry group, which is either empty or an affine subspace.
Flag algebra	Extremal CombinatoricsGraph Limits and Densities	Razborov's algebraic framework for asymptotic densities of partially labeled finite structures, turning extremal combinatorics inequalities into positive semidefinite and semidefinite-programming certificates.
Flat convergence	Geometric Measure Theory	Convergence of geometric chains or currents in the flat norm, permitting their difference to be decomposed into a small-mass current plus the boundary of another small-mass current.
Flat Vector Bundle	—	A vector bundle with a zero-curvature linear connection has homotopy-invariant parallel transport, locally constant transition data, and a monodromy representation.
Flattening	Geometry	A dimensionless axial-compression measure for an ellipse or spheroid, ordinarily the semiaxis difference divided by the semimajor axis, with explicitly convertible alternative normalizations.
Flexible Algebra	Nonassociative Algebra	Retain the reassociation identity `(xy)x = x(yx)` when full associativity is absent, forcing the associator to vanish whenever its first and third arguments coincide.
Floquet Theory	Ordinary Differential EquationsDynamical Systems	A theory for linear differential systems with periodic coefficients that decomposes evolution into periodic motion and exponential growth or decay governed by monodromy multipliers.
Flow (mathematics)	Dynamical SystemsOrdinary Differential Equations	The notion of flow is basic to the study of ordinary differential equations.
Fluid Limit	Applied ProbabilityQueueing Theory	In queueing theory, a discipline within the mathematical theory of probability, a fluid limit, fluid approximation or fluid analysis of a stochastic model is a deterministic real-valued process which approximates the evolution of a given stochastic process, usually subject to some scaling or limiting criteria.
Flux Limiter	—	A local nonlinear control function limits high-order numerical flux corrections near nonsmooth data while retaining more accurate transport in smooth regions.
Foliation	Differential GeometryFoliation Theory	A decomposition of a manifold into connected immersed submanifolds of equal dimension that locally look like parallel coordinate slices.
Formal ball	Domain Theory	An ordered pair of a metric-space point and a nonnegative radius, ordered so one pair computationally approximates another; generalized formal balls allow negative radii.
Formal derivative	Polynomial and Formal Power Series Algebra	Differentiate polynomials or formal power series algebraically by multiplying each coefficient by its exponent and lowering that exponent, without invoking limits or convergence.
Formal Model	Mathematical ModelingFormal ModelingSystems Modeling	An explicit symbolic representation of a target whose entities, states, relations, parameters, and transformations are governed by mathematical or logical rules so consequences can be derived, simulated, or checked.
Formal power series	Commutative Algebra	An infinite coefficient sequence manipulated as an algebraic series in an indeterminate, without any requirement that numerical substitution converge.
Formal scheme	Algebraic Geometry	A locally ringed space locally modeled by the formal spectrum of an adic topological ring, retaining infinitesimal neighborhoods through completion.
Formal theorem	Mathematical Logic	In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven.
Formal Theory	Mathematical LogicFormal Theories and Model Theory	A set of sentences in a formal language, commonly closed under a specified consequence relation, that serves as the asserted or derivable content interpreted within models.
Formally real field	AlgebraReal Algebra	A field that admits an ordering compatible with its operations, equivalently one in which minus one cannot be expressed as a finite sum of squares.
Formally smooth map	Algebraic Geometry	A ring map with the infinitesimal lifting property against nilpotent quotient extensions.
Fortunate number	Number TheoryPrime Gaps Near Primorials	For each positive index n, select the least integer m greater than one for which the nth primorial plus m is prime, producing the sequence governed by Fortune's still-open primality conjecture.
Fourier algebra	Harmonic Analysis	The commutative Banach algebra of coefficient functions of the left regular representation of a locally compact group, under pointwise multiplication.
Fourier analysis	Harmonic Analysis	The representation and study of functions or signals through sinusoidal or character components indexed by frequency.
Fourier Sine Transform	Integral TransformsHarmonic Analysis	In mathematics, the Fourier sine and cosine transforms are integral equations that decompose arbitrary functions into a sum of sine waves representing the odd component of the function plus cosine waves representing the even component of the function.
Fourier Transform	Harmonic Analysis	Decompose a function into a weighted superposition of complex exponentials, recording each frequency's amplitude and phase — an invertible, energy-preserving change of basis that diagonalizes every translation-invariant operation, so convolution becomes pointwise multiplication.
Fourier transform on finite groups	Harmonic AnalysisFinite Group Fourier Analysis	A harmonic transform mapping a function on a finite group to matrices indexed by irreducible representations, generalizing the scalar discrete Fourier transform beyond abelian groups.
Fourier–Bros–Iagolnitzer Transform	Analytic Microlocal Analysis	A Gaussian-windowed complex Fourier integral transform that embeds a function or distribution into phase space and detects local analytic regularity through exponential decay, providing a characterization of the analytic wave-front set.
Fourier–Motzkin Elimination	Linear Inequalities and Polyhedral Projection	Project a finite system of linear inequalities onto fewer variables by separating one variable's lower and upper bounds and emitting every cross-bound consistency inequality.
Fox–Wright function	Special Functions	A generalized hypergeometric-type function whose series allows affine step sizes in gamma-function parameters.
Fractal analysis	Data AnalysisScale Invariant Methods	A family of methods that estimates scale-dependent self-similarity, dimension, lacunarity or multifractal structure from geometric, temporal or spatial data while testing finite-range and sampling limitations.
Fractional Brownian Motion	Probability TheoryStochastic Processes	Model a Gaussian path with stationary increments whose scale and dependence are governed by a fixed Hurst parameter.
Fractional calculus	Mathematical Analysis	Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D.
Fractional Coloring	Graph TheoryGraph Coloring	Conflict-free assignment of multiple colors per graph vertex, optimized by palette size per assigned color.
Fractional-Order System	Dynamical SystemsFractional Differential Equations	A dynamical model whose governing relation uses a specified noninteger-order temporal operation.
Fragment (logic)	Mathematical LogicLogical Languages	A syntactically restricted sublanguage of a logic interpreted with the parent logic's semantics, often trading expressive power for decidability or lower computational complexity.
Frattini Subgroup	Group TheoryFinite Group Theory	The intersection of all maximal proper subgroups of a group, detecting generation redundancy in finite groups.
Fredholm Kernel	Fredholm and Nuclear Operator Theory	An element of the completed projective tensor product of a Banach-space dual with a Banach space, represented by an absolutely summable series of elementary tensors and canonically inducing a nuclear operator.
Free category	Category Theory	The category generated by a directed graph whose morphisms are finite composable paths and whose only equations are category axioms.
Free exact completion	Category Theory	Then the category of presheaves Set C op is equivalent to the exact completion of the coproduct completion of C.
Free Group	Group Theory	The group of reduced words on a basis and formal inverses, with no relations beyond group axioms and a universal extension property for maps from that basis.
Free monoid	Abstract AlgebraUniversal Algebra	The monoid of all finite words over an alphabet under concatenation with the empty word as identity, characterized by unique extension of every generator map to a monoid homomorphism.
Free presentation	Module Theory	An exact sequence of free modules mapping generators and relations onto a module.
Freiling's Axiom of Symmetry	Set TheoryAxioms of the Continuum	Every assignment of a countable forbidden set to each real admits two reals that avoid one another's assigned sets—a set-theoretic principle equivalent over ZFC to the negation of the continuum hypothesis.
Frenet–Serret formulas	Differential Geometry	The Frenet–Serret formulas relate the derivatives of a curve's tangent, normal, and binormal frame through curvature and torsion.
Friendly number	Number Theory	A positive integer sharing its abundancy index—the sum of divisors divided by the integer—with at least one distinct positive integer.
Friendly-index set	Graph Labeling	Collect every edge-label imbalance attainable from nearly balanced binary vertex labelings of a graph, yielding a set-valued graph invariant rather than one selected labeling.
Frink ideal	Order TheoryPoset Ideals	A subset I of a partially ordered set such that every common lower bound of the common upper bounds of each finite subset of I also belongs to I.
Frobenioid	Arithmetic Geometry	A category equipped with degree, divisor-like, and Frobenius structure that categorifies monoid actions arising from arithmetic line bundles.
Frobenius endomorphism	Commutative Algebra	The natural ring endomorphism x↦xᵖ on a commutative ring of prime characteristic p, becoming an automorphism precisely in important perfect cases.
Frobenius Formula	Representation Theory	Recover irreducible character values of a symmetric group from partitions and conjugacy-cycle data by extracting a specified monomial coefficient from a product of a Vandermonde factor and power sums.
Frobenius manifold	Differential GeometryFrobenius Geometry	A manifold whose tangent spaces carry smoothly varying commutative Frobenius-algebra products compatible with a flat metric and integrability conditions.
Frobenius normal form	Linear AlgebraMatrix Similarity and Module Structure	Replace a square matrix over a field by the unique block diagonal matrix of companion matrices determined by its divisibility-ordered invariant factors, thereby deciding similarity without splitting the characteristic polynomial.
Frobenius–Schur indicator	Representation Theory	An invariant distinguishing whether an irreducible complex representation is real, complex, or quaternionic in type.
Frontal Solver	Numerical Linear Algebra	Factor a sparse assembled system by advancing a dense active front, assembling each local contribution when it enters and eliminating a variable as soon as its last contribution has arrived.
Fréchet distance	Metric Geometry	In mathematics, the Fréchet distance is a measure of similarity between curves that takes into account the location and ordering of the points along the curves.
Fréchet manifold	Infinite Dimensional GeometryLocally Convex Manifolds	Build an infinite-dimensional smooth manifold from charts valued in Fréchet spaces, while making the chosen smooth calculus and the failure of general Banach inverse-function machinery explicit.
Fréchet Mean	Metric Geometry	Any point minimizing expected or empirical squared metric distance to observations, generalizing the Euclidean arithmetic mean to nonlinear metric spaces.
Fréchet space	Functional AnalysisTopological Vector Spaces	A complete metrizable locally convex topological vector space, often described by a countable separating family of seminorms and broad enough to include many function spaces that have no single adequate norm.
Function series	Mathematical AnalysisSeries of Functions	An infinite series whose terms are functions, with its sum and inherited properties determined by a specified mode and domain of convergence.
Functional Calculus	Functional AnalysisSpectral Theory	A spectrum-controlled homomorphism that extends scalar functions f to operator expressions f(T) while preserving the algebraic relations needed to reason about T through f.
Functional completeness	Mathematical Logic	The property of a set of Boolean connectives from which every Boolean function can be expressed by composition.
Functional determinant	Functional AnalysisSpectral Theory	The corresponding quantity det(S) is called the functional determinant of S.
Functional Integration	Infinite Dimensional AnalysisFunctional Analysis	Integration over spaces of functions, paths, or fields, defined through an infinite-dimensional measure or controlled limiting construction—or used as an explicitly formal regulated calculus.
Functor	Category Theory	A structure-preserving map between categories that carries objects and morphisms while respecting identities and composition — the two axioms that make it a licence to transport theorems from one category to another along its rails.
Functor Category	Category TheoryCategorical Constructions	For fixed categories C and D, the category whose objects are functors C→D and whose morphisms are natural transformations, with identities and composition defined componentwise.
Fundamental domain	Geometry and Group Actions	A representative region for a group action containing one representative from each orbit, whose translates reconstruct the acted-on space up to boundary identifications.
Fundamental Groupoid	Algebraic Topology	The groupoid whose objects are points of a space and whose arrows are fixed-endpoint homotopy classes of paths, retaining path components and all basepoint fundamental groups in one functorial invariant.
Fundamental theorem of algebra	Complex Analysis	Every nonconstant one-variable polynomial with complex coefficients has a complex root and therefore factors completely into linear terms.
Fundamental Theorem of Arithmetic	Number TheoryElementary Number Theory	The integer-specific theorem that every integer greater than one is a product of primes and that its prime multiset is unique up to order.
Fundamental theorem of Hilbert spaces	Functional AnalysisHilbert Space Duality	Characterize completeness of a Hausdorff pre-Hilbert space by surjectivity of its canonical inner-product isometry into the continuous anti-dual.
Fundamental unit (number theory)	Algebraic Number Theory	A generator, modulo roots of unity, of the rank-one unit group of a number field's ring of integers.
Fusion Category	Tensor Categories	Model finitely many particle-like object types with semisimple direct-sum decomposition, duals, and an associative tensor product whose decomposition coefficients form finite fusion rules.
Fuzzy measure theory	Measure TheoryFuzzy Mathematics	In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity.
Fuzzy number	Fuzzy Mathematics and Uncertain Quantities	A normalized convex fuzzy subset of the real line, usually with upper-semicontinuous membership and compact support, representing graded compatibility with numerical values.
G-Matrix	Linear AlgebraMatrix Theory	In linear algebra, a real invertible matrix A is called a G-matrix if A^{-T}=D_1AD_2 (where A^{-T} means (A^{-1})^T ) for some real diagonal matrices D_1 and D_2 .
Gabor atom	Harmonic Analysis	A time–frequency atom formed by translating, modulating and sometimes scaling a localized window function.
Gabriel–Rosenberg Reconstruction Theorem	Algebraic GeometryCategorical Algebraic Geometry	A categorical reconstruction theorem recovering a suitably separated scheme, including its topology and structure sheaf, from the abelian category of its quasi-coherent sheaves.
Gallai–Hasse–Roy–Vitaver theorem	Graph TheoryColoring and Orientation	A graph-theoretic duality stating that a graph's chromatic number equals one plus the minimum, over all edge orientations, of the longest directed-path length.
Galois Theory	Field Theory	Relate a field extension to its automorphism group so subgroup structure encodes intermediate fields and polynomial solvability becomes a symmetry question.
Gamas's theorem	Multilinear Algebra	A criterion stating that a tensor projected by an irreducible symmetric-group representation is nonzero exactly when its vectors can be partitioned into linearly independent subsets of sizes given by the partition’s columns.
Game form	Game Theory	A specification of players’ action sets and an outcome function, leaving each player’s preferences or utilities over outcomes unspecified.
Gamma Distribution	Probability TheoryStatistical Distributions	A positive continuous distribution family whose shape and scale govern skew and magnitude, with conditional waiting-time, sum, and rate-prior interpretations.
Gamma Function	Special Functions	The unique positive log-convex extension of shifted factorial on the positive reals, represented by Euler's integral and continued meromorphically to the complex plane with simple poles at the nonpositive integers and no zeros.
Gamma process	Stochastic Processes	A nondecreasing Levy process with independent stationary increments distributed according to a gamma law, used to model cumulative random growth, wear, or activity.
Gauss's lemma (number theory)	Elementary Number Theory	A criterion computing the Legendre symbol by counting how many least positive residues of a,2a,…,((p−1)/2)a modulo an odd prime exceed p/2.
Gaussian Free Field	Random Fields	The Gaussian free field is a Gaussian random field whose covariance is the Green operator of a specified Laplacian.
Gaussian probability space	Probability Theory	A probability space equipped with a closed Hilbert subspace of centered real Gaussian random variables, optionally separated from a transverse sigma-algebra.
Gauss–Jacobi Quadrature	Numerical AnalysisGaussian Quadrature and Orthogonal Polynomials	An n-node Gaussian rule for the Jacobi weight (1-x)^alpha(1+x)^beta on [-1,1], using roots of the degree-n Jacobi polynomial and integrating weighted polynomials through degree 2n-1 exactly.
Gauss–Lucas theorem	Complex AnalysisPolynomial Zeros	The roots of the derivative of a nonconstant complex polynomial lie in the convex hull of the polynomial's roots.
Gauss–Newton Algorithm	Numerical OptimizationNonlinear Least Squares	An iterative nonlinear least-squares algorithm that linearizes the residual vector and solves the resulting Jacobian least-squares subproblem for each parameter update.
Gauss–Seidel Method	Numerical Linear AlgebraStationary Iterative Methods	A stationary linear-system iteration that sweeps coordinates in order and immediately reuses each newly computed component within the same sweep.
Gegenbauer polynomials	Special Functions	An orthogonal-polynomial family on [−1,1] with weight (1−x²)^(alpha−1/2), generalizing Legendre and Chebyshev polynomials.
Gelfand representation	Functional AnalysisBanach Algebras	The homomorphism sending each element of a commutative Banach algebra to its evaluation function on the character space, becoming an isometric *-isomorphism for commutative C*-algebras.
Gelfand ring	Ring Theory	A ring satisfying separation conditions on distinct maximal ideals that generalize topological features of Gelfand duality.
Gelfand–Fuks cohomology	Differential Topology	Continuous Lie-algebra cohomology for topological Lie algebras of smooth vector fields.
Gelfand–Kirillov dimension	Noncommutative Algebra	An invariant measuring the polynomial growth rate of an algebra or module generated by finite-dimensional subspaces.
Gelfand–Naimark–Segal construction	C Algebras and Operator Theory	A construction sending a state on a C-star algebra to a cyclic star-representation on a Hilbert space, establishing the converse correspondence as well.
Gelfand–Shilov space	Functional Analysis	A space of smooth test functions whose derivatives and polynomially weighted values obey factorial growth bounds controlling simultaneous decay and regularity.
General Dirichlet Series	Complex AnalysisGeneral Dirichlet Series	A general Dirichlet series weights complex coefficients by exponential terms indexed by a freely chosen increasing frequency sequence.
Generalized chi-squared distribution	ProbabilityQuadratic Form Distributions	The probability distribution of a quadratic function of a multivariate normal vector, equivalently a weighted sum of independent noncentral chi-square variables with an optional normal term.
Generalized Dihedral Group	Group TheorySemidirect Products	A generalized dihedral group adjoins an involution to an abelian group so the new element acts on it by inversion.
Generalized inverse	AlgebraLinear AlgebraMatrix Theory	Generalized inverse denotes matrix satisfying some of the criteria of an inverse within algebra.
Generalized inverse Gaussian distribution	Probability and StatisticsContinuous Probability Distributions	A three-parameter positive continuous distribution with density proportional to x^{p−1}exp[−(ax+b/x)/2], normalized by a modified Bessel K function and containing inverse Gaussian and gamma-related limits.
Generalized Polynomial	AlgebraGeneralized PolynomialsCombinatorics	A generalized polynomial is a function or finite formal expression obtained by relaxing a specified ordinary-polynomial constraint—such as exponent set, coefficient periodicity, domain, basis, or piecewise dependence—while preserving an explicit algebraic rule that recovers ordinary polynomials as a special case.
Generalized Stokes theorem	Differential GeometryDifferential Forms	The orientation-sensitive equality ∫Ωdω=∫∂Ωω for an admissible differential form on a suitably regular manifold with boundary.
Generalized taxicab number	Number Theory	The least integer expressible as a sum of a fixed number of positive k-th powers in a specified number of distinct ways.
Generalized trigonometry	TrigonometryNon Euclidean Geometry	A family of extensions that adapts trigonometric functions and triangle relations beyond Euclidean plane geometry to other spaces, dimensions, algebras, and analytic settings.
Generalized Wiener process	Stochastic ProcessesDiffusion Models	A continuous-time diffusion formed by adding state- or time-dependent drift and volatility to Brownian noise, commonly written as a stochastic differential equation.
Generator (category theory)	Category Theory	An object or family of objects whose incoming probes distinguish every unequal pair of parallel morphisms in a category.
Generic fiber	Algebraic GeometryScheme Theory	The generic fiber of a morphism of schemes is the fiber obtained over the generic point of the base, capturing the behavior valid on a dense open part of that base.
Generic filter	Set Theory	A forcing filter that meets every dense subset of a partial order belonging to a specified ground model.
Generic property	—	A mathematical property holding outside a negligible exceptional set under a declared measure-theoretic, topological or algebraic notion of genericity.
Geodesic	—	A geometry-relative straight path whose tangent transports parallel to itself, yielding locally length-minimizing curves for a Riemannian metric while permitting broader affine and spacetime forms.
Geodesic convexity	Riemannian Geometry	The extension of convex sets and functions to curved spaces by replacing straight line segments with minimizing geodesics.
Geodetic Graph	Graph TheoryMetric Graph Theory	An undirected unweighted graph in which every vertex pair has exactly one shortest path, though longer alternate paths and some cycles may exist.
Geometric Brownian motion	Stochastic Calculus and Mathematical Finance	A positive continuous-time stochastic process whose logarithm follows Brownian motion with drift, equivalently solving a multiplicative-noise stochastic differential equation.
Geometric Langlands correspondence	Algebraic GeometryGeometric Representation Theory	A conjectural categorical correspondence relating local systems for a reductive group on an algebraic curve to sheaf-theoretic objects on the moduli stack of bundles for its Langlands dual group.
Geometric progression	Elementary Mathematics	A sequence in which every term after the first is obtained by multiplying the preceding term by one fixed common ratio.
Geometric quotient	Algebraic GeometryGroup Action Quotients	A quotient of an algebraic variety by a group action whose fibers are exactly orbits, topology is the quotient topology and regular functions are the invariant functions.
Geometric Transformation	GeometryTransformation Geometry	An invertible mapping of a geometric space that preserves the relations declared fundamental by a chosen geometry, thereby organizing transformations and figures by their invariants.
Geometrically Regular Ring	Commutative AlgebraAlgebraic Geometry	In algebraic geometry, a geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field.
Gerbe	Algebraic GeometryHigher Stacks and Cohomology	A stack locally equivalent to the classifying stack of a group, serving as a degree-two geometric analogue of a principal bundle and encoding obstruction and twisting data.
Gerstenhaber algebra	AlgebraGraded Algebra	A graded-commutative algebra equipped with a degree-minus-one graded Lie bracket that acts as a graded derivation of the product.
Giant Component	Random Graph TheoryNetwork Theory	In network theory, a giant component is a connected component of a given random graph that contains a significant fraction of the entire graph's vertices.
Gibbard–Satterthwaite Theorem	Game Theory	With at least three alternatives and unrestricted strict ordinal preferences, every deterministic onto single-winner rule that makes truthful reporting a dominant strategy is dictatorial.
Gilbert–Pollak conjecture	Discrete GeometrySteiner Tree Ratio	The assertion, now a theorem under accepted proof, that a Euclidean minimum spanning tree is at most 2/√3 times as long as a Steiner minimum tree on the same planar terminals.
Girth (Graph Theory)	Graph Theory	Assign an undirected graph the length of its shortest cycle, using infinity for an acyclic graph, to quantify how far local neighborhoods remain tree-like.
Giry monad	Categorical Probability	The probability monad on measurable spaces that sends each space to its measurable space of probability measures.
Giuga Number	Number TheorySpecial Classes of Composite Integers	A composite integer n for which every prime divisor p satisfies p dividing n/p minus 1, a restrictive factorwise congruence linked to Giuga's primality conjecture.
GJMS Operator	Conformal GeometryGeometric Analysis	A member of the conformally covariant differential-operator family with leading Laplacian power and dimension-bounded order.
Glicksberg's theorem	Game TheoryMinimax Theorems	A continuous zero-sum game result guaranteeing equality of maximin and minimax expected payoff over Borel mixed strategies on compact Hausdorff strategy spaces under the stated semicontinuity condition.
Global Dimension	Homological AlgebraRing Theory	A ring's global dimension is the supremum of projective-resolution lengths across all modules on a specified side.
Globally hyperbolic spacetime	Lorentzian Geometry	A spacetime with well-behaved causal structure admitting a Cauchy surface that every inextendible causal curve crosses exactly once.
Globular set	Higher Category Theory	A sequence of sets of n-cells with source and target maps satisfying globularity equations, forming the presheaf carrier for many higher-category structures.
Goldberg–Seymour Theorem	Graph TheoryMultigraph Edge Coloring	Every loopless multigraph can be properly edge-colored using no more than the larger of maximum degree plus one and the ceiling of its densest odd-set edge ratio.
Golden-section search	Numerical OptimizationUnivariate Search	A derivative-free interval-reduction algorithm for optimizing a unimodal function by placing interior evaluations in the golden ratio so one point can be reused each iteration.
Golomb sequence	Integer Sequences	The nondecreasing self-describing integer sequence in which each positive integer n occurs exactly as many times as the nth term states.
Good Spanning Tree	Graph Drawing	Select a rooted spanning tree of a fixed plane embedding whose non-tree edges avoid ancestor chords and occur in left, child-tree, and right blocks around every vertex, enabling ordered planar graph drawings and visibility representations.
Graceful labeling	Graph Labeling	An injective vertex labeling from zero through the edge count whose absolute edge differences are exactly one through that count.
Grade (ring theory)	Commutative and Homological Algebra	The least degree in which a module or ideal has nonzero Ext into the base ring, equivalently under suitable hypotheses the maximum length of a regular sequence in its annihilator or ideal.
Graded Ring	Abstract AlgebraAlgebraic Topology	A ring decomposed into additive homogeneous components whose products have the sum of their component degrees.
Gram Matrix	Linear AlgebraInner Product Spaces	Tabulate the pairwise inner products of a finite vector family in one square matrix, preserving its relative geometry and dependence.
Graph bandwidth	Graph Layout	The minimum possible largest label distance across an edge when a graph’s vertices are placed at distinct positions on a line.
Graph Coloring Game	Graph Theory	An adversarial sequential graph-coloring process in which players alternately make legal color assignments, one seeking a complete proper coloring and the other seeking to create an uncolorable position.
Graph Duality	Graph Theory	Pair a planar graph with a dual by placing a vertex inside each face and joining faces that share an edge, so vertices swap with faces and problems translate across a lossless table — paths become cuts, colorings become face colorings — letting you solve whichever side is easier.
Graph dynamical system	Discrete Dynamical SystemsGraph Theory	A discrete dynamical system built from a graph, local vertex states and neighborhood functions, and an update scheme that induces a global state-transition map.
Graph Embedding	Topological Graph Theory	A crossing-free placement of a graph's vertices and edge arcs on a specified topological surface that preserves their incidences.
Graph factorization	Graph TheoryFactor Theory	A decomposition of a graph's edge set into spanning regular subgraphs called factors, with a k-factorization partitioning all edges into k-regular factors.
Graph Invariant	Graph TheoryInvariants	A graph invariant is a value, polynomial, sequence, multiset, or other mathematical object assigned to a graph such that isomorphic graphs receive the same result, with its definition, graph category, and distinguishing power explicitly stated.
Graph isomorphism	Graph Theory	A bijection between two graph vertex sets that preserves adjacency and nonadjacency, showing that the graphs have the same structure up to relabeling.
Graph of a Function	Foundations and Analysis	Represent a function by the set of ordered input–output pairs selected by its evaluation rule, preserving every domain element with exactly one associated value while separating the graph from a plotted picture or graph-theoretic network.
Graph operations	Graph Theory	In the mathematical field of graph theory, graph operations are operations which produce new graphs from initial ones.
Graph Power	Graph Theory	The same-vertex graph transformation that makes every pair at original shortest-path distance at most k directly adjacent.
Graph Sphericity	Geometric Graph Theory	The least Euclidean dimension in which a graph can be represented as the intersection graph of congruent spheres, equivalently unit balls under the adopted convention.
Graph Toughness	Graph Theory	The minimum ratio of vertices removed to components left over all fragmenting vertex sets of a finite graph, with complete graphs assigned infinity.
Graph Vertex	Graph Theory	In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another.
Grassmannian	Algebraic Geometry	A parameter space whose points are the fixed-dimensional linear subspaces of a vector space.
Greatest Common Divisor	Number Theory	In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.
Grim Trigger	Game TheoryRepeated Games Strategies	A repeated-game strategy that cooperates every period until the opponent defects once, then switches to permanent defection forever — the severity-maximal, zero-forgiveness deterrent that sustains cooperation whenever the discount factor is high enough.
Gromov's compactness theorem (geometry)	Metric GeometryCompactness Theorems	A precompactness theorem for families of compact metric spaces with uniform diameter and covering-number bounds under Gromov–Hausdorff convergence.
Gromov–Hausdorff convergence	Metric Geometry and Geometric Convergence	A convergence notion for metric spaces defined by their Gromov–Hausdorff distance tending to zero after comparison in common ambient metrics.
Grothendieck category	Category Theory	An abelian category with arbitrary coproducts, exact filtered colimits, and a generator.
Grothendieck group	AlgebraGroup Completion	The universal abelian group completion of a commutative monoid, formally adjoining additive inverses while preserving every monoid homomorphism into an abelian group.
Grothendieck Local Duality	—	Over a Noetherian local ring with a normalized dualizing complex, Matlis duals of maximal-ideal local cohomology are identified with completed Ext groups into that dualizing object.
Grothendieck Space	Functional Analysis and Banach Space Theory	A Banach space whose continuous-dual sequences gain weak convergence whenever they converge weak-star, equivalently forcing every bounded operator into c0 or any separable Banach space to be weakly compact.
Grothendieck topology	Category TheorySites and Topoi	A categorical covering structure that designates compatible families of morphisms as covers, enabling sheaves and cohomology on categories whose objects need not be open subsets of a space.
Grothendieck–Riemann–Roch theorem	Intersection TheoryAlgebraic K Theory	For a proper map of smooth algebraic schemes, Grothendieck–Riemann–Roch equates a K-theoretic direct image with a rational Chow pushforward after Chern-character and Todd-class correction.
Ground expression	Mathematical Logic	A formal term or formula containing no free variables because every constituent is a constant, function application or fully closed construction.
Group algebra of a locally compact group	Harmonic AnalysisOperator Algebras	A convolution algebra built from functions on a locally compact group using Haar measure, with representations linked to representations of the group.
Group code	Coding Theory	A block-code subgroup of G^n over a finite group alphabet, optionally represented systematically by information symbols and homomorphic parity construction.
Group Ring	Abstract AlgebraRing Theory	In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group.
Groupoid object	Category TheoryInternal Categories	An internal category in which every arrow has an inverse, defined inside a category with suitable pullbacks rather than only inside sets.
Grundy Number	—	The largest number of colors that first-fit vertex coloring can be forced to use over all vertex orderings measures a graph's worst-case greedy order sensitivity.
Guess 2/3 of the Average	Game TheoryBehavioural Game Theory	A single-shot game where each player picks a number closest to two-thirds of the group's mean — its Nash equilibrium of zero is never reached, and because each level of iterated best-response leaves a distinct numerical signature, the modal guess reads off a population's depth of strategic reasoning.
Gödel numbering	Mathematical Logic	An effective injective encoding of symbols, formulas, proofs, or other formal objects as natural numbers so syntax can be represented and reasoned about arithmetically.
Gödel sentence	Mathematical LogicIncompleteness Theorems	The unprovable statement referred to by the theorem is often referred to as "the Gödel sentence" for the system .
Gödel–Dummett Logic	Mathematical Logic	The prelinear family of intermediate and many-valued logics characterized by linearly ordered Heyting semantics, with conjunction and disjunction interpreted by minimum and maximum and implication by the Gödel residuum.
Gδ Set	General TopologyBorel Hierarchy	A subset of a topological space that can be represented as a countable intersection of open sets.
H-closed space	General Topology	A Hausdorff topological space that is closed in every Hausdorff space in which it embeds as a subspace.
Hadamard manifold	Riemannian Geometry	A complete, simply connected Riemannian manifold with everywhere nonpositive sectional curvature.
Hadamard matrix	Combinatorial Matrix Theory	Arrange signs in a square matrix so every pair of distinct rows, and therefore columns, is orthogonal, equivalently satisfying the exact Gram identity \(HH^{\mathsf T}=nI\).
Hadamard product (matrices)	Linear Algebra	The entrywise product of two matrices of identical shape, multiplying corresponding entries without summing across indices.
Hadamard space	Metric and Nonlinear Geometry	A complete geodesic metric space of nonpositive curvature in the CAT(0) sense, equivalently satisfying a strong midpoint convexity inequality.
Hadwiger number	Structural Graph Theory	The largest integer k such that the complete graph on k vertices occurs as a minor of a given undirected graph.
Haefliger structure	Differential TopologyFoliation Theory	Encode generalized codimension-q foliation data by local maps to transverse Euclidean space whose transition germs form a cocycle, permitting pullback and classification beyond regular foliations.
Hafnian	CombinatoricsLinear Algebra	In mathematics, the hafnian is a scalar function of a symmetric matrix that generalizes the permanent.
Hajós construction	Graph Theory	A graph-composition operation that deletes one edge from each of two graphs, identifies selected endpoints and joins the remaining endpoints.
Hales–Jewett theorem	Ramsey Theory	A Ramsey-theoretic theorem guaranteeing a monochromatic combinatorial line in every sufficiently high-dimensional finite word cube.
Hall algebra	Representation Theory and Homological Algebra	An associative algebra whose basis represents isomorphism classes of objects and whose multiplication counts extensions or subobjects with specified quotient and subobject types.
Halley's Method	Nonlinear Equations	Halley's method refines a scalar root estimate with a curvature-corrected rational step that converges cubically near a suitable simple root.
Halton sequence	Numerical AnalysisQuasi Monte Carlo	A deterministic low-discrepancy sequence in the unit cube formed by combining one-dimensional radical-inverse sequences in pairwise coprime bases.
Hamburger moment problem	Analysis Probability and Moment Problems	The problem of deciding whether a given sequence is the sequence of moments of a positive Borel measure on the whole real line, and whether that representing measure is unique.
Hamming Scheme	Association Schemes and Coding Theory	The association scheme on fixed-length words over a finite alphabet whose relation classes are indexed by Hamming distance.
Hanani–Tutte theorem	Topological Graph Theory	A planarity theorem stating that a graph is planar when it has a plane drawing in which every pair of independent edges crosses an even number of times.
Handlebody	Geometric TopologyHandle Decompositions	A manifold piece obtained from a ball or collar by attaching standard handles, used to decompose manifolds according to topology and critical-point index.
Hankel matrix	Linear Algebra and Structured Matrices	A matrix whose entries are constant along every anti-diagonal, so each entry depends only on the sum of its row and column indices.
Hardy field	Asymptotic Analysis	A field of germs of real functions at positive infinity that is closed under differentiation.
Hardy–Littlewood maximal function	Harmonic Analysis	The pointwise supremum of local absolute-value averages of a function over all balls or cubes containing the evaluation point.
Harmonic Balance	Applied MathematicsNonlinear Periodic Systems	Approximate a nonlinear system's periodic steady state by a truncated Fourier series and solve for coefficients whose retained residual harmonics balance to zero.
Harmonic conjugate	Conjugate Harmonic Functions	Pair real harmonic functions whose gradients satisfy the Cauchy-Riemann rotation so they form the real and imaginary parts of one holomorphic function, subject to global topological existence and additive-constant ambiguity.
Harmonic measure	Potential Theory	A boundary probability measure giving the likelihood that Brownian motion started inside a domain first exits through each boundary subset, equivalently representing solutions of the Dirichlet problem.
Harmonic Spectrum	Acoustics and Signal AnalysisPeriodic Signal Spectra	A line spectrum whose nonzero component frequencies lie at integer multiples of a common fundamental, with amplitudes and phases determining waveform and timbre while the harmonic grid encodes periodicity.
Harrop Formula	Intuitionistic Logic	An intuitionistic formula built so disjunctions and existential quantifiers occur only in negative positions, yielding a stable, computationally disciplined fragment and inspiring hereditary-Harrop logic programming.
Hartman–Grobman Theorem	Dynamical SystemsLocal Stability Theory	Near a hyperbolic equilibrium or fixed point, a differentiable nonlinear dynamical system is locally topologically conjugate to its linearization, so qualitative orbit structure can be read from the linear model.
Hartogs's Extension Theorem	Complex AnalysisSeveral Complex Variables	Extend a holomorphic function uniquely across a compact hole in a connected domain of at least two complex dimensions.
Hasse diagram	Order Theory	A drawing of a finite partially ordered set using vertices for elements and upward cover edges while omitting reflexive and transitively implied relations.
Hasse–Schmidt derivation	Commutative Algebra	A sequence of additive maps encoding a formal higher-order derivation through a multiplicative generating-series identity.
Hausdorff completion	Topological Algebra	The inverse-limit completion of a filtered group formed from its discrete quotients, with the filtration intersection removed by the canonical map.
Hausdorff density	Geometric Measure Theory	The small-scale upper, lower or exact ratio of a Radon measure's mass in balls around a point to the radius raised to a declared dimension.
Hausdorff Maximal Principle	Set Theory	The choice-equivalent principle that every chain in any partially ordered set extends to an inclusion-maximal chain.
Hausdorff moment problem	Analysis and Probability	The problem of characterizing sequences that are moments of a positive measure on the unit interval, with a unique representing measure whenever one exists.
Hausdorff Space	TopologySeparation Axioms	A topological space in which every two distinct points admit disjoint open neighborhoods, equivalently one whose diagonal is closed and whose convergent nets have unique limits.
Haven (Graph Theory)	Structural Graph Theory	Assign every deletion set smaller than a stated order to a surviving connected component in a nested or pairwise-touching way, certifying an evader's coherent refuge and dualizing bounded treewidth.
Haversine Formula	Spherical TrigonometryGreat Circle Distance Computation	A half-angle spherical-trigonometry relation that converts two latitude–longitude positions into their central angle and great-circle arc distance, with explicit radius, angle-unit, and numerical-boundary controls.
Heaviside step function	Mathematical Analysis	A threshold function equal to zero below an origin and one above it, with a declared convention at the discontinuity.
Hecke Character	Algebraic Number Theory	A continuous complex quasicharacter of a global field's idèle class group whose local components, conductor, and infinity type generate a Hecke L-function.
Hedgehog (geometry)	Differential GeometrySupport Function Envelopes	A plane curve or higher-dimensional hypersurface represented as the envelope of oriented support lines or hyperplanes supplied by a differentiable support function, extending convex support geometry to self-crossing and projective cases.
Hedgehog Space	General Topology	Join a cardinal-indexed family of unit intervals at one origin and metrize the result so paths on different spines pass through the common center.
Heegner's lemma	Arithmetic Geometry	A descent lemma stating that a quartic curve with nonsquare leading coefficient has a rational point if it has a point over an odd-degree extension.
Heisenberg group	Lie Groups Harmonic Analysis and Quantum Mechanics	A two-step nilpotent group that centrally extends a position-momentum vector space, canonically realized by upper-unitriangular matrices with a bilinear cross term.
Helffer–Sjöstrand Formula	Spectral TheoryFunctional Analysis	The Helffer–Sjöstrand formula is a mathematical tool used in spectral theory and functional analysis to represent functions of self-adjoint operators.
Helix	Geometry	A space curve whose tangent maintains a constant angle with a fixed axis, with the circular helix as the constant-radius canonical case.
Helly family	Combinatorics	A set family in which global intersection follows whenever every sufficiently small subfamily has nonempty intersection.
Helly's selection theorem	Real AnalysisCompactness and Bounded Variation	Every uniformly bounded sequence of monotone real functions on a compact interval has a subsequence converging pointwise, with bounded-variation generalizations providing compactness for measure and weak-convergence arguments.
Hemiperfect number	Number Theory	A positive integer whose sum-of-divisors function divided by the integer is a half-integer with odd numerator.
Hensel's Lemma	P Adic Number Theory	A nondegenerate polynomial root or coprime monic factorization modulo a prime lifts compatibly and uniquely through higher prime-power precision.
Hereditarily normal space	General Topology	In topology and related branches of mathematics, a normal space is a topological space in which any two disjoint closed sets have disjoint open neighborhoods.
Hermite normal form	Integer Linear Algebra and Lattice Algorithms	A canonical echelon-like matrix form over the integers used to represent lattices and solve integer-coordinate linear systems.
Hermite reciprocity	Invariant Theory	A representation-theoretic isomorphism exchanging degree and symmetric-power indices for binary forms: Sym^m(Sym^n V) is isomorphic to Sym^n(Sym^m V) for two-dimensional V.
Hermite spline	Numerical AnalysisInterpolation	In the mathematical subfield of numerical analysis, a Hermite spline is a spline curve where each polynomial of the spline is in Hermite form.
Hermitian function	Fourier Analysis and Signal Theory	A complex-valued function satisfying conjugate symmetry f(-x)=conjugate(f(x)), equivalently having an even real part and an odd imaginary part.
Hermitian matrix	Linear Algebra	In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose.
Heronian mean	Mathematical Means	The symmetric mean of two nonnegative numbers equal to one third of their sum plus their geometric mean.
Heronian triangle	GeometryNumber Theory	In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths , , and and area are all positive integers.
Herz–Schur multiplier	Harmonic Analysis and Operator Algebras	A group function whose invariant Schur kernel induces a completely bounded Fourier multiplier, not necessarily a positive one.
Hessenberg Matrix	Numerical Linear Algebra	Constrain a square matrix to be triangular except for one adjacent off-diagonal band, yielding a similarity-reachable form that preserves eigenvalues while making QR iteration and related computations substantially cheaper.
Heteroclinic Cycle	Dynamical Systems	An invariant cycle in phase space formed by equilibrium points joined by heteroclinic trajectories, along which nearby dynamics may visit successive equilibria for increasingly long intervals.
Heteroclinic orbit	Dynamical Systems	In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins two different equilibrium points.
Hexagon	GeometryPolygons	In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon.
Heyting Algebra	Order TheoryAlgebraic Logic	A bounded lattice whose implication is the right adjoint of meet, giving an algebraic semantics for intuitionistic reasoning.
Heyting arithmetic	Intuitionistic Logic and Foundations of Arithmetic	A first-order theory of natural-number arithmetic using intuitionistic rather than classical logic while retaining arithmetic axioms and induction.
High (computability)	Computability TheoryMathematical Logic	In computability theory, a Turing degree [X] is high if it is computable in 0, and the Turing jump ['] is 0, which is the greatest possible degree in terms of Turing reducibility for the jump of a set which is computable in 0.
Higher local field	Arithmetic Geometry	A field equipped with an iterated tower of complete discrete valuations whose final residue field is finite or otherwise specified at dimension zero.
Higher spin alternating sign matrix	Algebraic Combinatorics and Lattice Models	A square integer matrix whose row and column sums equal a fixed positive spin r and whose running partial sums along every row and column remain between zero and r.
Higher Stack	Higher Category GeometryAlgebraic GeometryHomotopy Theory	A higher-categorical stack: a presheaf valued in spaces, infinity-groupoids, or higher categories that satisfies descent by gluing compatible local objects together with all levels of equivalence and coherence data.
Highly composite number	Number Theory	A positive integer whose divisor count strictly exceeds that of every smaller positive integer.
Highly cototient number	Number TheoryTotient Inverse Records	A positive integer k>1 having more solutions to x−φ(x)=k than any smaller integer greater than one, where φ is Euler's totient function.
Highly irregular graph	Graph TheoryDegree Properties	A graph in which the neighbors of every vertex all have pairwise distinct degrees.
Highly powerful number	Number Theory	A powerful integer setting a new record for the product of its prime exponents among all smaller powerful integers.
Highly totient number	Number Theory	An integer whose number of preimages under Euler’s totient function exceeds that of every smaller integer.
Hilbert metric	GeometryProjective Metrics	A projectively invariant metric on the interior of a bounded convex domain, defined by a logarithmic cross ratio of the two boundary intersections on the line through a point pair.
Hilbert system	Proof TheoryFormal Deductive Systems	An axiomatic proof calculus in which theorems are generated from axiom schemata by a small set of inference rules, often only modus ponens plus a rule for quantification.
Hilbert–Carleman determinant	Operator Theory	A second-regularized determinant det₂(I+A) for a Hilbert–Schmidt operator, related to the ordinary Fredholm determinant by a trace correction when that trace exists.
Hilbert–Poincaré Series	Graded AlgebraCommutative Algebra	Encode the dimension or rank of each graded component as the corresponding coefficient of a formal power series.
Hilbert–Schmidt integral operator	Functional Analysis	An integral operator whose square-integrable kernel makes it a compact Hilbert–Schmidt operator.
Hiptmair–Xu preconditioner	Numerical Partial Differential Equations	An auxiliary-space preconditioner for finite-element discretizations of H(curl) and H(div) problems that decomposes difficult vector fields into smoother scalar, gradient or curl components.
HN group	Group Theory	A group in which every subnormal subgroup has the whole group as its hypernormalizer.
Hochschild homology	Homological Algebra	In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings.
Hodge–Arakelov theory	Arithmetic Geometry	A proposed Arakelov-geometric analogue of Hodge comparison theory for elliptic curves, centered on functions on universal extensions and torsion points.
Holomorphic functional calculus	Operator Theory	A calculus assigning f(T) to a bounded operator T for every function holomorphic near its spectrum through a contour integral of the resolvent.
Holomorphic tangent bundle	Complex Geometry	The complex vector bundle of type-(1,0) tangent directions on a complex manifold, with holomorphic transition functions.
Holomorphic vector bundle	Complex Differential and Algebraic Geometry	A complex vector bundle over a complex manifold whose total space is complex and whose projection map is holomorphic.
Holonomic Basis	Differential Geometry	A holonomic basis is a smooth local frame whose vector fields are the partial-derivative directions of one coordinate chart, equivalently a commuting frame.
Homeomorphism (graph theory)	Graph Theory	The equivalence of graphs that become isomorphic after subdividing edges by degree-two vertices.
Homeotopy	Algebraic Topology	A homotopy group of the topological group of self-homeomorphisms of a space.
HOMFLY polynomial	Knot Theory	A two-variable oriented-link invariant defined by a skein relation and normalization that specializes to the Alexander and Jones polynomials.
Homoclinic bifurcation	Dynamical SystemsBifurcation Theory	A homoclinic bifurcation is a global change in a dynamical system caused when a periodic orbit or invariant manifold forms or loses a trajectory connecting a saddle point to itself.
Homogeneous Graph	Graph Theory	A graph whose every isomorphism between finite induced subgraphs extends to an automorphism of the entire graph, making all finite local copies globally interchangeable.
Homological Stability	Algebraic TopologyStable Topology	Eventual degreewise invariance in an indexed family: stabilization maps induce homology isomorphisms once the size parameter enters a degree-dependent stable range.
Homology Manifold	Geometric TopologyAlgebraic Topology	A finite-dimensional ANR whose local relative homology at every point matches Euclidean space in a stated dimension and coefficient system, whether or not Euclidean charts exist.
Homomorphism	Algebra	A map between algebraic structures of the same signature that preserves each distinguished operation and constant.
Homotopy associative algebra	Homotopy TheoryAlgebraic TopologyHomological Algebra	There is a notion of algebras, called A_\infty -algebras, which still have a property on the multiplication which still acts like the first relation, meaning associativity holds, but only holds up to a homotopy, which is a way to say after an operation "compressing" the information in the algebra, the multiplication is associative.
Homotopy Category	Algebraic TopologyHomotopy Theory	A category that retains objects while replacing maps by homotopy classes or, more generally, formally inverting a designated class of weak equivalences.
Homotopy fiber	Homotopy TheoryAlgebraic Topology	In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces f:A \to B .
Homotopy group with coefficients	Algebraic TopologyHomotopy Theory	In topology, a branch of mathematics, for i \ge 2 , the i-th homotopy group with coefficients in an abelian group G of a based space X is the pointed set of homotopy classes of based maps from the Moore space of type (G, i) to X, and is denoted by \pi_i(X; G) .
Homotopy Hypothesis	Higher Category Theory	Assert that homotopy types of spaces and suitably weak infinity-groupoids present equivalent homotopy theories, with paths, homotopies, and all higher homotopies represented as invertible higher morphisms.
Hook Length Formula	Algebraic CombinatoricsYoung Tableaux	Count standard Young tableaux of a partition shape by dividing n! by the product of its cell hook lengths.
Hopfian group	Group TheoryEndomorphism Rigidity	A group for which every surjective endomorphism is an automorphism, equivalently a group not isomorphic to any proper quotient of itself.
Hopkins statistic	Cluster Analysis	A nearest-neighbor statistic comparing observed data with uniform reference points to assess spatial cluster tendency.
Horocycle	Hyperbolic GeometryIdeal Boundary Geometry	A curve in the hyperbolic plane orthogonal to geodesics converging to one ideal boundary point, equivalently a limiting circle tangent to the boundary at that point.
Hosoya Index	Chemical Graph Theory	Count every matching of a graph, including the empty matching, to obtain a graph invariant used in matching theory and as a molecular topological descriptor.
Howson Property	Group Theory	A group has the Howson property when the intersection of every two finitely generated subgroups is again finitely generated, making finite generation closed under binary subgroup intersection.
Huge Cardinal	Set TheoryLarge Cardinal Axioms	A huge cardinal is the critical point of an elementary embedding whose target is closed under sequences of length equal to the image of that cardinal.
Humbert Polynomials	Special FunctionsPolynomial Sequences	A parameterized polynomial family defined by the generating function (1−mxt+t^m)^−λ, generalizing Pincherle polynomials through coefficients of powers of t.
Hurst Exponent	Time Series AnalysisLong Range Dependence	A model-indexed scaling exponent that describes how fluctuations, partial sums, or dependence persist across increasing temporal or spatial scales.
Hurwitz quaternion	Number TheoryQuaternion Algebra	In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers is excluded).
Hurwitz scheme	Algebraic GeometryModuli of Curves	An algebraic moduli scheme parameterizing branched covers of a fixed target curve, commonly degree-d genus-g covers of the projective line with specified ramification data.
Hyers–Ulam–Rassias stability	Functional Equations	A functional-equation stability property stating that an approximate solution satisfying a controlled error bound lies near an exact solution.
Hyperbolic distribution	Probability Distribution	A continuous probability distribution whose log-density traces a hyperbola, yielding exponential but heavier-than-normal tails.
Hyperbolic Geometric Graph	Geometric Graph TheoryNetwork Science	A graph model that positions vertices in hyperbolic space and makes edges depend on the vertices' hyperbolic distance.
Hyperbolic growth	Mathematical Modeling	Growth following an inverse-distance-to-a-finite-singularity form, diverging as the independent variable approaches a finite critical value.
Hyperbolic quaternion	Historical Hypercomplex Number Systems	Extend real scalars by three anticommuting square-\(+1\) units, producing a four-dimensional unital nonassociative algebra whose associator and quadratic form distinguish it from Hamilton and split quaternions.
Hyperbolization Procedures	Geometric Topology	Hyperbolization procedures replace and glue cells of a complex to build a controlled nonpositively or negatively curved space.
Hyperboloid	Analytic and Projective Geometry	A nondegenerate central quadric whose real canonical equation has either one sheet or two sheets according to the signs of its squared-coordinate terms.
Hypercube Graph	Graph TheoryCombinatorics	The graph on all n-bit strings with an edge exactly when two strings differ in one bit.
Hypercycle (Geometry)	Hyperbolic Geometry	A one-sided curve in the hyperbolic plane whose points all lie at the same positive perpendicular distance from a fixed geodesic axis.
Hyperharmonic number	Enumerative Combinatorics	A recursively iterated family of partial sums beginning with reciprocals and extending the ordinary harmonic numbers by an order parameter.
Hyperperfect number	Multiplicative Number Theory	Classify a natural number by the parameterized divisor-sum constraint n = 1 + k(σ(n) − n − 1), with perfect numbers as the k = 1 boundary case.
Hyperreal number	Nonstandard Analysis	An element of a proper ordered-field extension of the real numbers containing infinitesimal and infinite elements and satisfying a transfer principle.
Hypersphere	GeometryTopology	Hypersphere is a recurring identity in mathematics, logic, and statistics defined by this frozen evidence: Multidimensional generalization of a sphere in Euclidean space; an n-dimensional object embedded in an (n + 1)-dimensional Euclidean space.
Hypertopology	Set Valued Topology	A topology placed on a hyperspace of subsets, commonly the nonempty closed subsets of a topological space, so sets themselves become continuously varying points.
Hypograph (mathematics)	Convex Analysis	In mathematics, the hypograph or subgraph of a function f:\R^{n}\rightarrow \R is the set of points lying on or below its graph.
Hyponormal operator	Operator Theory	A bounded Hilbert-space operator whose self-commutator T*T−TT* is positive semidefinite.
I-bundle	Geometric TopologyFiber Bundles	A fiber bundle over a manifold whose fiber is an interval, classified as trivial when it is a product and twisted when its global gluing cannot be reduced to that product form.
IA automorphism	Group Theory	In mathematics, in the realm of group theory, an IA automorphism of a group is an automorphism that acts as identity on the abelianization.
Ideal (order theory)	Order and Domain Theory	A nonempty directed lower set of a partially ordered set, equivalently in a lattice a lower set closed under finite joins.
Ideal on a set	Set Theory	Ideal on a set denotes collection of sets regarded as "small" or "negligible" in set theory.
Ideal polyhedron	Hyperbolic Geometry	In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space.
Ideal sheaf	Algebraic GeometrySheaves of Ideals and Closed Subschemes	Assign an ideal of functions to every open set compatibly with restriction, so local vanishing conditions glue into a global sheaf and quasi-coherent ideal sheaves determine closed subschemes.
Idealizer	Abstract AlgebraSemigroup Theory	In abstract algebra, the idealizer of a subsemigroup T of a semigroup S is the largest subsemigroup of S in which T is an ideal.
Idempotent (ring theory)	Ring TheorySpecial Elements	A ring element e satisfying e²=e, whose multiplication acts as a projection and whose presence can encode decompositions of rings, modules and spectra.
Idempotent measure	Harmonic Analysis	A probability measure on a topological group that is unchanged by convolution with itself.
Idempotent Relation	Relation AlgebraDiscrete Mathematics	A binary relation on one set is idempotent when composing it with itself yields exactly the same related pairs.
Identity type	Type Theory	A type-theoretic proposition whose inhabitants witness equality between two terms of a type.
Iitaka dimension	Algebraic Geometry	An invariant measuring asymptotic growth of sections of powers of a line bundle, equivalently the dimension of the image of its associated rational maps.
Illumination problem	Computational GeometryMathematical Billiards	Illumination problems are a class of mathematical problems that study the illumination of rooms with mirrored walls by point light sources.
Image (category theory)	Category Theory	A universal monomorphism through which a morphism factors, generalizing the subset of attained values of a function.
Image (of a Function)	Set Theory and Function Theory	The set of values a function actually produces — im(f) = {f(x) : x in X} — a subset of the declared codomain that records what a map reaches rather than what it could in principle reach.
Imperfect Group	Group Theory	A group with no nontrivial perfect quotient, including itself among the quotients tested.
Implicit function	Mathematical Analysis	A function locally or globally determined by a relation among variables even when its dependent variable is not isolated by an explicit formula.
Imprecise probability	Probability and Decision TheoryImprecise Uncertainty Models	Represent incomplete probabilistic commitment by a coherent set of admissible probability measures or equivalent lower and upper expectations, so conclusions expose a range and distinguish robust agreement from decisions that depend on an unresolved model choice.
Improper integral	Mathematical Analysis	A limit-defined extension of a definite integral to an unbounded interval, an unbounded integrand or another excluded endpoint configuration.
Inaccessible cardinal	Set TheoryLarge Cardinals	A strongly inaccessible cardinal is an uncountable regular strong-limit cardinal; in modern usage, the unqualified term inaccessible cardinal normally denotes this strong form.
Incidence (geometry)	Incidence Geometry	In geometry, an incidence relation is a heterogeneous relation that captures the idea being expressed when phrases such as "a point lies on a line" or "a line is contained in a plane" are used.
Incidence algebra	Algebraic Combinatorics	An algebra of interval-indexed functions on a locally finite partially ordered set, with multiplication given by convolution over intermediate elements.
Inclusion (Boolean algebra)	Boolean Algebra	The canonical partial order on a Boolean algebra, where a≤b exactly when a∧¬b=0, equivalently a∧b=a or a∨b=b.
Inclusion map	Canonical Maps	The canonical injective function from a subset or subobject into its containing object that sends every element to itself viewed in the larger context.
Inclusion–exclusion principle	CombinatoricsEnumerative Methods	A counting identity that obtains the size or measure of a union by alternating sums over intersections, correcting repeated counting at every overlap order.
Incomplete polylogarithm	Special Functions	A special function obtained by truncating the integral representation of the polylogarithm at a nonzero lower limit, also related to incomplete Fermi–Dirac and Bose–Einstein integrals.
Incompressible surface	Three Manifold Topology	A properly embedded surface in a three-manifold with no essential loop that bounds a compressing disk in the ambient manifold.
Ind-completion	Category Theory	The free completion of a category under small filtered colimits, whose objects can be represented by filtered diagrams in the original category.
Ind-Scheme	Algebraic Geometry	A functor presented as a filtered direct limit of schemes along closed immersions.
Indecomposable distribution	Probability Theory	An indecomposable probability distribution cannot be represented as the convolution of two non-degenerate probability distributions, making it irreducible with respect to addition of independent random variables.
Indecomposable module	Module Theory	A nonzero module that cannot be expressed as a direct sum of two nonzero submodules.
Indefinite inner product space	Functional Analysis	A vector space with a Hermitian sesquilinear form that can assign positive, negative or zero squared length to nonzero vectors.
Indefinite orthogonal group	Linear and Lie Groups	The real linear group O(p,q) preserving a nondegenerate symmetric form with both positive and negative directions.
Independence of premise	Proof Theory	A constructive-logical principle allowing an existential witness to be moved outside an implication when the premise does not contain that witness variable.
Independence system	Combinatorics	A finite ground set paired with a downward-closed family of feasible subsets containing the empty set.
Independent and Identically Distributed Random Variables	Probability Theory	Model a collection of random variables as mutually independent draws from one common probability distribution, separating repeated sampling from dependence and distributional drift.
Independent increments	Probability TheoryStochastic Processes	In probability theory, independent increments are a property of stochastic processes and random measures.
Independent set (graph theory)	Graph Theory	A vertex subset of a graph in which no two selected vertices are adjacent.
Indeterminate form	Calculus	A limiting-value pattern whose component limits do not by themselves determine the limit of their arithmetic combination.
Index Group	Operator Theory	Take the discrete component group of the invertible elements of a unital Banach algebra by quotienting them by the connected component containing the identity.
Indicator function (complex analysis)	Complex Analysis	A directional asymptotic function recording an entire function’s exponential growth rate.
Indicator function (convex analysis)	Convex Analysis	An extended-real function that is zero on a selected set and positive infinity outside it, encoding membership as a hard optimization constraint.
Indiscernibles	Model Theory	Choose elements or tuples whose finite subconfigurations satisfy exactly the same formulas over a declared parameter set whenever their index patterns agree, creating model-theoretic symmetry that supports controlled constructions and automorphisms.
Indiscrete space	General Topology	A topological space whose only open sets are the empty set and the entire underlying set, giving the coarsest topology and making distinct points topologically indistinguishable.
Individual-Pieces Set	Fair DivisionMeasure Theory	In fair cake-cutting, the set of all utility vectors attainable by partitions of the cake among named agents; its boundary exposes feasible and Pareto-efficient allocations under declared valuations.
Induced homomorphism	Algebraic Topology	A homomorphism obtained canonically by applying a functorial algebraic construction to an underlying map.
Induced Metric	Differential Geometry	Transfer an ambient metric to an immersed manifold by pairing its tangent vectors after the immersion's differential, subject to the restricted form's signature.
Induced Path	Graph Theory	A graph path whose selected vertices have no host-graph edges except the consecutive edges of the path.
Induced representation	Representation Induction	Extend a subgroup representation to the ambient group through a universal construction on cosets or tensoring over group algebras.
Inertia stack	Algebraic Geometry and Stack Theory	A stack whose objects pair an object of an algebraic or differentiable stack with one of its automorphisms, thereby recording isotropy and conjugation data.
Inertial manifold	Dynamical SystemsDissipative Pdes	In mathematics, inertial manifolds are concerned with the long term behavior of the solutions of dissipative dynamical systems.
Inference Rule	LogicProof Theory	An inference rule is a formally specified, substitution-invariant schema that licenses deriving an expression of a conclusion form from expressions of designated premise forms within a proof system, with side conditions, variable restrictions, and validity or admissibility semantics declared.
Infinitary Logic	Mathematical LogicModel Theory	An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs.
Infinite expression	Mathematical LogicInfinitary Expressions	A mathematical expression with infinitely many operands or unbounded nesting whose meaning is defined only through a limit, fixed point, formal topology or other explicit semantic construction.
Infinitesimal	Nonstandard Analysis and Differential Methods	A nonzero mathematical quantity smaller in magnitude than every positive standard real scale, made rigorous only relative to a specified non-Archimedean, nilpotent, or formal framework.
Infinitesimal transformation	Lie Theory	A first-order generator describing the tangent direction of a continuous one-parameter family of transformations at the identity.
Inflation-restriction exact sequence	Group CohomologyHomological Algebra	The low-degree exact sequence in group cohomology that relates a group's cohomology to that of a normal subgroup and quotient through inflation, restriction, and transgression maps.
Inflection point	Differential Calculus and Geometry	A point on a sufficiently smooth curve where signed curvature changes sign, or on a function graph where local concavity changes from one side to the other.
Influence Diagram	Decision AnalysisGraphical Decision Modeling Under Uncertainty	A typed directed acyclic graph that joins uncertain variables, choices, available information, and value functions so a decision problem can be evaluated for an expected-utility-maximizing policy.
Infrabarrelled space	Functional AnalysisTopological Vector Spaces	In functional analysis, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled (also spelled infrabarreled) if every bounded barrel is a neighborhood of the origin.
Inhabited set	Constructive Mathematics	A set for which an element can be constructively exhibited or otherwise supplied as a witness, a stronger datum than double-negated nonemptiness in intuitionistic logic.
Initial and terminal objects	Category Theory	Initial and terminal objects are category-theoretic universal endpoints: an initial object has exactly one morphism to every object, while a terminal object has exactly one morphism from every object.
Injective and Projective Model Structure	Abstract Homotopy TheoryModel Categories	Dual model structures on a diagram category Fun(I,C) with objectwise weak equivalences; injective cofibrations or projective fibrations are objectwise, and the complementary class is determined by lifting when existence hypotheses hold.
Injective metric space	Metric Geometry	A metric space into which every nonexpansive map from a subspace extends nonexpansively over the containing space, equivalently a hyperconvex space.
Injective object	Category Theory	A categorical object into which every morphism defined on a subobject extends across the containing monomorphism.
Injective Tensor Product	Functional Analysis	A topological tensor construction that measures tensors by their action on pairs of continuous dual tests, using the ε norm in the Banach-space case.
Inner measure	Measure Theory	A set function giving lower size estimates through measurable subsets and satisfying inner-measure axioms.
Inscribed Square Problem	GeometryInscribed Configurations	The open square-peg conjecture that every planar Jordan curve contains four distinct points forming a nondegenerate Euclidean square, without requiring square order to match curve order.
Inserter category	Category Theory	For parallel functors F and G from C to D, the category whose objects are arrows F(X) to G(X) and whose morphisms are C-arrows making the corresponding naturality square commute.
Integer factorization	Number TheoryComputational Arithmetic	The decomposition of a positive integer into integer factors, canonically into a unique multiset of primes up to ordering, with computational difficulty depending strongly on input size and structure.
Integer matrix	Linear Algebra	In mathematics, an integer matrix is a matrix whose entries are all integers.
Integer points in convex polyhedra	Discrete GeometryPolyhedral Combinatorics	The study of integer points in convex polyhedra is motivated by questions such as "how many nonnegative integer-valued solutions does a system of linear equations with nonnegative coefficients have" or "how many solutions does an integer linear program have".
Integral curve	Differential Equations	A parametrized curve whose tangent at every point equals a specified vector field, representing a solution trajectory of an ordinary differential equation.
Integral graph	Spectral Graph Theory	A finite graph whose adjacency matrix has only integer eigenvalues.
Integral part	Elementary FunctionsNumber Theory	The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations).
Integral Transform	Functional AnalysisIntegral Transforms	An integral transform is an operator that maps a function or generalized function on one domain to a new representation by integrating it against a specified kernel over a declared measure space, with its meaning fixed by domain, codomain, convergence, regularity, and inversion conditions.
Integration by parts operator	Stochastic AnalysisMalliavin Calculus and Gaussian Analysis	Represent the adjoint of directional differentiation relative to a measure by mapping admissible vector fields to scalar divergences that satisfy a measure-specific integration-by-parts identity.
Interacting Particle System	Probability TheoryStochastic Processes	In probability theory, an interacting particle system (IPS) is a stochastic process (X(t))_{t \in \mathbb R^+} on some configuration space \Omega= S^G given by a site space, a countably-infinite-order graph G and a local state space, a compact metric space S .
Interchange law	Category TheoryTwo Dimensional Composition	The coherence equation stating that composing compatible 2-cells horizontally and then vertically gives the same result as composing vertically and then horizontally.
Interchange of limiting operations	Mathematical Analysis	The analysis problem of identifying conditions under which two limits, or a limit and integration, differentiation or summation, may be applied in either order with the same result.
Interior	Topology	Grade membership in a set by robustness rather than bare inclusion: a point lies in the interior only if some open neighborhood of it fits entirely inside the set, giving it room to spare in every direction.
Interior-Point Method	Numerical OptimizationOperations Research	A family of constrained-optimization algorithms that iteratively use a feasible region's interior geometry to advance toward an optimum.
Internal Set Theory	Mathematical LogicNonstandard Analysis	Nelson's conservative enrichment of ZFC with a standardness predicate and Transfer, Idealization, and Standardization schemes for internal nonstandard analysis.
Intersection Curve	Differential GeometrySurface Intersection	An intersection curve is a one-dimensional common locus of two surfaces, with regularity determined by how they meet.
Intersection graph	Graph Theory	A graph representing a family of sets or objects, with one vertex per object and an edge exactly when the corresponding pair intersects.
Intersection number (graph theory)	Graph Theory	The minimum ground-set size needed to represent a graph as intersections among finite vertex-associated sets, equivalently its minimum edge-clique cover size.
Intersection type	Type Theory	A type assigned to values satisfying multiple type descriptions simultaneously, written as an intersection such as sigma ∩ tau.
Interval	Order TheoryReal Analysis	An order-convex subset containing every ambient element between any two of its members.
Interval Arithmetic	Numerical AnalysisValidated Numerics	Arithmetic on interval enclosures that preserves inclusion of every admissible exact result, with outward rounding in finite-precision implementations.
Interval Contractor	Validated NumericsNumerical Constraint Processing	A box-to-box operator that narrows an interval search region without removing any point satisfying its declared target constraints.
Interval order	Order Theory	A partial order representable by real intervals where one element precedes another exactly when its interval lies completely to the left.
Intrinsic Equation of a Curve	Differential GeometryCurve Theory	A representation of curve shape by relations among arc length, tangent angle, curvature, and torsion that suppresses arbitrary position and coordinate frame.
Intuitionistic Type Theory	Constructive LogicType Theory	A constructive dependent type-theory family that treats propositions as types and proofs as terms governed by explicit formation, use, and computation rules.
Invariant basis number	Ring TheoryModule Rank	A ring property ensuring that isomorphic finitely generated free modules have the same finite rank, so basis cardinality is well defined.
Invariant factorization of LPDOs	Partial Differential EquationsDifferential Algebra	Each Laplace invariant is an explicit polynomial condition of factorization; coefficients of this polynomial are explicit functions of the coefficients of the initial LPDO.
Invariant measure	Ergodic Theory	A measure preserved by a specified transformation or group action, assigning every measurable set the same measure as its preimage or transformed image.
Invariant polynomial	Invariant Theory	In mathematics, an invariant polynomial is a polynomial P that is invariant under a group \Gamma acting on a vector space V .
Invariant Sigma-Algebra	Ergodic Theory and ProbabilityInvariant Events and Ergodic Decomposition	The sub-sigma-algebra of measurable events unchanged by a specified measurable transformation or action—exactly or modulo null sets—encoding all event-level information that the dynamics cannot alter.
Invariant Subspace	Linear AlgebraOperator Theory	A linear subspace mapped into itself by a specified operator or operator family.
Inverse Bundle	Vector Bundle TheoryAlgebraic Topology	A finite-rank vector bundle whose Whitney sum with a specified same-base bundle is isomorphic to a finite-rank trivial bundle.
Inverse distribution	Probability Theory	The probability distribution of the reciprocal of a random variable.
Inverse Element	Abstract AlgebraMonoid Theory	An element of a specified monoid that composes with another element on both sides to give that monoid's identity.
Inverse image functor	Sheaf Theory	The functor that pulls sheaves on a target space back along a continuous map to sheaves on the source space.
Inverse Semigroup	Semigroup TheoryAbstract Algebra	A semigroup in which every element has a unique generalized inverse, admitting partial-bijection models and commuting idempotents.
Inverse tangent integral	Special FunctionsPolylogarithmic Integrals	The special function Ti₂(x)=∫₀ˣ arctan(t)/t dt, equivalently an odd dilogarithmic combination with a characteristic alternating odd-power series.
Inverse Trigonometric Functions	Mathematical AnalysisTrigonometry	Restrict each periodic trigonometric function to an injective branch and invert it, returning a principal angle while retaining rules for other periodic or complex branches.
Invex Function	Generalized ConvexityMathematical Optimization	A differentiable function admitting a comparison map that bounds every value gap below by a gradient pairing at the base point.
IP set	Ramsey TheoryCombinatorics	A subset of the natural numbers or a semigroup that contains all nonempty finite sums drawn from some infinite generating set or sequence.
Irrationality measure	Diophantine ApproximationRational Approximation Exponents	A quantitative bound on how closely an irrational real or complex number can be approximated by rational numbers as denominator size grows.
Irreducible Component	Algebraic GeometryCommutative Algebra	Identify a maximal topological piece that cannot be expressed as the union of two proper closed pieces.
Irreducible polynomial	Polynomial Factorization	Classify a nonzero nonunit polynomial as irreducible relative to a declared coefficient ring when every factorization forces at least one factor to be a unit, with field and primitive-polynomial conventions kept explicit.
Irrelevant Ideal	Algebraic GeometryGraded Algebra	A homogeneous ideal that identifies the coordinate locus excluded when a specified graded presentation is turned into projective geometry.
Isbell duality	Enriched Category Theory	An enriched categorical adjunction between presheaves and copresheaves induced by hom-pairing with representable functors.
Isochron	Dynamical SystemsAsymptotic Phase and Model Reduction	Collect initial states that share one asymptotic phase or reduced long-term trajectory, forming a level set of the system's asymptotic-state map across transient directions.
Isolated point	General Topology	A point of a subset having a neighborhood that contains no other point of that subset.
Isolating Neighborhood	Dynamical SystemsConley Index Theory	A compact region isolates dynamics when every complete trajectory staying inside it lies strictly away from its boundary.
Isometry group	Metric GeometrySymmetry Groups	The group of all bijective self-maps of a metric space that preserve every distance, with composition encoding the space's exact metric symmetries.
Isomorphism of categories	Category TheoryCategorical Equivalence	A strict equivalence between categories given by functors whose two composites are exactly the identity functors, producing one-to-one correspondence of objects and morphisms without merely natural isomorphism.
Isomorphism theorem	Abstract Algebra	In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship among quotients, homomorphisms, and subobjects.
Isoparametric manifold	Riemannian GeometrySubmanifold Geometry	Define a Euclidean submanifold with flat normal bundle whose shape operators have constant eigenvalues along every parallel normal field.
Isoperimetric Inequality	Geometric AnalysisCalculus of Variations	The sharp planar inequality L² ≥ 4πA for a closed curve of length L enclosing area A, with equality exactly for a circle.
Isothermal coordinates	Differential Geometry	In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric.
Isotropic Measure	Measure Theory	A measure on Euclidean space invariant under the stipulated linear isometries, so its radial density depends on distance rather than direction.
Iterated Function System	Fractal GeometrySelf Similar and Self Affine Set Construction	Use a finite family of contractions on a complete metric space to induce a contraction on nonempty compact subsets, whose unique fixed set is approached by deterministic set iteration and sampled by coded or random map compositions.
Iterative method	Numerical Analysis	A numerical procedure that repeatedly updates an approximation from prior iterates toward a solution under a convergence rule.
Itô isometry	Stochastic Calculus	The equality identifying the second moment of an Itô integral with the expected time integral of the squared adapted integrand.
Iwasawa decomposition	Lie TheorySemisimple Lie Groups	A factorization G=KAN of a connected real semisimple Lie group into maximal compact, abelian and nilpotent subgroups, generalizing matrix QR decomposition.
Iwasawa group	Group Theory	In mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular.
J-homomorphism	Stable Homotopy TheoryAlgebraic Topology	In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres.
J-multiplicity	Commutative Algebra	A local-algebra multiplicity for an ideal obtained from maximal-ideal-supported graded data, extending Hilbert–Samuel multiplicity beyond m-primary ideals.
J-structure	AlgebraJordan Structures	An algebraic structure taking a rational inversion map and Hua-type identities as primitive, providing a linear-algebraic-group formulation closely equivalent to Jordan algebra theory in suitable characteristic.
Jacobi Method	Numerical Linear AlgebraStationary Iterative Linear Solvers	Solve a linear system by isolating its diagonal and synchronously recomputing every component from the same previous iterate, with the resulting iteration matrix governing convergence.
Jacobi operator	Spectral Theory	A self-adjoint or symmetric tridiagonal operator on a sequence space determined by positive off-diagonal and real diagonal coefficient sequences.
Jacobian variety	Algebraic GeometryCurves	A curve's Jacobian is the abelian variety of its degree-zero line-bundle classes, turning divisor equivalence into algebraic group geometry.
Jaffard ring	Commutative Algebra	In commutative algebra, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions.
Jensen's Inequality	Convex AnalysisProbability	For a convex function, the function of a mean is no greater than the mean of the function's values, under the required domain and expectation conditions.
Jet Group	Differential GeometryJet Bundles	The group of fixed-origin invertible Taylor germs truncated at order k, with group law induced by composition of local coordinate changes.
John Ellipsoid	Convex GeometryGeometry of Banach Spaces	The unique maximum-volume ellipsoid inscribed in a full-dimensional compact convex body, providing a centered geometric approximation whose bound depends on symmetry.
Join (graph theory)	Graph Theory	A graph operation that takes two disjoint graphs and adds every possible edge between their vertex sets while retaining their internal edges.
Join (simplicial sets)	Simplicial and Higher Category Theory	A monoidal operation combining two simplicial sets so simplices consist of an ordered simplex from the first followed by one from the second, corresponding under realization to topological join.
Join and meet	Order and Lattice Theory	The least upper bound and greatest lower bound, respectively, of a subset in a partially ordered set when those bounds exist.
Join of Categories	Category TheoryHigher Category Theory	The small-category construction that preserves two input categories, adds one morphism from every left object to every right object and none backward, and forms an associative monoidal product.
Jordan Identity	Jordan Algebra TheoryNonassociative Algebra	The nonassociative polynomial law (x²∘y)∘x=x²∘(y∘x), imposed with commutativity to define Jordan algebras and support coherent powers.
Jordan operator algebra	Operator Algebras	A normed Jordan algebra modeled on self-adjoint operators with the symmetrized product a circle b equal to one half of ab plus ba.
Jordan's totient function	Number TheoryArithmetic Functions	The multiplicative arithmetic function J_k(n) counting ordered k-tuples modulo n that are jointly coprime with n, equivalently n^k times the product of 1−p^(−k) over primes dividing n.
Josephus problem	Discrete Mathematics	The recurrence problem of locating the survivor or elimination order when positions in a circle are removed at a fixed counting interval.
Joyal Model Structure	Higher Category TheoryQuasi Categories	The model structure on simplicial sets whose cofibrations are monomorphisms, weak equivalences are categorical equivalences, and fibrant objects are quasi-categories.
JSJ decomposition	Geometric Topology	A canonical decomposition of an irreducible orientable compact 3-manifold along incompressible tori into atoroidal and Seifert-fibered pieces.
Julia set	Complex Dynamics	In complex dynamics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function.
K-Distribution	ProbabilityCompound Distributions	A compound positive-valued distribution obtained by mixing fast speckle with a gamma-distributed local mean, producing a Bessel-K density with convention-dependent tail behavior.
K-function	Special Functions	The special function extending the hyperfactorial to complex arguments through a functional equation involving powers and the gamma function.
K-Homology	K Theory and HomologyNoncommutative GeometryOperator Algebras	A generalized homology theory for locally compact Hausdorff spaces, represented analytically by equivalence classes of even or odd Fredholm modules over associated C*-algebras.
K-noid	Minimal Surface Theory	A genus-zero minimal surface with k catenoidal ends, topologically a sphere punctured at k points.
K-space (functional analysis)	Functional Analysis	In mathematics, more specifically in functional analysis, a K-space is an F-space V such that every extension of F-spaces (or twisted sum) of the form.
K-theory	Algebraic TopologyAlgebraic GeometryHomological Algebra	A family of functorial invariants that forms groups or spectra from stable classes of vector bundles, projective modules, automorphisms, or related structures, connecting topology, algebraic geometry, operator algebras, and index theory.
Kakeya Set	Geometric Measure TheoryKakeya Problems	A subset of Euclidean space containing a unit line segment in every direction, whose directional coverage can coexist with vanishing measure and extreme geometric overlap.
Kan extension	Category TheoryUniversal Constructions	A universal way to extend a functor along another functor, with left and right Kan extensions respectively initial and terminal among compatible factorizations.
Kan Fibration	Simplicial Homotopy TheoryAlgebraic Topology	A Kan fibration is a simplicial-set map that lifts every compatible horn over a specified simplex in its target.
Kaplan–Yorke map	Dynamical SystemsChaotic Maps	A two-dimensional skew-product chaotic map coupling the doubling map x↦2x mod 1 to a driven contraction or expansion y↦αy+cos(4πx), with dynamics controlled by one parameter α.
Kaprekar number	Number TheoryDigit Defined Integer Sequences	Classify a base-b natural number whose square can be split at a declared digit position into two parts that sum back to the number.
Karhunen–Loève theorem	Probability TheoryStochastic Processes	The importance of the Karhunen–Loève theorem is that it yields the best such basis in the sense that it minimizes the total mean squared error.
Karlsruhe Metric	Metric GeometryRadial Orbital Path Metrics	A planar path metric with a distinguished origin in which travel is restricted to radial segments and origin-centered circular arcs, so shortest routes switch at two radians between an inner-radius arc route and a route through the origin.
Karoubi envelope	Category TheoryCategorical Completions	The universal idempotent completion of a category, adjoining an image object for every idempotent morphism so that all idempotents split.
Keith number	Recreational Number Theory	A natural number whose base-b digits seed a k-step Fibonacci-like recurrence that later reproduces the number.
Keller–Segel System	Mathematical BiologyPartial Differential Equations	A family of coupled nonlinear partial differential-equation models in which a population diffuses and moves along a chemical gradient while the chemoattractant diffuses, is produced, and may decay.
Kelly's Lemma	Stochastic ProcessesQueueing Theory	Certify a continuous-time Markov chain's stationary law by matching each forward transition's weighted flow to a candidate reverse transition and matching their statewise exit rates.
Kelvin functions	Special FunctionsBessel Functions	Special functions ber, bei, ker, and kei defined as real and imaginary parts of Bessel or modified-Bessel functions on specified rotated complex arguments.
Kelvin transform	Potential TheoryHarmonic Functions	The Kelvin transform is a device used in classical potential theory to extend the concept of a harmonic function, by allowing the definition of a function which is 'harmonic at infinity'.
Kemnitz's Conjecture	Additive Number TheoryCombinatorics	In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point.
Kernel	Linear Algebra	The set of inputs a structure-preserving map sends to the identity of its target — the map's null directions — collected as a genuine subobject that carries the map's information loss and, via triviality, rank-nullity, and the isomorphism theorem, controls injectivity and reconstructs the faithful part.
Kernel (category theory)	Category Theory and Homological Algebra	The universal morphism into an object's domain that is annihilated by a given morphism, equivalently the equalizer of that morphism and zero in a category with zero morphisms.
Khintchine inequality	Probability and AnalysisRandom Series Inequalities	Two-sided bounds comparing the Lp norm of a Rademacher random sum with the ℓ2 norm of its coefficients, using constants depending only on p.
Killing form	Lie TheoryInvariant Bilinear Forms on Lie Algebras	Pair two elements of a finite-dimensional Lie algebra by tracing the composition of their adjoint endomorphisms, obtaining a canonical symmetric invariant bilinear form whose degeneracy diagnoses structure.
Killing spinor	Spin GeometrySpecial Spinor Fields	A spinor field on a spin manifold satisfying ∇Xψ=λX·ψ for every tangent vector, linking special curvature and holonomy to preserved supersymmetry when interpreted in supergravity.
Kinematic Wave	Continuum TransportNonlinear Conservation Laws	A disturbance in a conserved one-dimensional flow that propagates according to the slope of an approximate local flux–concentration relation.
Kirwan map	Symplectic GeometryHamiltonian Reduction and Equivariant Cohomology	Restrict equivariant cohomology classes of a Hamiltonian group space to a regular moment-map level set and identify them with ordinary cohomology classes on the resulting symplectic quotient.
Kissing number	Discrete Geometry	The maximum number of nonoverlapping congruent spheres that can simultaneously touch one congruent central sphere in a specified space or dimension.
Kite (geometry)	Euclidean Geometry	In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal.
KK-Theory	Operator AlgebrasNoncommutative Geometry	KK-theory connects pairs of C*-algebras through equivalence classes of analytic cycles and composes those classes by the Kasparov product.
Kleene–Brouwer Order	Mathematical Logic	Linearize finite sequences so every proper extension precedes its prefix and incomparable sequences follow the first differing coordinate, turning branch structure into an order whose well-foundedness detects infinite paths.
Kline sphere characterization	Geometric Topology	A theorem characterizing the topological two-sphere by how simple closed curves and pairs of points separate a compact connected space.
Knaster's condition	Set TheoryForcing Chain Conditions	A chain condition on a partial order requiring every uncountable subset to contain an uncountable pairwise-compatible or linked subset.
Knaster–Kuratowski–Mazurkiewicz Lemma	Fixed Point and Convexity Theory	A simplex-covering intersection theorem: if each face is covered by the closed sets indexed by that face’s vertices, then every indexed set shares a common point, converting boundary-compatible local coverage into global coexistence.
Knot invariant	Knot TheoryKnot Classification	A quantity, algebraic object or property assigned to a knot that is unchanged under the chosen knot-equivalence relation and can distinguish some inequivalent knots.
Knot Polynomial	Knot TheoryPolynomial Invariants	Compress a knot or link's ambient-isotopy class into a normalized polynomial or Laurent polynomial whose equality can obstruct nonequivalence, while never treating polynomial equality as a complete classification.
Knuth–Eve Algorithm	Computer AlgebraPolynomial Evaluation	A preconditioned polynomial-evaluation scheme that rewrites a fixed polynomial into nested quadratic factors to reduce runtime multiplications.
Knödel number	Number TheoryPseudoprime Classes	For a fixed positive integer n, a composite integer m such that a^(m−n) is congruent to one modulo m for every integer a coprime to m.
Koenigs function	Complex Dynamics	A holomorphic change of coordinate that conjugates iteration near an attracting or repelling fixed point to multiplication by the fixed-point multiplier.
Kolmogorov Equations for Continuous-Time Markov Chains	Probability TheoryMarkov ProcessesStochastic Processes	Forward and backward generator equations governing how transition probabilities, distributions, or expectations evolve in continuous-time Markov jump and diffusion processes.
Kolmogorov's criterion	Markov Chain Theory	A cycle-product condition characterizing when a Markov chain is reversible with respect to a positive stationary measure.
Kolmogorov's Three-Series Theorem	Probability Theory	An if-and-only-if test for almost-sure convergence of an independent random series using large-jump probabilities, truncated means, and truncated variances.
Kolmogorov's Two-Series Theorem	Probability Theory	Independent finite-variance random terms have an almost-surely convergent sum when their mean series converges and their total variance is finite.
Korn's inequality	Mathematical AnalysisElasticity Inequalities	A rigidity inequality bounding the full gradient of a vector field, modulo rigid motions, by its symmetric gradient under specified domain and boundary conditions.
Kosmann lift	Differential Geometry	The canonical metric-dependent lift of a vector field on a Riemannian manifold to the orthonormal frame bundle, enabling a Lie derivative of spinors.
Kostant partition function	Representation Theory	A function counting the ways a weight can be expressed as a nonnegative integer combination of the positive roots of a root system.
Kostka number	Algebraic CombinatoricsSymmetric FunctionsRepresentation Theory	The nonnegative integer K_{λμ} counting semistandard Young tableaux of shape λ and weight μ, also a Schur-to-monomial coefficient.
Koszul algebra	Homological Algebra	A graded algebra whose ground field admits a minimal graded free resolution that is linear in every homological degree.
Koszul–Tate resolution	Homological Algebra	A differential graded commutative algebra resolution of a quotient ring that generalizes the Koszul complex by adding generators to kill successive homology.
KR-theory	Algebraic Topology	A real-equivariant form of topological K-theory classifying complex vector bundles equipped with compatible conjugate-linear involution over an involutive space.
Kramers–Moyal expansion	Stochastic Processes	An expansion of a Markov process master equation into an infinite series of state derivatives weighted by conditional jump moments.
Kripke–Platek set theory	Mathematical LogicAxiomatic Set Theory	A weak axiomatic set theory centered on bounded separation and collection, used to formalize admissible sets and the predicative or recursion-theoretic fragment of set theory.
Kripke–Platek set theory with urelements	Set TheoryAdmissible Recursion TheoryProof Theory	The Kripke–Platek set theory with urelements (KPU) is an axiom system for set theory with urelements, based on the traditional (urelement-free) Kripke–Platek set theory.
Krull–Schmidt category	Category Theory	An additive category in which every object decomposes into finitely many indecomposables uniquely up to permutation and isomorphism.
Kurepa tree	Set Theory	An uncountable tree of height omega-one with countable levels and at least omega-two many cofinal branches.
Kushner–Stratonovich Equation	Nonlinear Stochastic Filtering	Evolve a hidden continuous-time state's normalized conditional law by combining generator-driven prediction with an observation-filtration innovation correction weighted by conditional covariance.
Kähler differential	Commutative Algebra	The universal module-valued derivation that algebraically represents first-order differentiation for a ring map.
Kōmura's theorem	Functional Analysis	An almost-everywhere differentiability theorem for absolutely continuous functions taking values in a reflexive Banach space.
L(R)	Set TheoryInner Models and Determinacy	The smallest transitive inner model of ZF containing every ordinal and every real, constructed by iterating definability from the real numbers and used to study determinacy under large-cardinal assumptions.
L-infinity	Functional Analysis	The Banach space of essentially bounded measurable functions under the essential-supremum norm, with bounded sequences as the counting-measure case.
L-Matrix	Matrix TheorySign Pattern Matrices	A matrix whose diagonal entries are positive and whose off-diagonal entries are nonpositive.
L-notation	Computational Number TheorySubexponential Algorithm Analysis	A two-parameter asymptotic scale that records the subexponential exponent and leading logarithmic constant of number-theoretic algorithms.
L-semi-inner product	Functional Analysis and Banach Space Geometry	A Banach-space generalization of inner product that is linear in one argument and positive but need not be conjugate symmetric or additive in the other.
L-theory	Algebraic Topology	An algebraic theory of quadratic and symmetric forms whose L-groups classify surgery obstructions and stable equivalence over rings with involution.
Laakso Space	Analysis on Metric Spaces	A Laakso-construction metric-measure space that is Ahlfors Q-regular for prescribed Q greater than one and supports a weak Poincare inequality despite potentially non-integer Hausdorff dimension.
Labeled graph	Graph Theory	A graph equipped with one or more functions assigning labels from declared sets to vertices, edges, or both, under explicit semantic and constraint conventions.
Lacunary value	Complex Analysis	A complex number omitted from the image of a complex-valued function on its stated domain.
Lady Windermere's Fan	Numerical AnalysisError Analysis	A telescoping identity decomposing global numerical error into locally introduced defects carried forward by later evolution.
Lagrange Stability	Dynamical SystemsOrbit Boundedness and Recurrence Theory	Classify a dynamical motion as Lagrange stable when the closure of its relevant forward, backward, or two-sided orbit is compact—equivalently, in Euclidean state space, when that orbit remains bounded.
Laguerre Formula	Projective Geometry	A projective-geometry formula recovering the acute angle between two real lines from the cross-ratio of their ideal points and absolute-conic intersections.
Lah number	CombinatoricsConnection Coefficients	A signed or unsigned combinatorial coefficient converting rising factorials to falling factorials and counting partitions of n labeled elements into k nonempty linearly ordered lists.
Lambek–Moser theorem	Number Theory	A theorem constructing complementary integer sequences from generalized inverse nondecreasing functions.
Lambert series	Analytic Number Theory	A generating series of the form sum a_n q^n divided by one minus q^n, whose expanded coefficients are divisor sums of the original sequence.
Lanczos Approximation	Numerical Analysis	Evaluate the gamma function at fixed precision by factoring out its dominant asymptotic behavior and approximating the remaining analytic factor with a short precomputed rational sum.
Laplace functional	Probability TheoryPoint Processes	A probability functional—most commonly f↦E[e^(−∫f dN)] for a point process—that transforms test functions to characterize a random measure, with a distinct named variant in metric concentration.
Large deviations of Gaussian random functions	Probability Theory	The asymptotic study of rare high excursions of Gaussian processes or fields over large domains or thresholds.
Large deviations theory	Probability Theory	Thus to estimate the premium you have to ask the following question: "What should we choose as the premium q such that over N months the total claim C = \Sigma X_i should be less than This is clearly the same question asked by the large deviations theory.
Large Set (Combinatorics)	Additive Combinatorics	A large set of positive integers has a divergent sum of reciprocals, a criterion finer than merely being infinite or having positive density.
Large Sieve	Analytic Number Theory	Bound the collective size of finite exponential sums at separated arithmetic frequencies, then use that mean-square control to limit modularly sifted sets.
LaSalle's Invariance Principle	Dynamical Systems	Conclude that trajectories approach the largest invariant subset of the region where a nonincreasing Lyapunov function stops decreasing, extending asymptotic-stability proofs beyond strictly negative derivatives.
Lattice (discrete subgroup)	Topological GroupsLie Groups	A lattice in a locally compact topological group is a discrete subgroup whose quotient has finite invariant measure, with uniform lattices distinguished by compact quotient.
Lattice graph	Graph TheoryGeometric and Grid Graphs	Represent a regular Euclidean lattice or tiling by vertices at lattice sites and edges joining sites under a fixed local-neighbor rule.
Laurent Polynomial	Commutative Algebra	A Laurent polynomial is a finite linear combination of integer powers of one or more variables, allowing negative as well as nonnegative exponents.
Law of continuity	History and Philosophy of Mathematics	Leibniz's heuristic that rules valid across finite cases may be extended consistently to limiting, infinitesimal or infinite cases, provided the resulting transition preserves the relevant relations.
Law of sines	Trigonometry and Triangle Geometry	Relate each side of a triangle to the sine of its opposite angle through one common ratio equal to the circumdiameter in Euclidean geometry, enabling triangle solution while preserving the side-side-angle ambiguous case and curvature-specific variants.
Law of total covariance	Probability Theory	The identity decomposing covariance into expected conditional covariance plus covariance of conditional expectations.
Law of total probability	ProbabilityConditioning Identities	A probability identity expressing an event's probability as the sum or integral of its conditional probabilities over a mutually exclusive exhaustive partition.
Law of trichotomy	Order TheoryLinear Orders	The order principle that for every two elements exactly one of x<y, x=y or y<x holds, equivalently combining connectedness with asymmetry for a strict order.
Lawson topology	Domain TheoryOrder Topologies	The common refinement of the Scott topology and lower topology on a poset, combining approximation-sensitive opens with complements of principal upper sets.
Least Common Multiple	Number TheoryElementary Arithmetic	Select the unique positive common multiple that divides every other common multiple of given positive integers.
Lebesgue covering dimension	Dimension Theory	The least integer n such that every open cover of a topological space has an open refinement of order at most n+1.
Lebesgue's Density Theorem	Measure Theory	Lebesgue's density theorem says that a measurable set occupies almost all sufficiently small neighborhoods at almost every point inside it, and almost none at almost every point outside it.
Lebesgue's lemma	Approximation Theory	An approximation bound stating that a bounded linear projection's error is at most one plus its operator norm times the best attainable error from the target subspace.
Lefschetz duality	Algebraic TopologyManifold Duality Theorems	A manifold-with-boundary extension of Poincaré duality pairing absolute cohomology with relative homology, and relative cohomology with absolute homology, through the relative fundamental class.
Lefschetz zeta function	Fixed Point TheoryDynamical SystemsAlgebraic Topology	In mathematics, the Lefschetz zeta-function is a tool used in topological periodic and fixed point theory, and dynamical systems.
Leibniz Operator	Abstract Algebraic Logic	An operator assigning each designated set in an algebra its greatest compatible congruence.
Leimkuhler–Matthews method	Stochastic Numerics	A discretization of overdamped Langevin dynamics using correlated noise to improve configurational sampling accuracy.
Length of a module	Commutative AlgebraModule Theory	In algebra, the length of a module over a ring R is a generalization of the dimension of a vector space which measures its size. page 153 It is defined to be the length of the longest chain of submodules.
Lens space	Geometric TopologyThree Manifolds	A closed manifold obtained by a cyclic quotient of an odd-dimensional sphere, or in dimension three by gluing two solid tori with a slope specified by coprime integers p and q.
Leonardo number	Discrete Mathematics	A recurrence sequence beginning with one and one in which each later term is the sum of the previous two plus one.
Levitzky's theorem	Ring TheoryNoncommutative Algebra	In a right Noetherian ring, every nil left or right ideal has one power that vanishes as an ideal.
Lie Algebra Extension	Lie Theory	A Lie algebra fitting into a short exact sequence with a specified ideal as kernel and a specified Lie algebra as quotient, with splitting and cohomology encoding how the parts combine.
Lie Bracket of Vector Fields	Differential GeometryLie Theory	The Lie bracket is an R-bilinear operation and turns the set of all smooth vector fields on the manifold M into an (infinite-dimensional) Lie algebra.
Lie coalgebra	Algebra	A vector space with a skew-symmetric cobracket satisfying the co-Jacobi identity, dual to a Lie algebra in finite dimensions.
Lie group	Lie TheoryDifferential Geometry	In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
Lie Sphere Geometry	Lie Sphere Geometry	Lie sphere geometry encodes oriented spheres, oriented hyperplanes and point spheres on a projective quadric so transformations preserve oriented contact.
Lie Superalgebra	Lie TheoryGraded Algebra	A parity-graded Lie structure whose bracket obeys graded skew-symmetry and the graded Jacobi identity.
Lifting Scheme	Wavelet Analysis	Construct or implement a wavelet transform through ordered, locally reversible updates between complementary coefficient subsets.
Lifting theory	Measure Theory	The study of selectors that choose pointwise measurable representatives of equivalence classes modulo null sets while preserving algebraic and order structure.
Lightface Pointclass	Descriptive Set Theory	In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a point is ordinarily understood to be an element of some perfect Polish space.
Limit (Category Theory)	Category Theory	A terminal cone over a diagram, through which every other cone factors by a unique mediating morphism.
Limited-Memory BFGS	Numerical OptimizationQuasi Newton Methods	A quasi-Newton optimizer that stores a short history of step and gradient-difference pairs and applies the implied inverse-Hessian approximation by two-loop recursion.
Limiting Absorption Principle	Spectral TheoryScattering Theory	A conditional spectral theorem obtaining resolvent boundary values at real energies by approaching from nonreal parameters in adapted operator spaces.
Lindenbaum–Tarski algebra	Algebraic Logic	The quotient algebra of formulas or sentences of a logical theory by provable equivalence, with logical connectives inducing well-defined algebraic operations on equivalence classes.
Lindström quantifier	Mathematical LogicGeneralized Quantifier TheoryModel Theory	A Lindström quantifier generalizes first-order quantification by declaring a class of relational structures and treating satisfaction of a formula-generated structure's membership in that class as a logical quantifier operation.
Lindström–Gessel–Viennot Lemma	Enumerative CombinatoricsAlgebraic Combinatorics	A determinant identity equating path-matrix minors in a weighted directed acyclic graph with a signed sum of vertex-disjoint path families between chosen sources and destinations.
Line Bundle	Geometry and TopologyAlgebraic GeometryDifferential Geometry	A locally trivial rank-one vector bundle whose one-dimensional fibers are glued over a base by invertible scalar transition functions, allowing local products to carry nontrivial global twisting.
Line graph of a hypergraph	Hypergraph Theory	The graph whose vertices are a hypergraph’s hyperedges and whose adjacency records nonempty intersection between the corresponding hyperedges.
Line group	Mathematical CrystallographyOne Dimensional Periodic Symmetry	A discrete symmetry group of a structure periodic along one spatial axis, combining axial translations or screw operations with compatible point symmetries.
Linear algebraic group	Algebraic Geometry	A matrix group defined over a field by polynomial equations in its entries and inverse determinant conditions.
Linear Canonical Transformation	Integral TransformsSymplectic Analysis	A symplectic-matrix-indexed family of wave operators that transforms functions while preserving the canonical structure of a conjugate phase plane.
Linear complex structure	Linear Algebra	A real-linear endomorphism J of a real vector space satisfying J squared equals minus the identity, thereby defining multiplication by complex scalars.
Linear Disjointness	Field TheoryCommutative Algebra	In mathematics, algebras A, B over a field k inside some field extension \Omega of k are said to be linearly disjoint over k if the following equivalent conditions are met.
Linear dynamical system	Dynamical Systems	A dynamical system whose state evolution and output laws are linear, enabling superposition and analysis through matrices, spectra, and modes.
Linear fractional transformation	Complex Analysis and Projective Linear Actions	An invertible map represented by a ratio of two linear expressions, including Mobius transformations over suitable scalar or algebraic settings.
Linear group	Group TheoryLinear and Matrix Groups	Characterize a group by the existence of a faithful finite-dimensional representation over a specified field, equivalently by its realization as a subgroup of a general linear matrix group.
Linear least squares	Numerical Linear Algebra	Approximation of an overdetermined or rank-deficient linear system by choosing parameters that minimize a quadratic residual norm.
Linear map	Linear AlgebraVector Space Morphisms	A function between vector spaces that preserves vector addition and scalar multiplication, equivalently preserving every finite linear combination.
Linear matrix inequality	Convex Optimization	A convex constraint requiring an affine combination of symmetric or Hermitian matrices to be positive semidefinite.
Linear Operator	Functional AnalysisLinear Operators	A linear operator is a map from a linear subspace of a vector space to another vector space that preserves vector addition and scalar multiplication, with domain, codomain, topology, boundedness, closure, and adjoint conditions declared when relevant.
Linear order	Order Theory	In mathematics, a total order or linear order is a partial order in which any two elements are comparable.
Linearly ordered group	Abstract AlgebraOrdered Algebraic Structures	In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant.
Line–line intersection	Computational and Euclidean Geometry	The classification and computation of the common-point set of two lines under a declared geometry, yielding no point, one point, or coincident lines in ordinary Euclidean settings.
Link (knot theory)	Knot TheoryLinks	A finite disjoint union of smoothly or tamely embedded circles in three-dimensional space, considered up to ambient isotopy, with a knot as the one-component case.
Link concordance	Knot Theory	An equivalence between links whose components cobound disjoint embedded cylinders in one higher-dimensional spacetime.
Linking number	Knot Theory	An oriented integer invariant measuring how many times one disjoint closed curve winds around another in three-dimensional space.
Linnik distribution	Probability TheoryProbability Distributions	A symmetric heavy-tailed probability distribution whose characteristic function has Linnik form.
Liouville's theorem (differential algebra)	Symbolic Integration	A differential-algebra theorem restricting the form of an elementary antiderivative and thereby proving many elementary functions have no elementary primitive.
Liouvillian function	Differential Algebra	A function obtainable through a finite tower of algebraic extensions, exponentials, logarithms and antiderivatives over a differential field.
Lipschitz continuity	Mathematical Analysis	In mathematical analysis, Lipschitz continuity is a regularity property of functions between metric spaces that is stronger than uniform continuity, and hence also stronger than continuity.
List coloring	Graph ColoringGraph Theory	List coloring denotes generalization of graph coloring in which each vertex has a list of allowed colors in graph coloring.
Literal (Mathematical Logic)	Mathematical Logic	Use an atomic formula or its negation as a signed atomic unit whose polarity supports clauses, normal forms, resolution, satisfiability assignments, and complementary-pair reasoning.
Lobb number	Enumerative Combinatorics	A two-parameter ballot number counting prefixes of balanced-parenthesis paths with a specified excess of open parentheses, generalizing Catalan numbers.
Local Analysis	Number TheoryAlgebraic Geometry	In algebraic geometry and related areas of mathematics, local analysis is the practice of looking at a problem relative to each prime number p first, and then later trying to integrate the information gained at each prime into a 'global' picture.
Local boundedness	Mathematical AnalysisLocal Properties	A property requiring a function, family or operator to remain bounded on some neighborhood of every point, without requiring one global bound over the whole domain.
Local class field theory	Algebraic Number Theory	The theory that classifies finite abelian extensions of a local field through a reciprocity map from the field's multiplicative group to its abelian Galois group.
Local complementation	Graph TheoryGraph Transformations	A graph operation that toggles every adjacency among neighbors of a selected vertex while leaving the selected vertex and all other adjacencies unchanged.
Local diffeomorphism	Differential Topology	Map smooth manifolds so that every source point has a neighborhood carried diffeomorphically onto an open target neighborhood, without requiring global injectivity.
Local field	Algebraic Number TheoryTopological Fields	A nondiscrete locally compact Hausdorff topological field, equivalently in the non-Archimedean case a complete discretely valued field with finite residue field, serving as a completion-scale model of global arithmetic.
Local martingale	Stochastic Processes	A stochastic process that becomes a martingale when stopped along an increasing sequence of stopping times tending to the time horizon.
Local property	—	A property that holds around each point or on sufficiently small neighborhoods, even if a corresponding global property may fail.
Local Tate Duality	Arithmetic Geometry	Local Tate duality perfectly pairs complementary-degree Galois cohomology of a finite module and its dual over a p-adic local field.
Local Time (Mathematics)	Stochastic Calculus	The selected level density of a continuous semimartingale's quadratic-variation-weighted occupation, tracked as time advances.
Localization (commutative algebra)	Commutative Algebra	The construction that formally inverts a multiplicative subset of a commutative ring or module, creating fractions that focus algebra on a chosen region or prime.
Localization formula for equivariant cohomology	Equivariant CohomologyAlgebraic Topology	The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers.
Localization of a category	Category Theory	A universal construction that formally makes a chosen class of morphisms invertible in a category.
Localizing Subcategory	Category TheoryAbelian Categories and Gabriel Localization	A Serre subcategory whose exact quotient functor admits a right-adjoint section, so the objects it annihilates define a recoverable Gabriel localization of an abelian category.
Locally catenative sequence	Combinatorics on Words	A locally catenative sequence is a word sequence governed by a fixed recurrence in which each sufficiently late word is the concatenation, in fixed order, of specified earlier words.
Locally Closed Subset	Topology	A subset of a topological space that is open within its own closure, equivalently the intersection of an ambient-open and ambient-closed set.
Locally Discrete Collection	General TopologyMetrization Theory	A family of subsets of a topological space for which every point has an open neighborhood meeting at most one family member.
Locally finite measure	Measure Theory	In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.
Locally Hausdorff space	General TopologySeparation Axioms	A topological space in which every point has a neighborhood that is Hausdorff in its subspace topology, allowing locally unique limits while global point separation can still fail.
Locally integrable function	Real AnalysisFunction Spaces	A measurable function whose absolute value has finite integral on every compact subset of its open domain, without requiring finite integral over the whole domain.
Locally nilpotent	Algebra	A local finiteness condition under which every finitely generated subobject is nilpotent, or an ideal becomes nilpotent after localization at a specified prime.
Locally Optimal Block Preconditioned Conjugate Gradient	Numerical Linear AlgebraIterative Eigenvalue Methods	Compute a few extreme eigenpairs of a large Hermitian-definite problem by repeatedly Rayleigh–Ritz optimizing a block over current Ritz vectors, preconditioned eigen-residuals, and compressed prior directions.
Locally profinite group	Topological GroupsRepresentation Theory	In mathematics, a locally profinite group is a Hausdorff topological group in which every neighborhood of the identity element contains a compact open subgroup.
Locally simply connected space	Algebraic Topology	Require every point to have a neighborhood basis of simply connected open sets, separating local loop triviality from global simple connectedness and weaker semilocal conditions.
Location–scale family	Probability Theory	A family of probability distributions closed under positive affine transformations of a fixed standardized random variable.
Loeb space	Nonstandard Measure Theory	A standard countably additive measure space constructed from an internal finitely additive measure in nonstandard analysis by taking standard parts and completing the induced measure.
Log-polar coordinates	Coordinate Geometry	A planar coordinate system whose angular coordinate is polar angle and whose radial coordinate is the logarithm of distance from a chosen origin.
Logarithm	Elementary and Complex Analysis	The inverse of exponentiation with a fixed admissible base, assigning to a positive input the exponent needed to produce it.
Logarithmic Form	Complex Geometry	A complex meromorphic differential form whose form and exterior derivative have controlled first-order behavior along a specified reduced divisor.
Logarithmic mean	Mathematical AnalysisTwo Variable Means	Average two positive numbers by their difference divided by the difference of their logarithms, using the continuous value x when the arguments coincide.
Logarithmic pair	Algebraic GeometryBirational Geometry	In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities.
Logarithmically concave measure	Convex GeometryMeasure Theory	In mathematics, a Borel measure μ on n-dimensional Euclidean space \mathbb{R}^{n} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of \mathbb{R}^{n} and 0 < λ < 1, one has.
Logic Puzzle	Recreational Mathematics	A logic puzzle is a deliberately constructed problem that presents entities, states, clues, and explicit or inferable constraints and asks a solver to derive a required configuration, classification, quantity, or explanation primarily through valid deduction and exhaustive consistency rather than hidden factual knowledge.
Logical equality	Propositional Logic	A truth-functional connective that is true exactly when its two propositions have the same truth value.
Logical NOR	Boolean Logic	The Boolean connective that is true exactly when all its operands are false, equivalently the negation of disjunction; repeated NOR alone is functionally complete for propositional logic.
Logical Operation	Mathematical LogicFormal Semantics	A logical operation is a rule-governed transformation or interpretation that maps typed truth values, propositions, formulas, terms, or formally specified program values to an output according to declared semantic or inferential rules.
Logical or	Propositional Logic	The inclusive disjunction connective: classically false only when every disjunct is false, and otherwise governed by the declared logic’s semantic and proof rules.
Logrank test	Survival AnalysisTwo Sample Tests	A nonparametric hypothesis test comparing the event-time distributions of groups by accumulating observed-minus-expected events across ordered failure times while accounting for right censoring.
Lollipop Graph	Graph Theory	In the mathematical discipline of graph theory, the (m,n)-lollipop graph is a special type of graph consisting of a complete graph (clique) on m vertices and a path graph on n vertices, connected with a bridge.
Lommel polynomial	Special Functions	A polynomial in the reciprocal argument that expresses shifted-order Bessel functions through a two-term basis of neighboring Bessel orders.
Lomonosov's invariant subspace theorem	Functional Analysis and Operator Theory	A theorem stating that every bounded operator on an infinite-dimensional complex Banach space that commutes with a nonzero compact operator has a nontrivial closed invariant subspace.
Long and Short Scales	Number Nomenclature	Two large-number naming conventions in which successive -illion names advance by factors of one thousand or one million after the shared name million.
Loop (Algebra)	Abstract AlgebraQuasigroup Theory	A quasigroup with a two-sided identity: multiplication has uniquely solvable left and right division without requiring associativity.
Loop (Graph Theory)	Graph Theory	A graph edge whose two endpoints are the same vertex, admitted or excluded according to the chosen graph category.
Loop Group	Lie TheoryInfinite Dimensional Groups	A group of maps from a circle into a Lie group under pointwise multiplication, often equipped with smoothness, based-loop, and central-extension structure.
Lorden's Inequality	Renewal TheorySequential Analysis	Lorden's inequality bounds the mean excess at first crossing of any nonnegative threshold by a positive-part second moment divided by the positive drift of iid increments.
Lorenz System	Dynamical SystemsNonlinear Dynamics	A parameterized three-state nonlinear rate-law model that separates deterministic dynamics from parameter-dependent instability and chaos.
Loss Function	Decision Theory	A real-valued rule assigning penalty to an action or prediction under a realized state or target, whose expectation or sample aggregate defines the risk to minimize.
Lottery mathematics	Applied Probability	The combinatorial and probabilistic analysis of lottery drawings, prize tiers, expected returns and apparent coincidences under declared game rules.
Low-rank matrix approximations	Numerical Linear AlgebraMatrix Approximation	A rank-limited matrix surrogate evaluated by its residual error and computational purpose.
LU Decomposition	Numerical Linear Algebra	A square-matrix factorization into lower- and upper-triangular factors, with permutations when needed.
Lucky number	Number Theory	A natural number surviving an iterative positional sieve that repeatedly deletes every kth remaining number.
Ludics	Proof TheoryLinear Logic	Reconstruct logical propositions and proofs from address-based designs, polarized actions, and successful interaction, defining meaning extensionally through orthogonality rather than presupposed formulas.
Luhn formula	—	The Luhn formula is the decimal modulus-10 check-digit algorithm that alternately doubles digits, reduces two-digit products, and selects a final digit making the transformed sum divisible by ten.
Lunar arithmetic	Recreational Number TheoryCarryless Idempotent Arithmetic	Replace digit addition by maximum and digit multiplication by minimum, extending them positionally without carries so nonnegative base-b numerals form a closed idempotent arithmetic with altered sums, products, factors, and primes.
Lyapunov Exponent	Dynamical SystemsMultiplicative Ergodic Theory	The asymptotic logarithmic growth or contraction rate of infinitesimal tangent perturbations along a dynamical trajectory, yielding a directional spectrum whose largest member measures the fastest local loss of predictability under stated existence and sampling conditions.
Lyndon–Hochschild–Serre spectral sequence	Homological AlgebraGroup Cohomology	A spectral sequence that computes or constrains the homology or cohomology of a group from a normal subgroup, the quotient group and the quotient action on the subgroup's (co)homology.
Lévy's continuity theorem	Probability Theory	Lévy's continuity theorem equates convergence in distribution of probability measures with pointwise convergence of their characteristic functions, subject to continuity of the limiting function at zero.
Lévy's modulus of continuity theorem	Probability TheoryStochastic Processes	Lévy's modulus of continuity theorem is a theorem that gives a result about an almost sure behaviour of an estimate of the modulus of continuity for Wiener process, that is used to model what's known as Brownian motion.
Lévy–Prokhorov metric	Probability TheoryMetrics on Probability Measures	A distance between probability measures on a metric space that permits both spatial enlargement and a matching probability slack, metrizing weak convergence on separable spaces and connecting compactness to tightness.
M-matrix	Linear Algebra	A real Z-matrix expressible as a nonnegative scalar multiple of the identity minus a nonnegative matrix with scalar at least its spectral radius.
Mac Lane's coherence theorem	Category Theory	In a monoidal category, every well-formed diagram composed only of associators and left or right unitors commutes; equivalently, every monoidal category is monoidally equivalent to a strict one.
Macbeath Region	Convex Geometry	The centrally symmetric neighborhood of a point inside a convex body, formed by intersecting the body with its reflection about that point.
Maclaurin's Inequality	Inequality TheorySymmetric Functions	The descending chain of root-normalized elementary symmetric means of a finite nonnegative real vector.
Magic square	Recreational MathematicsCombinatorics	In mathematics, especially historical and recreational mathematics, a magic square is a square array of numbers, usually positive integers, where the sums of the numbers in each row, each column, and both main diagonals are the same.
Maharam Algebra	Measure TheorySet Theory	A complete Boolean algebra admitting a strictly positive continuous submeasure, whether or not it admits a measure.
Mahler measure	Algebraic Number Theory	A multiplicative height-like measure of a polynomial equal to its leading coefficient magnitude times the moduli of roots outside the unit circle.
Mal'cev's criterion	Lie TheoryLattices in Lie Groups	In differential geometry, Mal'cev's criterion, proved by Anatoly Mal'cev, states that a simply connected nilpotent Lie group admits a lattice, i.e., a discrete co-compact subgroup, if and only if the associated Lie algebra admits a basis such that the structure constants are rational.
Malcev-admissible algebra	Nonassociative Algebra	A possibly nonassociative algebra whose commutator product satisfies the Malcev identity and therefore forms a Malcev algebra.
Malkus waterwheel	Dynamical SystemsChaos Theory	A driven, dissipative wheel with filling and leaking rim containers whose evolving mass imbalance can produce steady, periodic, reversing, or chaotic Lorenz-type dynamics.
Malliavin Derivative	Malliavin Calculus	Differentiate a random functional with respect to infinitesimal perturbations of its underlying Gaussian noise, producing a Hilbert-valued gradient whose adjoint is the Skorokhod divergence.
Manin matrix	Noncommutative Algebra and Representation Theory	A matrix over a possibly noncommutative ring whose column entries commute and whose cross commutators satisfy relations sufficient to recover many classical determinant identities.
Many-sorted logic	Mathematical Logic	A formal logic whose language and structures partition objects into multiple sorts, restricting constants, variables, functions, predicates, substitution, and quantification to declared sort signatures.
Map graph	Structural and Geometric Graph Theory	The intersection graph of finitely many interior-disjoint simply connected regions in the plane, with vertices for regions and adjacency whenever two regions meet at any boundary point.
Mapping Cylinder	Algebraic TopologyHomotopy Theory	The quotient space formed by gluing one end of X×[0,1] to Y through a continuous map f:X→Y.
Mapping Space	Algebraic TopologyFunction Space Topology	A topological or enriched space whose points are maps between fixed spaces, with topology chosen so families, homotopies, and evaluation become structural.
Marcinkiewicz interpolation theorem	Functional Analysis	An interpolation theorem deriving strong intermediate Lp bounds for a sublinear operator from suitable weak-type endpoint bounds.
Marginal probability	Probability Theory	In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset.
Markov kernel	Probability TheoryStochastic Transition Structures	A measurable assignment sending each source point to a probability measure on a target space, generalizing a stochastic transition matrix to arbitrary measurable spaces.
Markov operator	Probability Theory	A positive mass-preserving operator that propagates probability densities, measures or observables through a stochastic transition.
Markov Renewal Process	Stochastic Processes	Model a sequence of jump states and jump times with a kernel whose joint next-state and holding-time law depends only on the current embedded state, yielding a semi-Markov process between jumps.
Markov strategy	Game Theory	A dynamic-game strategy whose action depends only on the current payoff-relevant state rather than the full history.
Markov's Principle	Constructive Logic and Computability Theory	A constructive-logical schema allowing double-negated existence of a witness for a decidable predicate on the natural numbers to be converted into positive existence, reflecting the legitimacy of unbounded search.
Martingale (probability theory)	Probability TheoryStochastic Processes and Filtrations	Model an adapted integrable stochastic process whose conditional expected future value, given present information, equals its current value.
Maschke's theorem	Representation TheoryFinite Groups	Every finite-dimensional representation of a finite group over a field whose characteristic does not divide the group order decomposes as a direct sum of irreducible representations.
Maslov index	Symplectic GeometryMicrolocal Analysis	Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis.
Mass Point Geometry	Elementary GeometryEuclidean Geometry	A triangle-geometry method that represents segment ratios and cevian intersections by assigning balancing masses to points.
Matching	Graph Theory Combinatorics	Cast a pairing problem as a largest (or minimum-cost, or perfect) set of pairwise vertex-disjoint edges on an explicit graph, then split on bipartiteness to select the theorems and polynomial-time algorithm that solve it.
Matching (graph theory)	Graph Theory	A set of graph edges with no shared endpoint.
Matching pennies	Game TheoryZero Sum Games	Force randomization with the smallest strictly-competitive game — a matcher wins on agreement, a mismatcher on difference — where the best-response cycle admits no pure equilibrium and each player must mix to leave the opponent indifferent.
Material conditional	Propositional Logic	The truth-functional binary connective P→Q that is false only when P is true and Q is false, and otherwise true, classically equivalent to ¬P∨Q.
Material Nonimplication	Propositional LogicBoolean Algebra	Material nonimplication is the classical binary truth function P ∧ ¬Q, true only when P is true and Q is false.
Mathai–Quillen formalism	Differential Geometry and Mathematical PhysicsThom Forms and Localization	Represent a vector bundle's Thom class by a canonical Gaussian differential form built with a connection and curvature, linking cohomological localization, superconnections and topological quantum field theory.
Mathematical Category	Category Theory	A mathematical category is a structure consisting of objects, morphisms between objects, identity morphisms, and an associative partial composition law with matching domains and codomains, optionally enriched or equipped with additional categorical structure.
Mathematical Coordinate System	GeometryAnalytic Geometry	A mathematical coordinate system is a rule-governed representation that assigns coordinate tuples to points in a specified region of a space relative to declared origins, axes, charts, bases, singularities, and transition conventions.
Mathematical diagram	Mathematical Visualization	A visual inscription whose spatial marks and conventions represent mathematical objects, relations, transformations, or proofs.
Mathematical fallacy	Mathematical ProofRecreational Mathematics	In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy.
Mathematical Flow Graph	Systems MathematicsSignal Flow Analysis	Encode coupled linear equations as a weighted directed graph whose declared readback and path, loop, determinant, or elimination rules preserve and solve the represented system.
Mathematical Functional	Functional Analysis	A mathematical functional is a function whose input is itself a function, vector, operator, state, measure, or other structured mathematical object and whose output lies in a declared codomain, commonly a scalar field; linearity, continuity, locality, and variational role are additional properties rather than the genus.
Mathematical Invariant	—	A mathematical invariant is a property, quantity, equivalence class, or algebraic object assigned to a mathematical structure that remains unchanged under a specified class of transformations, representations, equivalences, or admissible evolution.
Mathematical Modeling	Applied Mathematics	The process of developing a mathematical model is termed mathematical modeling.
Mathematical Operator	—	A mathematical operator is a rule with a declared domain and codomain that acts on mathematical objects such as functions, forms, vectors, sets, or algebraic structures to produce an object, relation, or structure according to specified algebraic, analytic, or logical laws.
Mathematical Relation	—	A mathematical relation is a formally specified subset of a Cartesian product, predicate over typed objects, equivalence or order structure, or equation constraining quantities, with arity, domain, parameters, and satisfaction conditions declared.
Mathematical Space	—	A mathematical space is a set or class of mathematical objects equipped with declared structure—such as topology, metric, order, linear operations, measure, geometry, or parameter interpretation—that determines how its elements relate, vary, converge, or transform.
Mathematical structure	Foundations and Structural Mathematics	Endow one or more carrier sets with declared operations, relations, distinguished elements, topology, measure, or other typed data satisfying axioms, so objects are compared by morphisms and isomorphisms that preserve the selected structure rather than incidental presentation.
Mathieu wavelet	Wavelet Theory	A wavelet family constructed from periodic Mathieu functions and their associated filter coefficients.
Matrix	Linear Algebra	Encode a linear map, a system Ax=b, a bilinear form, or a graph's adjacencies as one rectangular array under a single arithmetic, so derived quantities like rank and a menu of factorizations (LU, QR, spectral, SVD) read the structure off directly.
Matrix Analytic Method	Applied ProbabilityQueueing Theory	Solves structured Markov models by exploiting repeating transition blocks through class-specific matrix equations and boundary conditions.
Matrix Chernoff Bound	ProbabilityRandom Matrix Theory	A spectral concentration inequality controlling extreme eigenvalues of sums of independent bounded positive-semidefinite random matrices.
Matrix congruence	Linear Algebra	An equivalence relation on square matrices in which B equals transpose-P times A times P for an invertible change-of-basis matrix P.
Matrix Difference Equation	Discrete Dynamical SystemsLinear Systems	Propagate a vector-valued state at discrete indices through matrix-valued lag operators, using transition products, fixed points, and spectral structure to solve and diagnose the evolution.
Matrix equivalence	Linear Algebra	The two-sided change-of-basis equivalence B=Q⁻¹AP for same-sized rectangular matrices, classifying representations of one linear map under independent domain and codomain bases and determined solely by rank.
Matrix exponential	Linear AlgebraMatrix Functions and Linear Systems	Map a square matrix to the absolutely convergent power series exp(A), yielding the fundamental linear flow while preserving noncommutative ordering and commutation conditions.
Matrix factorization (algebra)	Homological Algebra	A pair of finite free-module maps whose two composites equal multiplication by a fixed potential, yielding a two-periodic resolution over the hypersurface quotient.
Matrix factorization of a polynomial	Commutative Algebra	A pair of square matrices over a polynomial ring whose two products both equal multiplication by a fixed polynomial times the identity.
Matrix Multiplication	Linear AlgebraMatrix Operations	The ordered contraction of compatible matrices, producing cᵢⱼ=Σₖaᵢₖbₖⱼ and representing composition of linear transformations.
Matrix Pencil	Linear Algebra	Treat a pair of same-sized matrices as the affine one-parameter family A−λB, whose finite and infinite generalized eigenvalues and singular structure remain meaningful even when B cannot be inverted.
Matrix Similarity	Linear AlgebraMatrix Canonical Form Theory	Treat square matrices A and B over the same field as equivalent exactly when B = P⁻¹AP for an invertible P, so they represent one linear operator in different bases.
Matrix-Free Methods	Numerical Analysis	Solve large linear, eigenvalue, or nonlinear subproblems through an operator-application interface that computes matrix–vector products on demand without assembling or storing the full coefficient or Jacobian matrix.
Matroid	Combinatorics	A matroid is a combinatorial structure on a ground set whose independent subsets satisfy nonemptiness, heredity, and exchange axioms, equivalently representable through bases, circuits, rank, closure, or other axiom systems, thereby abstracting dependence shared by linear algebra, graphs, and related settings.
Matroid Rank	Matroid Theory	Matroid rank assigns each subset the maximum size of an independent subset, encoding a finite matroid as an integer-valued set function.
Matroid-Constrained Number Partitioning	Combinatorial Optimization	A multiway number-partitioning problem in which every assigned subset must be independent in a corresponding matroid, while a declared aggregate objective balances or optimizes the subsets' weights.
Maurer–Cartan form	Differential Geometry	The canonical Lie-algebra-valued one-form that translates each tangent vector on a Lie group back to the identity.
Maximal and minimal elements	Order Theory	Elements of a subset in a preorder that have no strictly greater or strictly lesser comparable member in that subset, without necessarily dominating or being dominated by every member.
Maximal function	Harmonic Analysis	A harmonic-analysis operator assigning each point the supremum of local averages of a function over neighborhoods containing or centered there.
Maximal Ideal	Ring Theory	A proper two-sided ideal maximal under inclusion, equivalently one whose quotient ring is simple—and, for a commutative unital ring, a field.
Maximal independent set	Graph Theory	In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set.
Maximal torus	Lie TheoryCompact Lie Groups	In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.
Maximising measure	Ergodic OptimizationErgodic Theory	A transformation-invariant probability measure that maximizes the integral of a specified observable over all invariant probability measures.
Maximum matching	Graph Theory	In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts.
Maximum Theorem	Parametric Optimization	Guarantee continuity of a parametric optimum's value and upper hemicontinuity of its maximizer correspondence when objective and compact feasible correspondence vary continuously.
May spectral sequence	Algebraic Topology	A spectral sequence used to compute the cohomology of the Steenrod algebra and the input to the Adams spectral sequence.
Mayer–Vietoris sequence	Algebraic Topology	A natural long exact sequence relating the homology or cohomology of a space to those of two covering subspaces and their intersection.
Mazur's lemma	Functional Analysis	A result stating that convex combinations of tails of a weakly convergent sequence in a normed space can be chosen to converge in norm to the same limit.
McDiarmid's inequality	Probability TheoryConcentration Inequalities	A concentration bound for a function of independent variables whose value can change by at most cᵢ when only coordinate i is replaced.
McKay Graph	Representation Theory	Encode tensoring by a fixed finite-group representation as a multiplicity-weighted quiver on irreducible representations, turning fusion rules into adjacency, paths, spectra, and—in the SU(2) case—affine ADE structure.
McShane integral	Real AnalysisGauge Integration	A gauge integral using free tagged partitions whose tags may lie outside their associated subintervals, yielding for real-valued functions exactly the Lebesgue-integrable class.
Mean curvature	Differential Geometry	The average of a hypersurface’s principal curvatures at a point, measuring its local extrinsic bending in an ambient manifold.
Mean Dimension	Topological Dynamics	Mean dimension measures asymptotic topological degrees of freedom per iterate in a compact dynamical system, remaining informative when entropy is infinite.
Mean of a function	Analysis	The domain-normalized integral of a function, giving the constant value with the same total integral over a set of finite nonzero measure.
Mean Value Theorem	Real AnalysisCalculus	In calculus and real analysis, the mean value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions, roughly stating that the average rate of change of such a function over an interval is equal to the instantaneous rate of change at some point in the interval.
Mean-field game theory	Game Theory	A framework for strategic control in very large populations where each negligible agent responds to the aggregate state distribution generated by all agents.
Measurable space	Measure Theory	A set equipped with a sigma-algebra specifying which subsets are admissible as measurable events or regions.
Measure space	Measure Theory	Bind a set, a sigma-algebra of measurable subsets, and a countably additive nonnegative measure into the ambient structure on which almost-everywhere reasoning and integration are defined.
Medial magma	Universal Algebra	A magma whose binary operation satisfies (ab)(cd)=(ac)(bd) for all four elements.
Median	StatisticsOrder Statistics	A central cut of ordered data or a distribution with at least half the observations or probability at or below it and at least half at or above it.
Medoid	Data AnalysisClustering Representatives	An observed member of a dataset or cluster minimizing total dissimilarity to the other members, used as a robust representative when an arithmetic centroid is unavailable or inappropriate.
Mehler Kernel	Special FunctionsHarmonic Analysis	The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator.
Mehler–Fock transform	Harmonic AnalysisSpecial Function Transforms	An integral transform using conical Legendre functions as its kernel, with a weighted inverse transform on the half-line under suitable analytic conditions.
Meijer G-function	Special FunctionsMellin Barnes Functions	Define a highly general special function by a Mellin–Barnes contour integral whose gamma-factor parameters subsume many hypergeometric and classical functions.
Mellin transform	Integral Transforms	An integral transform mapping a function on positive scales to complex powers, making multiplicative scaling analogous to Fourier analysis of translation.
Menger space	Set Theoretic TopologySelection Principles	A topological space in which, from every sequence of open covers, finitely many sets can be selected from each cover so that all selected sets together still cover the space.
Mennicke symbol	Algebraic K Theory and Congruence Subgroups	A map from admissible element pairs of a Dedekind domain to an abelian group satisfying the Mennicke identities used in congruence-subgroup analysis.
Meromorphic function	Complex Analysis	A complex function holomorphic on a domain except at isolated poles, equivalently locally a quotient of holomorphic functions with nonzero denominator.
Mertens function	Analytic Number Theory	The summatory function of the Möbius function, equal to the running difference between square-free integers with even and odd numbers of prime factors.
Mertens-stable equilibrium	Game TheoryEquilibrium Refinements	A robust set-valued refinement of Nash equilibrium requiring strategically coherent equilibrium components to persist under admissible perturbations and satisfy invariance and rationality properties.
Meshedness coefficient	Graph TheoryNetwork Science	In graph theory, the meshedness coefficient is a graph invariant of planar graphs that measures the number of bounded faces of the graph, as a fraction of the possible number of faces for other planar graphs with the same number of vertices.
Mesocompact Space	General TopologyOpen Cover Properties	A topological space whose every open cover has an open refinement that meets each compact subset in only finitely many members.
Metalogic	Mathematical Logic	The formal study of logical languages and deductive systems as mathematical objects, including their semantics, proof theory and global properties.
Metanilpotent Group	Group Theory	In mathematics, in the field of group theory, a metanilpotent group is a group that is nilpotent by nilpotent.
Metatheorem	Mathematical Logic	A theorem proved in a metalanguage about the syntax, derivability, semantics or other properties of a formal object system.
Metric dimension (graph theory)	Graph Distances and Resolving Sets	The minimum size of a vertex subset whose distance vectors uniquely identify every graph vertex.
Metric Map (Nonexpansive Map)	Metric GeometryNonlinear Analysis	A function between metric spaces that never increases pairwise distance, equivalently a Lipschitz map with constant at most one.
Metric outer measure	Measure Theory	An outer measure additive on sets separated by a positive distance in a metric space.
Metric projection	Best Approximation	Map a point to the set of points in a designated subset that minimize its metric distance, retaining nonexistence and nonuniqueness unless geometry supplies stronger guarantees.
Metric Space Aimed at Its Subspace	Metric GeometryInjective Metric Spaces	A metric superspace whose distance differences to points of a distinguished subspace approximate every ambient pair distance arbitrarily closely.
Metric tensor	Differential Geometry	A smoothly varying nondegenerate bilinear form on tangent spaces that determines lengths, angles, volumes and causal or geodesic structure on a manifold.
Metrizable space	General Topology	A topological space whose open sets are exactly those generated by some metric on its underlying set.
Metrizable topological vector space	Functional AnalysisTopological Vector Spaces	In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric).
Microcontinuity	Nonstandard Analysis	The nonstandard-analysis form of continuity requiring infinitely close inputs to have infinitely close outputs.
Microdifferential operator	Microlocal AnalysisAlgebraic Analysis	A microlocal linear operator on cotangent phase space represented by a homogeneous formal symbol series, with analytic members selected by growth conditions on negative-order terms.
Midpoint	Affine GeometryMetric Geometry	The point that divides a specified segment between two endpoints into equal halves, expressed by affine averaging or half-length along a chosen geodesic when those structures apply.
Miller–Rabin primality test	Computational Number Theory	A randomized strong-probable-prime test that repeatedly checks modular-power witnesses and bounds the chance that a composite integer passes all selected bases.
Milman's reverse Brunn–Minkowski inequality	Convex GeometryGeometric Functional Analysis	Two centrally symmetric convex bodies admit a volume-preserving relative position in which their Minkowski sum has a dimension-free upper bound on its volume radius.
Milnor K-theory	Algebraic K Theory	The graded ring generated by nonzero field elements modulo Steinberg relations, linking symbols in algebraic K-theory with Galois cohomology.
Milnor number	Singularity TheoryHypersurface Singularities	Measure an isolated hypersurface singularity by the finite dimension of its local Jacobian algebra, equivalently the number of middle-dimensional spheres in its Milnor fiber.
Minakshisundaram–Pleijel zeta function	Spectral Geometry	The Minakshisundaram–Pleijel zeta function is a zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold.
Minimal Axioms for Boolean Algebra	Algebraic Logic	A Boolean-algebra axiom presentation shown complete and minimal by an explicitly stated size or redundancy criterion.
Minimal Model Program	Birational Algebraic GeometryMori Theory	A birational-classification program that follows canonical-divisor-negative extremal directions through contractions and flips, seeking a model with nef log canonical divisor or a Mori fiber-space endpoint while admitting only controlled singularities.
Minimal polynomial (field theory)	Field Theory	The unique monic polynomial of least positive degree over a base field having a given algebraic extension element as a root.
Minimal Polynomial (Linear Algebra)	Linear AlgebraOperator Theory and Canonical Forms	The unique monic generator of all polynomial identities satisfied by a finite-dimensional linear operator, encoding the least annihilating relation and the largest primary-block exponents.
Minimax Theorem	Game TheoryTwo Person Zero Sum Games	A theorem family giving hypotheses under which opposed max–min and min–max values coincide, thereby certifying a saddle value.
Minkowski space (number field)	Algebraic Number Theory	In mathematics, specifically the field of algebraic number theory, a Minkowski space is a Euclidean space associated with an algebraic number field.
Minlos's theorem	Topological Vector Spaces	Minlos's theorem turns a cylindrical measure on the dual of a nuclear space into a Radon measure when its Fourier transform is continuous.
Minor (linear algebra)	Linear Algebra	The determinant of a square submatrix obtained by selecting equal-size subsets of a matrix’s rows and columns.
Minority Game	Game Theory Econophysics	Model any anti-coordination setting as an odd population repeatedly choosing between two actions where only the minority side wins, reading collective volatility off one control parameter — strategy diversity relative to population — that fixes a sharp crowded-versus-dilute phase transition.
Mixed Chinese Postman Problem	Graph TheoryCombinatorial Optimization	Find a least-cost closed walk covering every link of a weighted graph with undirected edges and directed arcs.
Mixed Strategy Equilibrium	Game Theory	Solve a game with no stable deterministic play by having each player randomize over their actions in exactly the proportions that leave every opponent indifferent, so no one can profitably deviate.
Mixture Distribution	Probability TheoryMixture Models	A probability law generated by first selecting a latent component according to normalized weights and then sampling from that component.
Mnëv's universality theorem	Combinatorial GeometryOriented Matroid Realization	A theorem showing that realization spaces of oriented matroids can reproduce, up to stable equivalence, arbitrary integer-defined primary semialgebraic sets and therefore arbitrarily complicated topology.
Modal algebra	Algebraic LogicModal Algebras	A Boolean algebra equipped with a unary normal meet-preserving modal operator, providing algebraic semantics for normal propositional modal logics.
Modal matrix	Linear Algebra	A matrix whose columns are eigenvectors of a square matrix, used as the change of basis that diagonalizes it when a full eigenbasis exists.
Modified half-normal distribution	Probability Theory	A positive-support probability family extending the half-normal shape with power and exponential-tilt parameters.
Modified Morlet wavelet	Wavelet Analysis and Signal Processing	A localized oscillatory analyzing function formed by multiplying a cosine carrier by a hyperbolic-secant envelope and normalizing it for wavelet use.
Modular arithmetic	Number Theory	Arithmetic on congruence classes in which integers differing by a multiple of a fixed positive modulus are identified.
Modular exponentiation	Computational Number Theory	Computation of a power modulo a positive integer, returning the residue of a base raised to an integer exponent without constructing the full power.
Modular graph	Metric Graph Theory	An undirected graph in which every triple of vertices has at least one common median lying on a shortest path between each pair.
Modular multiplicative inverse	Modular Arithmetic	A residue x whose product with a given integer a is congruent to one modulo m, existing exactly when a and m are coprime.
Modular product of graphs	Graph Products and Common Subgraph Algorithms	A graph product on the Cartesian product of two vertex sets whose adjacency encodes agreement of adjacency or nonadjacency in the factor graphs.
Module (Algebra)	Abstract AlgebraModule Theory	An additive abelian group equipped with a compatible left or right action by a ring, generalizing vector spaces from field scalars to ring scalars.
Moduli Space	Algebraic Geometry	A geometric parameter object whose points represent objects or equivalence classes in a fixed classification problem and whose structure records how those objects vary in families.
Moduli Stack of Formal Group Laws	Algebraic TopologyChromatic Homotopy Theory	A coordinate-independent moduli stack obtained from formal group laws by quotienting coordinate changes, retaining isomorphisms and organizing formal groups by height for chromatic homotopy theory.
Modulus (algebraic number theory)	Algebraic Number Theory	A finite formal product of places of a global field encoding congruence and ramification conditions for ray class groups and abelian extensions.
Modulus of continuity	Real and Functional Analysis	A nonnegative function bounding output variation in terms of input distance and tending to zero at zero, thereby quantifying uniform continuity.
Modus Ponendo Tollens	Propositional Proof TheoryRules of Inference	From the incompatibility of A and B together with A, infer not-B: affirming one jointly forbidden proposition eliminates the other.
Moment matrix	Applied MathematicsMoment Problems	A symmetric matrix indexed by monomials whose entry at two indices is the moment associated with their product, encoding a moment sequence and positivity constraints.
Moment problem	Analysis and Probability	The inverse problem of deciding whether a sequence is represented by moments of a measure, and whether that representing measure is unique.
Monad (nonstandard analysis)	Nonstandard AnalysisInfinitesimal Neighborhoods	The set of hyperreal points infinitesimally close to a given hyperreal point, with a finite point's monad containing exactly one real standard part.
Monadic predicate calculus	Mathematical Logic	The function-free fragment of first-order logic whose predicate symbols all have arity one.
Monadic second-order logic	Mathematical LogicFinite Model Theory	The fragment of second-order logic that permits quantification over individual elements and unary predicates or sets, but not arbitrary higher-arity relations.
Monic Polynomial	Polynomial Algebra	A nonzero univariate polynomial whose coefficient on its highest-degree nonzero term is the multiplicative identity one in the declared coefficient ring.
Monogenic function	Complex Analysis	In classical complex analysis, a function possessing a unique finite complex derivative at a point or throughout a stated set.
Monoid (category theory)	Category Theory	An object in a monoidal category equipped with associative multiplication and a two-sided unit expressed by coherent morphism diagrams.
Monoidal Category	Category Theory	A category with a bifunctorial product, unit object, and coherent natural associativity and unit isomorphisms.
Monoidal category action	Category Theory and Higher Algebra	A functor from a monoidal category times another category to that category, equipped with coherent natural isomorphisms expressing associative action and a unit.
Monoidal Monad	Category Theory	A monad on a monoidal category equipped with coherent lax-monoidal comparison maps compatible with the monad unit and multiplication.
Monoidal Natural Transformation	Category Theory	A natural transformation between monoidal functors whose components also commute with their tensor comparison and unit maps.
Monoidal t-Norm Logic	Mathematical LogicT Norm Fuzzy Logics	The propositional many-valued logic common to all left-continuous t-norms, coupling strong conjunction to residual implication and enforcing prelinearity.
Monomial Ideal	Commutative Algebra	An ideal generated by monomials in a multivariate polynomial ring, equivalently one with termwise membership determined by divisibility by a finite minimal set of monomial generators.
Monopole moduli space	Gauge TheoryModuli Spaces	In mathematics, the monopole moduli space is a space parametrizing monopoles (solutions of the Bogomolny equations). studied the moduli space for 2 monopoles in detail and used it to describe the scattering of monopoles.
Monotone matrix	Matrix Analysis and Ordered Linear Algebra	A real square matrix A for which componentwise nonnegativity of Ax implies componentwise nonnegativity of x, equivalently an invertible matrix with a nonnegative inverse.
Monotonic Function	Order Theory	Map one ordered set into another while preserving comparison direction everywhere, or reverse that direction everywhere in the antitone variant, so input order constrains output order globally.
Monsky–Washnitzer cohomology	Arithmetic Geometry	A p-adic cohomology theory for smooth affine varieties in characteristic p, built from weakly completed lifts and de Rham forms.
Monte Carlo integration	Numerical Probability	A numerical-integration method estimating an integral from randomized samples and reporting sampling uncertainty that typically decreases with the square root of sample count.
Montel space	Functional AnalysisTopological Vector Spaces	A barrelled topological vector space in which every closed bounded subset is compact, extending Montel-type compactness from spaces of holomorphic functions.
Moore determinant of a Hermitian matrix	Linear AlgebraQuaternion Algebra	In mathematics, the Moore determinant is a determinant defined for Hermitian matrices over a quaternion algebra, introduced by .
Moore graph	Graph TheoryExtremal Graph Theory	In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices).
Moore space (topology)	General Topology	A regular Hausdorff topological space possessing a countable development of open covers that locally refines every neighborhood.
Morava K-theory	Stable Homotopy Theory	A prime- and height-indexed family of periodic generalized homology theories that isolates chromatic layers of stable homotopy theory.
Mordellic Variety	Arithmetic GeometryDiophantine Geometry	In mathematics, a Mordellic variety is an algebraic variety which has only finitely many points in any finitely generated field.
Mordell–Weil Rank of an Elliptic Curve	Arithmetic GeometryElliptic Curves	Count the independent infinite-order rational points on an elliptic curve over a declared number field by taking the free rank of its finitely generated Mordell–Weil group.
Mordell–Weil Theorem	Arithmetic GeometryAbelian Varieties	For an abelian variety over a number field, the group of rational points is finitely generated.
Moreau Envelope	Convex Analysis	Smooth a proper lower-semicontinuous convex function by infimizing its value plus a quadratic distance penalty, linking nonsmooth optimization to the proximal map.
Morita Equivalence	Abstract AlgebraRing Theory	In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties.
Morley Rank	Model Theory	Morley rank measures recursive infinite definable splitting of a set in a first-order structure.
Morphism of algebraic varieties	Algebraic Geometry	A map between algebraic varieties that is locally given by regular polynomial or rational-function expressions without poles.
Morphism of finite type	Algebraic GeometryScheme Morphisms	A scheme morphism that is locally induced by finitely generated algebras, expressing algebraic dependence on finitely many generators without requiring module finiteness.
Morphism of schemes	Algebraic Geometry	A morphism of locally ringed spaces between schemes, combining a continuous map of spectra with a compatible local homomorphism of structure sheaves.
Morse homology	Differential Topology	A homology theory whose chain groups are generated by critical points of a Morse function and whose boundary counts gradient-flow trajectories between adjacent indices.
Morse–Smale System	Smooth Dynamical Systems	A smooth complete flow with finitely many hyperbolic recurrent pieces whose stable and unstable manifolds meet transversely.
Motion (geometry)	GeometryMetric and Euclidean Isometries	Transform a metric space by a surjective distance-preserving map, with Euclidean motions further classified by orientation, fixed-point structure, and decomposition into translations, rotations, reflections, and glide operations.
Motive (algebraic geometry)	Algebraic Geometry	An object in a specified motivic category that packages algebraic-geometric correspondences and their cohomological realizations.
Motivic integration	Algebraic GeometryArc Spaces and Motives	An algebraic-geometric integration theory assigning classes in a Grothendieck ring to measurable subsets or functions on arc spaces, retaining geometric information beyond numerical measure.
Mott polynomials	Special FunctionsPolynomial Sequences	A polynomial sequence defined by an exponential generating function involving the Catalan-series expression (sqrt(1−t²)−1)/t, introduced in connection with electron theory.
Motzkin number	Enumerative Combinatorics	A sequence whose nth term counts noncrossing chord matchings on n labeled points in convex position, allowing unmatched points.
Mouse (set theory)	Inner Model Theory	A small iterable fine-structural model equipped with extender data, used to approximate large-cardinal universes and build core models.
Move by nature	Game Theory	A chance move in an extensive-form game that selects an outcome according to a specified probability distribution without strategic preferences.
Mrs. Miniver's Problem	Plane GeometryRecreational Mathematics	The placement problem for two circles of fixed radii in which the area of their intersection lens must equal the area of their symmetric difference.
Much-Greater-Than Relation	Mathematical NotationAsymptotic Reasoning	These relations are known as strict inequalities, meaning that a is strictly less than or strictly greater than b.
Muckenhoupt Weights	Harmonic Analysis	Positive Euclidean weights whose local mean and reciprocal-power mean satisfy a uniform, exponent-indexed A_p balance over all cubes.
Multilinear form	Multilinear Algebra	A scalar-valued map of several vector arguments that is linear separately in each argument.
Multiplicative Cascade	Fractal GeometryProbability Theory	In mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process.
Multiplicative Digital Root	Number TheoryDigit Dynamics	The terminal single base-b digit reached by repeatedly replacing a nonnegative integer with the product of its digits, paired with multiplicative persistence as the number of iterations required to reach that fixed point.
Multiplicative Function	Number Theory	Recognize an arithmetic function normalized at one whose value on a product of coprime positive integers factors into the product of their values, thereby reducing global behavior to independently specified prime-power data.
Multiplicative partition	Number Theory	An unordered factorization of a positive integer into integers greater than one, with products differing only by factor order identified.
Multiplicatively closed set	Commutative Algebra	A subset of a ring containing the multiplicative identity and closed under every finite product.
Multiply perfect number	Number Theory	A positive integer whose sum of positive divisors is an integer multiple k of the number itself.
Multiresolution Analysis	Harmonic AnalysisWavelet Theory	A dilation-linked nested sequence of approximation spaces whose successive orthogonal complements isolate wavelet detail across scales.
Multitree	Directed Graphs	A directed acyclic graph in which every ordered pair of vertices has at most one directed path, equivalently a DAG whose reachability order is diamond-free.
Multivalued function	Set Valued Analysis	A relation assigning each input a set of possible outputs, represented by a subset of the Cartesian product and often equipped with nonempty-value or regularity conditions.
Multivariate Pareto distribution	ProbabilityMultivariate Heavy Tails	A family of joint heavy-tailed distributions whose margins have Pareto-type behavior and whose dependence construction models simultaneous extremes across variables.
Murnaghan–Nakayama rule	Representation Theory	A signed rim-hook removal rule for computing irreducible character values of symmetric groups from partitions.
Mutual coherence (linear algebra)	Sparse Approximation and Compressed Sensing	The largest absolute normalized inner product between distinct columns of a matrix or atoms of a dictionary.
Möbius function	Number TheoryArithmetic Functions	The multiplicative arithmetic function μ(n) that is zero on numbers divisible by a prime square and otherwise equals minus one to the number of distinct prime factors.
N = 2 superconformal algebra	Conformal Field Theory	An infinite-dimensional Lie superalgebra extending the Virasoro algebra by a U(1) current and two fermionic supercurrents.
N-body problem	Dynamical Systems	The problem of determining the coupled motion of multiple bodies interacting through mutual forces, classically Newtonian gravitation.
N-flake	Fractal GeometryPolygonal Iterated Function Systems	Generate a family of polygonal self-similar attractors by placing contracted copies of a regular n-gon at its vertices, optionally adding a convention-controlled central copy, and iterating the resulting similarity system.
N-Square Identity	Quadratic Forms	An algebraic composition identity that represents a product of two sums of n squares as another sum of n squares, with its witness type and domain stated.
Nakayama's Lemma	Commutative AlgebraModule Theory	A finite-generation principle saying radical multiples cannot exhaust a nonzero module and that generators after quotienting by the Jacobson radical lift to generators before quotienting.
NAND Logic	Boolean AlgebraFunctionally Complete Connectives	A basis-restricted Boolean construction in which every target function is synthesized as a composition of NAND alone, exploiting NAND's functional completeness.
Napkin ring problem	Geometry	In geometry, the napkin-ring problem involves finding the volume of what remains after a circular hole is drilled through a sphere.
Narayana Polynomials	Enumerative Combinatorics	Package the Narayana distribution of a Catalan family into a polynomial whose coefficient of each power counts objects with a specified statistic, retaining the Catalan total at one while exposing symmetry, unimodality, real-rootedness, and specializations.
Narrow escape problem	Stochastic ProcessesFirst Passage Problems	The narrow escape problem asks for the first-passage behavior of a diffusing particle confined in a domain that can leave only through a small absorbing window in an otherwise reflecting boundary.
Narumi polynomials	CombinatoricsSpecial Polynomial Sequences	A parameterized Sheffer polynomial sequence defined by the exponential generating function (t/log(1+t))^a(1+t)^x.
Nash blowing-up	Algebraic GeometryBirational Modification of Singularities	Replace a singular variety by the closure of the graph of its smooth-point tangent-space map, retaining each singular point together with the limiting tangent spaces approached nearby.
Natural Exponential Family	Probability TheoryMathematical Statistics	A full one-parameter family of probability laws formed by exponentially tilting one fixed measure by the observed value and normalizing on its finite natural domain.
Natural filtration	Probability Theory	The smallest filtration that makes a given stochastic process adapted by recording exactly the events observable from its history up to each time.
Natural logarithm	Real and Complex Analysis	The logarithm to base e, equivalently the inverse of the natural exponential and the signed area under one-over-x on the positive real line.
Natural Number	Foundations of Arithmetic	The successor-generated arithmetic carrier for finite counting and ordinal indexing, equipped with induction and recursion from a distinguished first element.
Natural numbers object	Category TheoryTopos Theory	In category theory in mathematics, a natural numbers object (NNO) is an object endowed with a recursive structure similar to natural numbers.
Near-field (mathematics)	Abstract Algebra	In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws.
Nearest-Neighbor Chain Algorithm	Cluster AnalysisAgglomerative Hierarchical Clustering	An agglomerative-clustering accelerator that follows nearest-neighbor links until a reciprocal pair is found, then merges that pair under a reducible linkage rule.
Nearly completely decomposable Markov chain	Probability TheoryMarkov Chains	In probability theory, a nearly completely decomposable (NCD) Markov chain is a Markov chain where the state space can be partitioned in such a way that movement within a partition occurs much more frequently than movement between partitions.
Necklace ring	Algebra	A ring on infinite sequences over a commutative ring whose multiplication combines indices by least common multiple and weights products by greatest common divisor.
Negation introduction	Proof Theory	A rule of inference that derives not-P after assuming P and deriving a contradiction within a properly discharged subproof.
Negation normal form	Mathematical Logic	A logical formula form using only conjunction, disjunction and literals, with every negation applied directly to an atomic proposition.
Negative Hypergeometric Distribution	Probability TheoryFinite Population Sampling	The distribution of failures observed before a fixed number of successes when sampling without replacement from a finite population.
Negative volume index	Technical AnalysisConditional Cumulative Market Indicators	Conditionally accumulate price or breadth changes only on sessions whose trading volume falls from the previous session, producing a path-dependent technical-analysis series whose interpretation requires a declared variant and test period.
Neumann–Dirichlet method	Numerical Analysis	A nonoverlapping domain-decomposition preconditioner that alternates Neumann and Dirichlet subdomain solves across shared interfaces.
Neumann–Poincaré operator	Potential Theory	A boundary integral operator built from the normal derivative of the Laplace fundamental solution and used to reduce harmonic boundary-value problems to Fredholm integral equations.
Neuman–Sándor Mean	Mathematical AnalysisBivariate Means and Inequalities	Aggregate two positive numbers by dividing their difference by twice the inverse hyperbolic sine of their normalized difference, producing a symmetric homogeneous mean strictly between the arithmetic and second Seiffert means.
Neutral Geometry	Geometry Foundations	An axiomatic geometry retaining incidence, betweenness, and congruence while withholding any parallel postulate, so its theorems survive in both Euclidean and hyperbolic extensions.
Newest Vertex Bisection	Adaptive Mesh Refinement	A labeled triangle-mesh refinement rule that inherits midpoint vertices and completes neighbor splits to retain a conforming, shape-regular mesh.
Newton fractal	Complex Dynamics and Numerical Analysis	The basin boundary produced in the complex plane by applying Newton's root-finding iteration to a fixed polynomial or meromorphic function from varying initial points.
Newton Polynomial	Numerical Analysis	An interpolating polynomial expressed in a nested Newton basis with divided-difference coefficients at distinct nodes.
Newton polytope	Algebraic and Convex Geometry	The convex hull of exponent vectors of nonzero monomials in a multivariate polynomial, with polynomial products mapping to Minkowski sums.
Newton–Gauss line	Euclidean Geometry	In geometry, the Newton–Gauss line (or Gauss–Newton line) is the line joining the midpoints of the three diagonals of a complete quadrilateral.
Newton–Okounkov body	Algebraic GeometryConvex Geometry	A convex body associated, after choosing valuation or flag data, with a divisor or graded linear series on an algebraic variety, encoding asymptotic section growth and positivity in Euclidean geometry.
Nijenhuis–Richardson bracket	Differential AlgebraGraded Brackets	A graded Lie bracket on alternating vector-valued multilinear forms, defined by antisymmetrized insertion and used to encode Lie algebra structures and their deformations.
Nikodym Set	Geometric Measure Theory	Construct a full-measure planar set whose every point lies on a line that otherwise avoids the set, exposing an extreme mismatch between global size and linewise incidence.
Nilmanifold	Differential Geometry and Lie Groups	A homogeneous manifold represented as a quotient of a nilpotent Lie group by a closed subgroup, with compact nilmanifolds commonly arising from cocompact discrete lattices.
Nilpotent algebra	Nonassociative AlgebraRing Theory	In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer n every product containing at least n elements of the algebra is zero.
Nilpotent Operator	Linear AlgebraFunctional Analysis	A linear self-map whose one fixed finite power is exactly the zero operator on its entire space.
Nilradical of a ring	Commutative AlgebraRing Radicals	The ideal of all nilpotent elements in a commutative ring, equivalently the radical of the zero ideal and the intersection of all prime ideals.
Nilsemigroup	Semigroup Theory	A semigroup with a zero element in which every individual element has some positive power equal to zero.
Nimber	Impartial Normal Play Game Values	Assign an impartial normal-play game position the unique Nim-heap value determined recursively by the minimum excluded values of its options and composed by nim-sum.
Nine lemma	Homological Algebra	A diagram lemma stating conditions under which exactness of rows and columns in a commutative three-by-three diagram forces exactness of the remaining row or column.
Nine-Point Conic	Euclidean and Projective Geometry	The conic through the six side midpoints and three diagonal points determined by a complete quadrangle, with circle and hyperbola cases governed by the quadrangle geometry.
Niven's constant	Analytic Number Theory	The limiting average, over positive integers, of the largest exponent in each integer’s prime factorization.
Nodal decomposition	Category Theory	A category-theoretic factorization of a morphism as a strong epimorphism, followed by a bimorphism, followed by a strong monomorphism.
Noether's Second Theorem	Variational CalculusMathematical Physics	A local variational symmetry with arbitrary function parameters yields an off-shell differential identity among the Euler–Lagrange expressions.
Nome (mathematics)	Elliptic FunctionsSpecial Functions	In mathematics, specifically the theory of elliptic functions, the nome is a special function that belongs to the non-elementary functions.
Non-Archimedean geometry	Foundations of Geometry	In mathematics, non-Archimedean geometry is any of a number of forms of geometry in which the axiom of Archimedes is negated.
Non-Archimedean Ordered Field	Ordered Fields	An ordered field whose scale outruns every natural-number bound, yielding infinitely large elements, their infinitesimal reciprocals, and a natural hierarchy of finite and infinite magnitudes.
Non-cooperative game theory	Game Theory	The study of strategic interaction modeled through individual players’ actions and incentives without externally enforced binding coalition agreements.
Non-Euclidean geometry	Axiomatic GeometryDifferential Geometry	Classical formal plane geometries with unique point-line incidence and a hyperbolic or elliptic alternative to Euclidean unique parallels.
Non-integer base of numeration	Number RepresentationBeta and Complex Radix Expansions	Represent numbers positionally with a real or complex radix that is not an integer, making admissible digits, expansion algorithms, and nonuniqueness depend on the radix.
Non-logical symbol	Mathematical Logic	A constant, function or relation symbol in a formal language whose denotation varies with the chosen interpretation or model.
Non-measurable set	Measure TheoryMeasurability Pathologies	A subset lying outside a specified sigma-algebra, so the chosen measure cannot consistently assign it a value while preserving the measure axioms.
Non-standard model of arithmetic	Models of Arithmetic	A non-standard model of arithmetic satisfies first-order Peano arithmetic while containing elements beyond every standard numeral.
Non-well-founded set theory	Mathematical LogicAxiomatic Set Theory and Hypersets	Study axiomatic set universes in which membership may contain infinite descent or cycles because Foundation is omitted or replaced by a declared anti-foundation principle, with graph decoration and bisimulation specifying which circular presentations denote equal sets.
Noncototient	Number TheoryTotient Value Sets	A positive integer that is not equal to n−φ(n) for any positive integer n, where φ is Euler's totient function.
Nonhypotenuse number	Number Theory	A natural number that is not the hypotenuse length of any integer-sided right triangle.
Nonnegative Matrix	Matrix Theory	A real matrix constrained entrywise to the nonnegative orthant, forming a convex cone closed under addition and multiplication and linking directed-graph structure to Perron–Frobenius spectral behavior.
Nori motive	Algebraic Geometry	A mixed-motive construction obtained from a diagram of algebraic varieties, pairs and cohomology through Nori’s universal abelian category and coalgebra representation.
Norm	Linear Algebra Functional Analysis	Extract a single non-negative magnitude from a vector obeying three axioms (positive definiteness, absolute homogeneity, triangle inequality), where the choice among L¹, L², and L^∞ is a consequential modeling decision read off the unit ball's geometry.
Norm Form	Algebraic Number Theory	The homogeneous degree-n polynomial obtained by expressing the field norm of a degree-n extension in coordinates relative to a chosen base-field basis.
Norm group	Local Class Field Theory	The subgroup of a local field’s multiplicative group consisting of field norms from a finite abelian extension.
Normal automorphism	Group Theory	A group automorphism that maps every normal subgroup onto itself and therefore induces an automorphism on every quotient by a normal subgroup.
Normal closure (group theory)	Group Theory	The smallest normal subgroup of a group containing a specified subset, equivalently the subgroup generated by all conjugates of that subset and their inverses.
Normal Cone	Convex Analysis	At a point of a convex set, collect every vector making a nonpositive inner product with every feasible displacement from that point.
Normal convergence	Functional Analysis	Convergence of a function series whose sum of termwise uniform norms is finite.
Normal function	Set TheoryOrdinal Functions	An ordinal-valued function that is strictly increasing and continuous at limit ordinals, so its value at a limit is the supremum of all earlier values.
Normal measure	Large Cardinal Set Theory	A kappa-complete nonprincipal ultrafilter on a measurable cardinal that is closed under diagonal intersections, equivalently makes every regressive function constant on a measure-one set.
Normal Morphism	Category Theory	A normal monomorphism is a subobject map that actually arises as a kernel; dually, a conormal epimorphism arises as a cokernel.
Normal Number	Number TheoryNormal Numbers and Uniform Distribution	A real number whose base-b expansion gives every finite length-k digit block its uniform limiting frequency b^-k, for every k, with the base kept explicit.
Normal operator	Functional Analysis	A bounded linear operator on a complex Hilbert space that commutes with its adjoint.
Normal Order of an Arithmetic Function	Analytic Number Theory	Describe the typical size of an arithmetic function by relative approximation on a natural-density-one set, while allowing infinitely many sparse and arbitrarily large exceptions.
Normal scheme	Algebraic Geometry	A scheme whose local rings are integrally closed domains at every point.
Normal space	TopologySeparation Axioms	A topological space in which every pair of disjoint closed sets can be enclosed in disjoint open neighborhoods, with Hausdorffness required separately for the T4 convention.
Normal Surface	3 Manifold TopologyNormal Surface Theory	A properly embedded surface put into standard triangle-and-quadrilateral position relative to a triangulation of a 3-manifold.
Normed division algebra	Abstract AlgebraDivision AlgebrasComposition Algebras	If the underlying coefficient field is the reals and is positive-definite, so that is an inner product, then is called a Euclidean Hurwitz algebra or (finite-dimensional) normed division algebra.
Novikov conjecture	Geometric Topology	The conjecture that higher signatures of closed oriented manifolds are invariant under oriented homotopy equivalence.
Nowhere commutative semigroup	Semigroup Theory	A semigroup in which two elements commute only when they are equal.
Nowhere continuous function	Real Analysis	A function that fails the continuity condition at every point of its domain.
Nuclear C*-algebra	Operator Algebras	A C*-algebra whose algebraic tensor product with every C*-algebra has a unique C*-norm, equivalently whose identity approximately factors through matrix algebras by completely positive maps.
Nuclear operators between Banach spaces	Functional Analysis	Linear operators admitting a summable rank-one representation through dual functionals and target vectors.
Nuclear space	Functional Analysis	A locally convex topological vector space whose connecting maps between suitable seminorm completions are nuclear, giving strong finite-dimensional-like compactness and tensor properties.
Null Set	Measure Theory	Classify a measurable subset as negligible when its measure is zero, allowing it to be ignored by almost-everywhere statements without requiring it to be empty.
Number line	Mathematics Education and Geometry	A spatial representation of an ordered number system on a directed line, with a chosen origin, unit interval, and monotone coordinate correspondence.
Number of groups of a given order	Group TheoryFinite GroupsEnumeration	The function assigning each positive integer n the number of isomorphism classes of finite groups with exactly n elements.
Numbering (Computability Theory)	Computability Theory and Effective Enumerations	A surjective coding from natural numbers onto a countable class of mathematical objects, used to transport computability, reducibility, and effective enumeration questions from objects to their indices.
Numeration System	History of MathematicsNumeral Systems	A numeration system is a conventional system for representing numbers or numerical order with an inventory of signs and rules that assign values and compose sign sequences through additive, subtractive, multiplicative, positional, ciphered, alphabetic, or hybrid principles.
Numerical certification	Validated Numerics	The production of a mathematically checkable guarantee that a numerically computed candidate corresponds to an actual solution of a stated problem.
Numerical Method	Applied MathematicsNumerical Analysis	A numerical method is a specified computational procedure that represents a mathematical problem in finite form and produces an approximate solution or trajectory while making accuracy, stability, convergence, and computational cost assessable.
Nut Graph	Spectral Graph Theory	A finite simple graph whose adjacency matrix has a one-dimensional kernel generated by a vector nonzero at every vertex, making zero a simple eigenvalue with full vertex support.
Néron–Severi Group	Algebraic Geometry	The Picard-group quotient that identifies algebraically equivalent line bundles on a smooth projective complex variety.
Néron–Tate height	Arithmetic Geometry	A canonical quadratic height on rational points of an abelian variety over a global field.
Oblate Spheroidal Coordinates	Coordinate SystemsDifferential Geometry	Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci.
Obstruction theory	Algebraic Topology	A family of topological methods that assigns cohomological classes whose vanishing determines whether a partial construction extends to the next dimension.
Octahedral symmetry	Finite Symmetry Groups	The finite symmetry group of a regular octahedron, equivalently a cube, comprising 24 rotations and 48 full isometries when reflections are included.
Octonion	Abstract AlgebraHypercomplex Numbers	In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system.
Odds	Probability and Betting	The ratio of an event’s probability to the probability of its complement, with betting formats translating that ratio into stake and payout conventions.
Odds Algorithm	Optimal StoppingLast Success Problems	An optimal-stopping algorithm that sums independent Bernoulli success odds backward to a unit threshold, then stops on the first later success to maximize catching the final success.
Odious number	Number Theory	A nonnegative integer whose binary expansion contains an odd number of one bits.
Ohsawa–Takegoshi L2 extension theorem	Several Complex Variables	A theorem extending square-integrable holomorphic functions from a complex submanifold with controlled norm.
Omega and agemo subgroup	Group Theory	Characteristic subgroup constructions in a finite p-group that collect elements annihilated by bounded p-powers and generate bounded p-power images, encoding its power structure.
One-form	Differential Geometry	A smooth covector field assigning a linear functional on each tangent space of a differentiable manifold.
One-parameter group	Lie Theory	A continuous homomorphism from the additive real numbers into a topological group, representing a continuously parameterized group action or flow.
One-seventh area triangle	Plane Geometry	The inner triangle formed by three one-third cevians of a triangle, whose area is exactly one seventh of the original.
One-sided limit	Calculus	The limiting value approached by a function when its argument tends to a point only through values on one specified side.
Opaque set	Discrete Geometry	A set of planar curves or segments intersecting every line that crosses a specified convex body.
Open and closed maps	Topology	Maps of topological spaces classified by whether images of every open set or every closed set retain the corresponding property.
Open Set	Topology	A subset in which every point has neighborhood room entirely inside it — and, via three axioms on the whole collection τ, the primitive that IS a space's topology, letting continuity, compactness, and connectedness be defined with no distance function.
Operational Calculus	Mathematical AnalysisOperator Methods	A family of methods that represents differentiation, integration, or related operators in an algebraic domain, solves the resulting operator equation, and interprets the inverse representation as a function.
Operator Algebra	Functional AnalysisOperator TheoryNoncommutative Geometry	An algebra of continuous linear operators on a common topological vector space, using composition as multiplication and usually carrying a specified operator topology and closure condition.
Operator Ideal	Operator Theory	A class of bounded linear maps between Banach spaces that contains finite-rank operators, is linear in each source–target component, and is stable under arbitrary bounded pre- and post-composition.
Operator topologies	Functional Analysis	Standard topologies on spaces of bounded linear operators that distinguish norm, strong, weak and weak-star modes of operator convergence.
Opposite category	Category Theory	The category obtained by retaining every object and reversing the direction of every morphism and composition order.
Opposite simplicial set	Higher Category Theory	The simplicial set obtained by precomposing with the order-reversing automorphism of the simplex category, extending categorical arrow reversal to higher categorical models.
Optimization Problem	OptimizationMathematical Programming	An optimization problem specifies decision variables, a feasible set determined by domains and constraints, and an objective function or preference ordering whose optimum is sought, optionally with uncertainty, multiple objectives, or approximation criteria.
Orbit (group theory)	Group TheoryGroup Actions	The class of 2-transitive groups (that is, subgroups of a finite symmetric group whose action is 2-transitive) and more generally multiply transitive groups is well-studied in finite group theory.
Order (group theory)	Group Theory	The cardinality of a group, and for a group element the cardinality of its generated cyclic subgroup or least positive exponent that returns it to the identity.
Order convergence	Functional Analysis	Convergence in an ordered vector lattice defined by eventual confinement between bounds that close monotonically on the limit.
Order Dual	Functional AnalysisOrdered Vector Spaces	The vector space generated by all positive linear functionals on an ordered vector space, equipped with its canonical pointwise order and serving as the first step toward an order bidual.
Order polynomial	Algebraic Combinatorics	The polynomial whose value at n counts order-preserving maps from a finite poset to an n-element chain.
Order polytope	Algebraic Combinatorics	The convex polytope of order-preserving maps from a finite poset into the unit interval.
Order type	Order TheoryOrdered Sets	The isomorphism class of an ordered set under order-preserving bijection, capturing its ordering structure independently of element names.
Ordered field	Algebra	A field with a total order preserved by addition and multiplication by positive elements.
Ordered geometry	Foundations of GeometryAxiomatic Geometry	Form of geometry without distances.
Ordinal collapsing function	Proof Theory	A notation-building function that introduces symbols for very large ordinals and systematically collapses them into canonical notations for large countable ordinals.
Ordinal definable set	Set Theory	A set uniquely definable in some rank-initial universe by a first-order formula using finitely many ordinal parameters.
Orientation (graph theory)	Graph Theory	An assignment of one direction to every edge of an undirected graph, producing a directed graph with the same vertices and underlying edges.
Orientation character	Geometric TopologyManifold Orientation and Surgery	Classify loops in a manifold by whether their transport preserves or reverses local orientation, encoding the result as a homomorphism from the fundamental group to the two-element sign group.
Ornstein–Uhlenbeck Process	Stochastic Processes	A continuous-time Gaussian process with positive linear mean reversion and constant Brownian forcing.
Orthant	GeometryEuclidean Geometry	In geometry, an orthant or hyperoctant is the analogue in n-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions.
Orthocentric system	Euclidean Geometry	A planar set of four points in which each point is the orthocenter of the triangle formed by the other three.
Orthocompact Space	General Topology	A topological space whose every open cover has an interior-preserving open refinement.
Orthodox Semigroup	Semigroup TheoryRegular Semigroups	A regular semigroup whose idempotents are closed under multiplication, equivalently one in which chosen inverses of two factors compose in reverse order to an inverse of their product.
Orthogonal coordinates	Differential Geometry and Mathematical Physics	A curvilinear coordinate system whose coordinate curves or hypersurfaces meet mutually at right angles, making the metric tensor diagonal in the coordinate basis.
Orthogonal Procrustes problem	Matrix Optimization	A least-squares alignment problem seeking the orthogonal transformation that best maps one matrix configuration to another.
Orthoptic (geometry)	Plane Geometry	The locus of points from which two tangents to a given curve meet at a right angle.
Orthostochastic matrix	Matrix Theory	A doubly stochastic matrix obtained by squaring the entries of a real orthogonal matrix componentwise.
Oscillation theory	Ordinary Differential Equations	The study of zeros and sign changes of differential-equation solutions and their relation to boundary-value spectra and comparison theorems.
Osculant	Invariant Theory	An invariant that vanishes when hypersurfaces have contact of unusually high order at a common point.
Osculating Curve	Differential GeometryProjective Geometry	An osculating curve is the member of a chosen curve family with maximal local contact at a target point.
Osculating plane	Differential Geometry	The plane through a space curve point spanned by its tangent and principal normal, giving second-order local contact when curvature is nonzero.
Outer automorphism group	Group Theory	The quotient of a group’s automorphism group by its normal subgroup of inner automorphisms.
Outer product	Linear AlgebraTensor and Matrix Products	Map two coordinate vectors to the rank-at-most-one matrix whose ij entry is the product of the first vector’s i component and the second vector’s j component.
Overlap coefficient	Set SimilarityOverlap Measures	A set-similarity measure equal to intersection size divided by the size of the smaller set, reaching one whenever either set contains the other.
Overlapping interval topology	General Topology	A topology on the interval minus-one to one generated by left and right half-open intervals whose overlap produces a standard counterexample with distinctive separation properties.
Overspill	Nonstandard Analysis	Infer that an internal property holding for every standard natural number must also hold at some unlimited hypernatural, because internal predicates cannot cut out exactly the external standard initial segment.
P-adic Hodge theory	Arithmetic Geometry	A comparison theory classifying p-adic Galois representations through period rings and their relations to de Rham, crystalline, semistable and Hodge–Tate cohomology.
P-adic number	Number Theory	An element of the completion of the rational numbers under the non-Archimedean absolute value determined by a prime p.
P-derivation	Arithmetic Geometry	A prime-indexed arithmetic analogue of derivation whose addition and product laws encode a lift of Frobenius modulo p.
P-Laplacian	Nonlinear Partial Differential EquationsCalculus of VariationsGeometric Analysis	The quasilinear second-order operator Δp u = div(|∇u|^(p−2)∇u), typically for p greater than one, which generalizes the Laplacian and arises as the Euler–Lagrange operator of p-energy.
P-Matrix	Matrix Theory	A real square matrix whose every principal minor is strictly positive, equivalently guaranteeing a unique solution to every associated linear complementarity problem.
p-Variation	Real AnalysisStochastic AnalysisRough Paths	In mathematical analysis, p-variation is a collection of seminorms on functions from an ordered set to a metric space, indexed by a real number p\geq 1 . p-variation is a measure of the regularity or smoothness of a function.
Packing density	Discrete GeometryPacking Problems	A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing.
Packing dimension	Fractal Geometry	A fractal dimension defined from the critical exponent of disjoint small-ball packings after a countable-cover regularization.
Packing Problem	OptimizationDiscrete Geometry	An optimization problem seeking an admissible arrangement of specified objects in containers while optimizing density, used extent, container count, or overlap under declared geometric or combinatorial constraints.
Padé Table	Approximation TheoryComplex AnalysisNumerical Analysis	A two-dimensional array whose entry at degree pair (m,n) is the Padé rational approximant with numerator degree at most m and denominator degree at most n matching a given formal power series to the highest prescribed order.
Painlevé transcendents	Differential EquationsIntegrable Systems	New special functions defined by the six canonical nonlinear second-order Painlevé equations, whose movable singularities are poles rather than movable branch points.
Pairing function	Mathematical Logic	A bijection that uniquely encodes an ordered pair of natural numbers as one natural number, with generalizations to higher arity or other infinite sets.
Palm calculus	Stochastic Processes	The probability calculus relating a stationary point process as seen from a typical event to its ordinary time- or space-average law.
Pancyclic graph	Graph Theory	A graph containing a cycle of every length from three through its number of vertices.
Paneitz Operator	Differential GeometryConformal Geometry	In the mathematical field of differential geometry, the Paneitz operator is a fourth-order differential operator defined on a Riemannian manifold of dimension n.
Pansu Derivative	Geometric AnalysisSubriemannian Geometry	The homogeneous group-homomorphism limit of a Carnot-group map's translated and dilated increments at a point, when that limit exists.
Parabola	Analytic Geometry	A conic curve whose points are equidistant from a fixed focus and directrix, equivalently a nondegenerate quadratic curve of eccentricity one.
Parabolic Cylindrical Coordinates	Orthogonal Curvilinear CoordinatesPotential Theory	An orthogonal 3D coordinate system formed by extruding a confocal parabolic coordinate web along a Cartesian axis, with a quadratic planar map and explicit branch domain.
Parabolic geometry (differential geometry)	Differential GeometryCartan Geometries	A Cartan geometry modeled on a homogeneous quotient G/P of a semisimple Lie group by a parabolic subgroup, unifying conformal, projective and related structures.
Parabolic Hausdorff dimension	Fractal GeometryGeometric Measure Theory	In fractal geometry, the parabolic Hausdorff dimension is a restricted version of the genuine Hausdorff dimension.
Parabolic line	Differential Geometry	The curve on a smooth surface where Gaussian curvature is zero and that generically separates elliptic from hyperbolic regions.
Paraboloidal coordinates	Coordinate GeometryOrthogonal Curvilinear Coordinates	Paraboloidal coordinates denotes three-dimensional orthogonal coordinate system in coordinate geometry.
Paradoxical set	Geometric Group TheoryParadoxical Decompositions	A set that can be partitioned into finitely many pieces and moved by a group action into two disjoint reconstructions of the whole, exposing nonamenability and the failure of finitely additive invariant size on all subsets.
Parallelogram Law	Functional Analysis	Test whether a norm comes from an inner product by requiring the squared lengths of every vector pair's sum and difference to equal twice the sum of their squared lengths, with polarization recovering the unique inner product.
Parameter space	Mathematical Modeling	The parameter space is the space of all possible parameter values that define a particular mathematical model.
Paranormal Operator	Operator Theory	A bounded Hilbert-space operator whose first and second iterates satisfy a norm-growth inequality, forming a class broader than hyponormal operators but narrower than arbitrary normaloid operators.
Paravector	Geometric AlgebraClifford Algebra Elements	An element formed by adding a scalar to a vector in a Clifford or geometric algebra, often used to encode spacetime events within a lower-dimensional algebra.
Paris–Harrington theorem	Proof Theory and Combinatorics	A strengthened finite Ramsey statement that is true in the standard natural numbers but not provable in first-order Peano arithmetic.
Parovicenko space	Set Theoretic Topology	A compact Hausdorff space of continuum weight satisfying characteristic separation and interior conditions modeled on the Stone–Čech remainder of the integers.
Parthasarathy's Theorem	Zero Sum GamesGame TheoryMeasure Theory	A minimax theorem for bounded unit-square games with finitely many curve discontinuities, obtaining a mixed value when one player is restricted to Lebesgue-absolutely-continuous strategies.
Partial cube	Graph TheoryIsometric Graph Embeddings	A graph that embeds isometrically into a hypercube, equivalently admitting equal-length bit labels whose Hamming distances exactly equal graph distances.
Partial differential algebraic equation	Differential EquationsNumerical Analysis	In mathematics a partial differential algebraic equation (PDAE) set is an incomplete system of partial differential equations that is closed with a set of algebraic equations.
Partial groupoid	Abstract Algebra	A set equipped with a binary operation that is defined only for a specified subset of ordered pairs.
Partial isometry	Functional AnalysisOperator Theory	A Hilbert-space operator that acts isometrically on the orthogonal complement of its kernel and vanishes on the kernel, mapping an initial subspace onto a final subspace.
Partial k-tree	Graph TheoryStructural Graph Theory	A graph that embeds as a subgraph of a k-tree, equivalently one whose treewidth is at most k.
Partially ordered set	Order Theory	A set equipped with a reflexive, antisymmetric and transitive binary relation whose elements need not all be comparable.
Partially Ordered Space	Ordered Topology	A partially ordered space is a topological space whose partial-order relation is closed in the product topology.
Partition algebra	Representation Theory	An associative diagram algebra whose basis elements are set partitions and whose product concatenates diagrams while weighting closed middle components.
Partition function (number theory)	Number TheoryInteger Partitions	The arithmetic function p(n) that counts unordered representations of a nonnegative integer as a sum of positive integers, with generating-function, recurrence, asymptotic and modular-congruence structure.
Partition of a set	Set Theory	A family of nonempty, pairwise disjoint subsets whose union is the whole underlying set, equivalently the classes of an equivalence relation.
Partition of unity	Differential Topology and Analysis	A locally finite family of nonnegative continuous or smooth functions whose pointwise sum is one, usually with each support subordinate to a member of an open cover.
Pascal matrix	Matrix Theory	A lower-triangular, upper-triangular or symmetric matrix whose entries are binomial coefficients arranged according to Pascal’s triangle.
Pascal's rule	Binomial Coefficients	Decompose the family of fixed-size subsets by whether they contain a distinguished element, yielding the binomial-coefficient recurrence that generates Pascal's triangle.
Path coloring	Graph TheoryPath Conflict Coloring	An assignment of colors to specified graph paths so any two paths sharing an edge receive different colors, commonly modeling wavelength allocation in optical networks.
Path space (algebraic topology)	Algebraic Topology	The mapping space of continuous interval paths in a topological space, either with a fixed starting point or with both endpoints free.
Pathwidth	Graph TheoryGraph Width Parameters	Measure how narrowly a graph can be arranged along a path by minimizing the largest bag size minus one over path decompositions with vertex continuity and edge coverage.
Pauli–Villars regularization	Quantum Field Theory	A field-theory regulator that subtracts auxiliary massive-field contributions to suppress ultraviolet divergences.
Peirce's Law	Propositional LogicProof Theory	An implication-only formula valid in classical logic but not intuitionistic logic: if (P→Q) implies P, then P.
Penrose graphical notation	Tensor Calculus	A diagrammatic tensor notation in which shapes, lines and contractions visually encode multilinear maps, indices and composition.
Penrose tiling	Tiling TheoryAperiodic Order	Cover the plane nonperiodically with a finite set of prototiles and matching rules that forbid translational periodicity yet produce repetitive local patches, inflation symmetry and long-range fivefold order.
Pentadiagonal Matrix	Matrix TheoryNumerical Linear Algebra	In mathematics, particularly matrix theory, a band matrix or banded matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side.
Pentagonal Planar Molecular Geometry	Molecular GeometryHypervalent Main Group Molecular Shapes	Classify a five-coordinate molecular center whose five bonded ligands and central atom lie in one plane around an approximately pentagonal perimeter, ideally giving AX5E2 and D5h geometry.
Perfect core	Group Theory	The largest perfect subgroup of a group, equivalently the stable term of its transfinite derived series.
Perfect digit-to-digit invariant	Recreational Number Theory	A perfect digit-to-digit invariant is a positive integer equal to the sum obtained by raising each of its decimal digits to the power of that same digit.
Perfect measure	Measure Theory	A finite measure whose real-valued measurable images admit Borel approximations with no measure gap.
Perfect number	Number TheoryDivisor Sums and Multiplicative Arithmetic	Classify a positive integer as perfect when the sum of all its proper positive divisors equals the integer itself, equivalently when its divisor-sum is exactly twice the integer.
Perfect obstruction theory	Derived and Enumerative Algebraic Geometry	A morphism from a perfect two-term complex to a space's cotangent complex that correctly captures first-order deformations and obstructions and supports a virtual fundamental class.
Perfect rectangle	Discrete GeometrySquared Rectangles	A rectangle tiled exactly by finitely many squares whose side lengths are all distinct.
Perfect ring	Ring Theory	A ring for which every module on the specified side has a projective cover, with equivalent chain and radical conditions under Bass's theorem.
Perfect ruler	CombinatoricsDifference Rulers	An integer-marked ruler for which every distance through a stated maximum occurs exactly once as a positive difference between two marks.
Perfect spline	Approximation Theory	A univariate spline of order m whose m-th derivative takes alternating values plus or minus one between successive knots.
Permutation group	Group Theory	A group whose elements are bijections of a set and whose operation is function composition, equivalently a group action represented faithfully by permutations.
Perpetuant	Classical Invariant Theory	A stable irreducible covariant in classical invariant theory whose degree–weight space persists once the binary form’s degree exceeds the covariant weight.
Perrin number	Number Theory	A doubly infinite integer sequence generated from initial values 3, 0 and 2 by adding terms two and three positions earlier.
Persistence of a number	Recreational Number Theory	The number of repeated applications of a specified digit operation needed for an integer to reach a fixed point, most commonly a single digit under repeated digit sums or products.
Peters Polynomials	Special FunctionsUmbral Calculus	The polynomial sequence whose exponential generating function is (1+t)^x/(1+(1+t)^λ)^μ.
Pettis integral	Weak Vector Integration	Integrate a Banach-space-valued function weakly by requiring every continuous linear functional to yield an ordinary scalar integral represented by one vector for each measurable set.
Phragmen–Brouwer theorem	General Topology and Separation Theory	A theorem schema equating a separation property for unions of disjoint closed sets with unicoherence under specified local and global connectedness hypotheses.
Phragmén–Lindelöf principle	Complex Analysis	A complex-analytic extension of the maximum-modulus principle that controls a holomorphic function on an unbounded domain using boundary bounds plus a growth restriction.
Pi	Mathematical Constants	The dimensionless mathematical constant equal to a Euclidean circle's circumference divided by its diameter, equivalently characterized through analysis, with an irrational transcendental value near 3.14159.
Picard Group	Algebraic Geometry	Classify invertible line-bundle classes on a fixed locally ringed base, with tensor product making them an abelian group.
Picard–Lefschetz Theory	Singularity Theory	Analyze how the topology of fibers changes around isolated critical values of a complex map by encoding vanishing cycles and the monodromy transformations generated by loops around those values.
Pidduck polynomials	Special Functions	A named polynomial sequence defined by an exponential generating function involving the ratio of one plus t to one minus t.
Piecewise function	Mathematical Functions and Modeling	A function specified by different formulas or rules on declared subsets that partition or cover its domain with consistent treatment of their boundaries.
Piecewise syndetic set	Combinatorial Number Theory	A subset of the natural numbers whose gaps are uniformly bounded on arbitrarily long intervals.
Pillai's Arithmetical Function	Number TheoryMultiplicative Arithmetic Functions	The multiplicative gcd-sum function P(n)=sum from k=1 to n of gcd(k,n), whose divisor-class decomposition P=id*phi exposes prime-power evaluation, Euler products, and average-order analysis.
Piola transformation	Applied MathematicsContinuum MechanicsFinite Element Methods	A Jacobian-scaled mapping of a reference vector field to a physical domain that preserves corresponding boundary flux in continuum and finite-element settings.
Planar ternary ring	Projective GeometryFoundations of Geometry	A coordinate algebra for projective planes consisting of a set with distinguished 0 and 1 and a ternary operation T satisfying five incidence-solving axioms, generalizing the field expression T(a,b,c)=ab+c.
Planarity	Graph Theory	The graph property of admitting a crossing-free drawing in the plane — pinned by Kuratowski/Wagner to a finite obstruction (no K₅ or K₃,₃) and by Euler's formula to a density bound that makes planar graphs sparse; planar structure also makes some NP-hard problems, such as max-cut, polynomial, though others, such as Hamiltonian cycle, stay NP-hard.
Plane of Rotation	Geometry	Represent a simple Euclidean rotation by the oriented two-dimensional subspace in which vectors turn, leaving its orthogonal complement fixed.
Pluriharmonic function	Several Complex Variables	A function on a complex manifold whose restriction to every complex line is harmonic, locally the real part of a holomorphic function in the real-valued case.
Pluripolar set	Pluripotential Theory	A subset of a complex domain contained in the negative-infinity locus of a nontrivial plurisubharmonic function.
Plurisubharmonic function	Complex Analysis	An upper-semicontinuous function on a complex domain whose restriction to every complex line is subharmonic.
Pocket set theory	Set TheoryAlternative Set Theories	Pocket set theory (PST) is an alternative set theory in which there are only two infinite cardinal numbers, ℵ 0 (aleph-naught, the cardinality of the set of all natural numbers) and c (the cardinality of the continuum).
Poincaré space	Algebraic Topology	A finite-type space equipped with a fundamental homology class whose cap product realizes Poincaré duality in every degree.
Point reflection	Geometry	An affine transformation that sends each point x to 2c−x about a fixed center c, preserving distances and reversing every displacement vector.
Pointed set	Category TheoryPointed Objects	A set equipped with one distinguished basepoint, with morphisms required to preserve that point, forming a category that adds a canonical zero-like reference to otherwise unstructured sets.
Poisson binomial distribution	Probability TheoryHeterogeneous Bernoulli Sums	Describe the success count obtained by summing independent Bernoulli variables whose success probabilities may differ, retaining the full heterogeneous probability vector.
Poisson boundary	ProbabilityRandom Walk Boundaries	A measure-theoretic boundary of a random walk that captures its asymptotic tail behavior and represents bounded harmonic functions by boundary data.
Poisson geometry	Differential GeometrySymplectic Geometry	In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure.
Polar space	Incidence Geometry	An incidence geometry of points and singular subspaces modeled by isotropic subspaces of a form and satisfying the one-or-all axiom.
Pole and polar	Projective Geometry	A reciprocal point-line correspondence induced by a nondegenerate conic or quadric that reverses incidence.
Polish notation	Formal NotationExpression Syntax	A prefix expression notation in which each fixed-arity operator precedes its operands, making parse structure unambiguous without parentheses.
Poly-Bernoulli number	Enumerative Combinatorics and Number Theory	A doubly indexed integer sequence defined by an exponential generating function involving the polylogarithm and generalizing Bernoulli numbers.
Polyad (mathematics)	Category Theory and Bicategories	A bicategorical generalization of a monad in which a locally punctual indexing bicategory maps into another bicategory, distributing monad-like data across multiple objects.
Polygon	Plane GeometryGeometry	A planar geometric figure whose boundary is a finite cyclic sequence of straight line segments joined endpoint to endpoint under a declared simplicity and interior convention.
Polygon partition	Computational GeometryPolygon Decomposition	A decomposition of a polygon into nonoverlapping primitive polygons whose union is the original polygon, often optimized by piece count, boundary length or another criterion.
Polyhedral Complex	Polyhedral Geometry	A face-closed collection of convex polyhedra whose pairwise intersections are common faces, assembling coherent geometric cells.
Polyhedral space	Metric Geometry	A metric space assembled by gluing constant-curvature simplices isometrically along compatible faces.
Polyhedron	Polyhedral GeometryGeometry	A three-dimensional geometric figure formed from finitely many flat polygonal faces joined edge-to-edge under declared surface, solid, manifold, boundedness, and self-intersection conventions.
Polynomial	Polynomial AlgebraAlgebra	A finite formal sum of monomials with coefficients in a declared ring and nonnegative integer exponents, distinguished from the function obtained by evaluating it.
Polynomial Content and Primitive Part	Commutative AlgebraPolynomial Rings	Separate a nonzero polynomial over a unique factorization domain into its coefficient gcd and a residual polynomial with unit coefficient gcd.
Polynomial differential form	Rational Homotopy Theory	An element of the commutative differential graded algebra of polynomial coordinate functions and their differentials on a standard simplex, varying simplicially with face and degeneracy maps.
Polynomial identity ring	Ring Theory	A ring on which some nonzero noncommutative polynomial vanishes under every substitution of ring elements.
Polynomial Matrix Spectral Factorization	Operator TheoryMatrix Polynomials	Polynomial matrix spectral factorization turns one-variable positive Hermitian polynomial-matrix data into a polynomial factor and its domain-appropriate adjoint.
Polynomial Ring	Abstract AlgebraCommutative Algebra	Adjoin one or more algebraically free commuting indeterminates to a coefficient ring, with finite coefficient support and the universal substitution property.
Polynomially Reflexive Space	Functional AnalysisBanach Space Theory	A Banach space X for which, at every positive degree n, the Banach space of continuous scalar-valued n-homogeneous polynomials on X is reflexive.
Polytope	Convex GeometryDiscrete Geometry	A finite-dimensional flat-sided geometric or ranked-incidence object that generalizes polygons and polyhedra under an explicit convexity, boundedness, realization, and face convention.
Pontryagin duality	Harmonic Analysis and Topological Groups	A duality for locally compact abelian groups that assigns each group its continuous characters into the circle group and naturally identifies the original group with its double dual.
Poset topology	Order TheoryTopological Combinatorics	In mathematics, the poset topology associated to a poset (S, ≤) is the Alexandrov topology (open sets are upper sets) on the poset of finite chains of (S, ≤), ordered by inclusion.
Positive element	Functional AnalysisC Star Algebras and Operator Order	Place a self-adjoint element of a C-star algebra in its positive cone when its spectrum is nonnegative, equivalently when it is a star-square b-star-b or has a unique positive square root, thereby inducing the order used throughout operator algebra.
Positive form	Complex Geometry	A real differential form of Hodge type (p,p) satisfying a declared positivity cone, with several inequivalent notions in higher bidegree.
Positive harmonic function	Harmonic and Complex Analysis	Characterize a nonnegative harmonic function on the unit disc as the Poisson integral of a unique finite positive boundary measure, with normalization at the origin fixing the measure's total mass.
Positive set theory	Mathematical LogicAlternative Set Theories	A family of alternative set theories permitting comprehension for positive membership formulas while restricting negation so broad set formation avoids classical paradoxes.
Positive-definite kernel	Functional AnalysisPositive Kernels and Reproducing Kernel Spaces	Assign a Hermitian pairwise function whose every finite Gram matrix has a nonnegative quadratic form, equivalently realizing the inputs as inner-product feature vectors in a Hilbert space.
Positively separated sets	Metric Geometry and Analysis	Two nonempty subsets of a metric space whose infimum pairwise distance is strictly greater than zero.
Postselection	Probability TheoryConditional Selection	Conditioning an experiment, probability model or computation on a specified event after outcomes are available, thereby replacing the original distribution with its conditional distribution.
Powell's method	Numerical Optimization	A derivative-free local optimization algorithm that performs successive line minimizations along a changing set of directions and replaces a direction with the net displacement to build approximate conjugacy.
Power Associativity	Nonassociative AlgebraPower Associative Algebras	Require associativity inside every one-generated substructure so repeated powers are independent of parenthesization even when mixed-element products are not.
Power Residue Symbol	Algebraic Number TheoryReciprocity Laws	An nth-root-of-unity-valued character that generalizes the Legendre symbol by recording whether and how an algebraic integer is an nth-power residue modulo a suitable prime ideal.
Pre-measure	Measure Theory	A countably additive nonnegative set function defined on an algebra or ring of sets, serving as the extendable precursor of a measure on a generated sigma-algebra.
Predicate abstraction	Mathematical Logic	In logic, predicate abstraction is the result of creating a predicate from a formula.
Predual	Functional Analysis	Predual denotes banach space of a dual within functional analysis.
Prenex normal form	Mathematical Logic	A first-order formula form in which all quantifiers occur in one leading prefix followed by a quantifier-free matrix.
Presheaf (category theory)	Category Theory	A contravariant set-valued functor on a category, assigning data to each object and restriction maps to each morphism.
Presheaf with transfers	Algebraic GeometryMotivic Cohomology	In algebraic geometry, a presheaf with transfers is, roughly, a presheaf that, like cohomology theory, comes with pushforwards, “transfer” maps.
Prestack	Algebraic Geometry and Category Theory	A category fibered in groupoids over a site whose isomorphisms satisfy descent, while objects need not yet glue effectively as they do in a stack.
Prevalent and shy sets	Infinite Dimensional Analysis	Translation-based analogues of full measure and measure zero for subsets of infinite-dimensional vector spaces.
Primal ideal	Ring TheoryCommutative Algebra	A proper ideal I of a commutative ring A is said to be primal if the elements that are not prime to it form an ideal.
Prime Graph	Finite Group TheoryElement Order Structure	Represent a finite group by making the prime divisors of its order the vertices and joining distinct primes p and q exactly when the group contains an element of order pq.
Prime ideal	Commutative AlgebraIdeal Theory	A proper ideal P of a commutative ring such that ab in P implies a in P or b in P, equivalently making the quotient ring an integral domain.
Prime k-tuple	Analytic Number TheoryPrime Numbers	A fixed finite offset set considered through common integer translates whose entries are all prime at an occurrence.
Prime manifold	Manifold Topology	In topology, a branch of mathematics, a prime manifold is an n-manifold that cannot be expressed as a non-trivial connected sum of two n-manifolds.
Prime Model (Model Theory)	Model TheoryModel Classification	A model of a complete first-order theory that admits an elementary embedding into every model of that theory, making it the theory's embedding-minimal representative when one exists.
Prime signature	Number TheoryMultiplicative Classification of Integers	Classify a positive integer by the unordered multiset of positive exponents in its unique prime factorization, discarding prime labels while preserving multiplicative shape.
Prime triplet	Number Theory	A set of three prime numbers spanning six integers, necessarily in one of two offset patterns apart from exceptional triples containing three.
Prime-counting function	Analytic Number Theory	The arithmetic function pi of x that counts prime numbers less than or equal to a real bound x.
Primefree Sequence	Number TheoryRecurrence Sequences	A nontrivial Fibonacci-type integer sequence begun from coprime composite seeds and proved to contain only composite terms, typically by a finite cover of periodic modular divisibility classes.
Primitive ideal	Ring TheoryNoncommutative Algebra	In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module.
Primitive ring	Ring Theory	A ring admitting a faithful simple left module or, separately, a faithful simple right module.
Primitive Semiperfect Number	Elementary Number Theory	A semiperfect integer with no smaller semiperfect divisor.
Principal homogeneous space	Algebra and Geometry	A nonempty set or space with a free and transitive action of a group, also called a torsor.
Principal Ideal	Ring Theory	Form the smallest ideal containing one ring element, using all allowed left, right, or two-sided ring multiples and additive closure.
Principal indecomposable module	Module Theory	An indecomposable direct summand of the regular module of a ring, equivalently an indecomposable projective cyclic module under standard hypotheses.
Principal Value	Complex AnalysisMultivalued Analytic Functions	A branch convention selects one reproducible value from a multivalued complex relation while making its cut, boundary, and analytic-continuation limits explicit.
Principle of distributivity	Propositional LogicBoolean Equivalences	The propositional-logic laws stating that conjunction distributes over disjunction and disjunction distributes over conjunction.
Prior Probability	Bayesian Statistics	Represent uncertainty about a parameter or hypothesis before the focal evidence is incorporated by assigning it a probability distribution that will be combined with a likelihood.
Prism graph	Graph Theory	The cubic graph C_n □ K_2 of an n-gonal prism, formed by two n-cycles joined by matching cross-layer edges.
Prismatic surface	GeometryRuled Polyhedral Surfaces	Generate a polyhedral ruled surface by translating every point of a polygonal-chain directrix along one fixed direction, producing parallel planar strips joined along parallel generators.
Probabilistic Graphical Model	Probability and StatisticsGraphical Models	A statistical model whose graph and declared Markov semantics encode conditional independences and a corresponding factorization of a joint probability law into local terms.
Probability axioms	Probability Theory	The foundational conditions requiring a probability measure to be nonnegative, assign one to the whole sample space and add over countably many disjoint events.
Probability Density Function	Probability Theory	A nonnegative function integrating to one relative to a declared measure, with probabilities of measurable regions obtained by integrating the function.
Probability Distribution	Probability Theory	The complete specification of how probability mass or density is spread over a random variable's possible values — a measure that, once compressed to a named parametric family, encodes shape, moments, tails, and a generative claim about the process producing the data.
Probability integral transform	Probability TheoryDistributional Transformations	The result that applying a continuous random variable's own cumulative distribution function produces a standard uniform random variable.
Probability Mass Function	Probability Theory	Represent a discrete random variable’s law by the nonnegative singleton probabilities p(x)=P(X=x), whose sum over its countable support equals one.
Probability measure	Probability Theory	A countably additive measure on a sigma-algebra that assigns total mass one to the sample space.
Probable prime	Computational Number Theory	Probable prime denotes number that satisfies a given necessary condition for primality within computational number theory.
Problem of Points	Classical Probability	Divide the stake of an interrupted race-to-a-target game by each player's conditional probability of eventually winning, computed from the current score and the agreed continuation model rather than from points already accumulated.
Procrustes transformation	Geometry	A similarity transformation using translation, rotation, reflection if allowed and uniform scaling while preserving shape.
Product measure	Measure TheoryProduct Spaces and Integration	Construct a measure on a product measurable space whose values on measurable rectangles multiply the component measures, with existence and uniqueness controlled by sigma-finiteness or related hypotheses.
Product of Rings	Ring Theory	The Cartesian product of a family of rings with coordinatewise operations, characterized by projection homomorphisms satisfying the categorical-product universal property.
Product order	Order Theory	In mathematics, given partial orders \preceq and \sqsubseteq on sets A and B , respectively, the product order (also called the coordinatewise order or componentwise order ) is a partial order \leq on the Cartesian product A \times B.
Product term	Boolean Algebra	A Boolean expression formed as the conjunction of one or more literals, with a minterm as the special case containing every variable exactly once.
Profinite group	Topological Group Theory	A compact totally disconnected Hausdorff topological group expressible as an inverse limit of finite discrete groups.
Profinite Integer	Profinite Groups and Arithmetic	An element of the inverse-limit ring of all finite residue rings of the integers—a compatible residue modulo every positive integer, equivalently one p-adic integer for every prime.
Profinite Word	Algebraic Automata Theory	An element of the all-finite-monoid completion of finite words over a finite alphabet, determined by its compatible images in every finite quotient.
Profunctor	Category Theory	In category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules.
Progressive function	Harmonic Analysis	An L2 signal whose Fourier transform is supported only on nonnegative frequencies, equivalently a boundary function in the upper-half-plane Hardy space under the stated convention.
Progressively measurable process	Stochastic Processes	A stochastic process whose restriction through every time t is jointly measurable with respect to Borel time and the information available by t.
Projection filters	Stochastic FilteringInformation Geometric Approximation	Nonlinear state-estimation algorithms that approximate an evolving conditional probability density by projecting infinite-dimensional filtering dynamics onto a finite-dimensional statistical manifold.
Projection-valued measure	Functional AnalysisSpectral Theory	In mathematics, particularly in functional analysis, a projection-valued measure, or spectral measure, is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space.
Projective bundle	Algebraic Geometry	A fiber bundle or scheme morphism locally modeled on projective space, often obtained by projectivizing a vector bundle.
Projective line	Projective Geometry and Algebraic Geometry	The one-dimensional projective space over a field or ring, commonly the one-dimensional subspaces of a two-dimensional vector space and equivalently an affine line completed by points at infinity.
Projective representation	Representation TheoryGroup Representations	A homomorphism from a group to a projective linear group, equivalently linear operators whose multiplication respects the group law only up to nonzero scalar factors.
Projective tensor product	Functional Analysis	The tensor product of locally convex spaces equipped with the strongest locally convex topology making the canonical bilinear map continuous.
Projective variety	Algebraic Geometry	In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space.
Projectively extended real line	Real and Projective Analysis	The real line completed by one unsigned point at infinity, yielding a topological circle and the real projective line.
Projectivization	Projective Geometry	The construction that maps a nonzero vector space, cone or vector bundle to its space of one-dimensional linear subspaces by quotienting nonzero vectors under scalar equivalence.
Prolate Spheroidal Coordinates	Orthogonal Coordinate Systems	Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located.
Proof calculus	Proof TheoryMathematical Logic	A formal proof framework specifying admissible formulas, axioms or assumptions, and inference-rule patterns for deriving theorems.
Proper Convex Function	Convex AnalysisExtended Real Valued Functions	An extended-real convex function whose effective domain is nonempty and which nowhere takes negative infinity, excluding the two degenerate functions that break convex-analytic operations.
Proper forcing axiom	Set TheoryForcing Axioms	A strong set-theoretic forcing axiom asserting that for any proper partial order and any family of ℵ₁ dense sets, a filter meeting every one of them exists.
Property of Baire	Descriptive Set Theory	The property of a set that it differs from some open set by a meager set.
Proportionality (mathematics)	—	A relation in which corresponding quantities maintain a constant ratio, or under inverse proportionality a constant product.
Propositional formula	Mathematical LogicPropositional Logic	Type of logical formula in propositional logic.
Propositional function	Mathematical Logic	An open sentence containing free variables that becomes true or false when admissible values are substituted.
Propositional logic	Mathematical Logic	A formal logic that treats whole propositions as truth-valued atoms and builds compound formulas with connectives, deriving validity, equivalence, satisfiability, and consequence from their truth-functional structure.
Prototile	Tiling TheoryDiscrete Geometry	In mathematics, a prototile is one of the shapes of a tile in a tessellation.
Prouhet–Thue–Morse constant	Number Theory	The real number whose binary expansion is the Thue–Morse sequence.
Proximal operator	Convex Optimization	The operator mapping a point to the unique minimizer of a function plus one-half the squared distance to that point, under standard proper lower-semicontinuous convex assumptions.
Pseudo algebraically closed field	Field Arithmetic and Model Theory	A field in which every absolutely irreducible variety defined over the field has a rational point, imitating a key geometric consequence of algebraic closure without requiring every polynomial to split.
Pseudo-abelian category	Category Theory and Homological Algebra	A preadditive category in which every idempotent splits, equivalently every idempotent has an appropriate kernel and cokernel decomposition.
Pseudo-canonical variety	Birational Geometry	An algebraic variety whose canonical divisor or class is pseudo-ample under the convention used, placing it in the general-type side of birational classification.
Pseudo-differential operator	Mathematical AnalysisMicrolocal Analysis	An operator defined through a position- and frequency-dependent symbol, extending differential operators to non-polynomial multipliers and supporting microlocal analysis of PDE.
Pseudo-Euclidean Space	GeometryQuadratic Form Geometry	A finite-dimensional real vector or affine space equipped with a nondegenerate symmetric bilinear form of mixed signature, admitting positive, negative, and nonzero null directions.
Pseudoanalytic function	Complex Analysis	A complex-valued function satisfying a generalized first-order Cauchy–Riemann system determined by an admissible coefficient.
Pseudometric space	Generalized Metric Spaces	Equip a set with a symmetric, nonnegative, triangle-inequality distance that may assign zero separation to distinct points, with metric quotient obtained by identifying zero-distance classes.
Pseudomonad (category theory)	Higher Category TheoryCategory Theory	A higher-categorical monad whose unit and multiplication laws hold up to coherent invertible cells.
Pseudoreflection	Invariant Theory	A finite-order nonidentity linear automorphism whose fixed subspace is a hyperplane.
Pseudospectrum	Operator Theory	For an operator and tolerance epsilon, the set of spectral values attainable under perturbations of size epsilon, equivalently points where the resolvent is large.
Pullback (category theory)	Category Theory	The categorical limit of two morphisms sharing a codomain.
Pullback (differential geometry)	Differential Geometry	The pullback transfers covariant geometric data from a map's codomain to its domain by precomposing with the differential, preserving functorial relations for functions, differential forms, metrics, and bundles.
Pullback attractor	Dynamical SystemsRandom Dynamical Systems	Importantly, in the case of a deterministic dynamical system (one without noise), the pullback limit coincides with the deterministic forward limit, so it is meaningful to compare deterministic and random omega-limit sets, attractors, and so forth.
Pursuit Curve	Differential GeometryOrdinary Differential Equations	Trace the path of a pursuer whose instantaneous forward tangent points toward a moving target's current position.
Pushforward measure	Measure TheoryMeasure Transport	The measure on a target measurable space obtained by assigning each target set the original measure of its preimage under a measurable map.
Pushout (category theory)	Category TheoryColimits	The colimit of a span X←Z→Y, giving the universal object formed by mapping X and Y together while identifying their images of Z.
Pyjama problem	Tiling Theory	In mathematics, the pyjama problem asks whether the plane can be covered by a finite number of rotated copies of a repeating pattern of stripes ("pyjama stripes"), no matter how thin the stripes are.
Pythagorean Hodograph Curve	Computer Aided Geometric Design	A planar polynomial curve whose derivative components have a polynomial-square norm, giving exact polynomial arc length on regular intervals.
q-Analog	Q Series and Algebraic Combinatorics	A mathematically natural q-parameter deformation of an object, identity, or theorem that recovers a designated classical counterpart, usually at q approaching 1.
Q-difference polynomial	CombinatoricsQ Calculus Polynomial Sequences	A polynomial sequence lowered by the q-derivative according to D_q p_n=[n]_q p_(n−1), generalizing Appell polynomials and ordinary differentiation.
Q-function	ProbabilityNormal Tail Probability	Map a real threshold to the upper-tail probability of a standard normal variable, equivalently one half of the complementary error function at the threshold divided by the square root of two.
Q-Gaussian process	Noncommutative ProbabilityFree Probability	A noncommutative multivariate stochastic process whose joint moments are generated by q-deformed commutation relations, interpolating among classical Gaussian, free, and other deformation regimes as the parameter q changes.
Q-Weibull distribution	Probability and StatisticsGeneralized Lifetime Distributions	A positive distribution formed by replacing the Weibull exponential with a q-exponential, recovering Weibull at q=1 and allowing compact-support or heavy-tail behavior according to q.
QR Algorithm	Numerical Linear AlgebraEigenvalue Algorithms	An eigenvalue iteration that repeatedly QR-factorizes a shifted matrix and reverses the factors, preserving similarity while driving it toward real or complex Schur form for deflation.
QR Decomposition	Numerical Linear Algebra	An exact matrix factorization into an orthogonal or unitary coordinate factor and an upper-triangular or trapezoidal coefficient factor.
Quadratic Assignment Problem	Combinatorial Optimization	Optimize a one-to-one placement when the cost of one item-position choice depends on other simultaneous placements.
Quadratic differential	Complex Geometry	A section of the square of a Riemann surface’s holomorphic cotangent bundle, locally written as a coefficient times the square of a coordinate differential.
Quadratic Equation	—	A one-variable degree-two polynomial equality whose coefficient triple, discriminant, and root formulas provide a complete solution classification over a declared scalar domain.
Quadratic extension	Field Theory	A quadratic extension is a field extension L/K of degree two, meaning that L is a two-dimensional vector space over K.
Quadratic Field	Algebraic Number Theory	A quadratic field is a degree-two extension of the rational field, equivalently a uniquely determined field Q(sqrt(d)) for a squarefree integer d other than 1, with d positive in the real case and negative in the imaginary case.
Quadratic function	Algebra	A polynomial function of degree exactly two, represented by a nonzero quadratic form plus lower-degree terms.
Quadratic Integer	Algebraic Number TheoryRings of Integers of Quadratic Fields	An algebraic integer in a quadratic number field—strictly, a nonrational algebraic integer of degree two—whose trace, norm, conjugation, and field-specific integer-ring lattice organize quadratic arithmetic.
Quadratic residuosity problem	Computational Number TheoryCryptographic Hardness Problems	The computational decision problem of determining whether a number with Jacobi symbol one is a square modulo a composite whose prime factorization is unknown.
Quadratic Space	—	A vector space equipped with a quadratic form, carrying isotropy, radical, orthogonality, and equivalence structure that depends essentially on the base field.
Quadratically Closed Field	Field TheoryQuadratic Form Theory	A quadratically closed field contains a square root of every one of its elements, eliminating missing roots of quadratic polynomials.
Quadrature domains	Complex AnalysisPotential Theory	It is known that quadrature domains exist for all values of k.
Quadrisecant	GeometrySpace Curves and Knot Geometry	A line meeting a spatial curve or related geometric set in four distinct points, a maximal generic multi-secant whose existence and ordering carry knot- and algebraic-geometric information.
Quantifier Elimination	Model TheoryMathematical Logic	The property that every formula of a specified theory and language has a quantifier-free equivalent valid in all its models.
Quantifier Rank	Mathematical LogicModel Theory	Assign a logical formula the maximum nesting depth of its quantifiers, stratifying expressibility so bounded-rank formulas, types, and structure-distinguishing games can be compared at a fixed logical depth.
Quantifier-Alternation Hierarchy	Definability HierarchiesQuantifier Complexity	Stratify definable properties by alternating existential and universal quantifier blocks over a declared base class, keeping the quantified objects and equivalence criterion explicit.
Quantile	Probability and Statistics	A distribution cut point at or below which a specified cumulative probability lies, defined through a generalized inverse when the distribution has jumps or flat regions.
Quantile function	Probability Theory	A generalized inverse of a cumulative distribution function that maps a probability level to the smallest value whose cumulative probability reaches that level.
Quantized State Systems Method	Numerical SimulationQuantization Based Integration	A family of event-driven numerical integrators that quantize state trajectories instead of time, scheduling each component's next update when its continuous state departs from its quantized surrogate by a prescribed quantum.
Quantum calculus	Mathematical AnalysisQ and H Calculus	Calculus built from finite q-ratio or h-shift difference operators and corresponding sums, recovering ordinary differential and integral calculus as q→1 or h→0 rather than using limits internally.
Quantum Cohomology	Algebraic GeometrySymplectic Geometry	A cohomology algebra whose cup product is deformed by curve-class-weighted genus-zero Gromov–Witten invariants.
Quantum graph	Spectral Geometry	A metric graph equipped with differential operators on its edges and vertex boundary conditions that couple edgewise wavefunctions.
Quarter period	Special FunctionsElliptic Functions	The complete elliptic-integral quantities K(m) and iK′(m) that generate the period lattice of Jacobi elliptic functions and locate their characteristic quarter-cycle values.
Quartic graph	Graph TheoryRegular Graphs	A graph in which every vertex has degree four, also called a 4-regular graph.
Quartic reciprocity	Algebraic and Elementary Number Theory	A family of reciprocity theorems relating the solvability of fourth-power congruences after interchanging the relevant prime moduli.
Quasi-algebraically closed field	Field TheoryDiophantine Dimension and C K Fields	Classify a field as C1 when every nonconstant homogeneous form of degree d in more than d variables has a nontrivial zero over that field.
Quasi-arithmetic mean	Theory of Means	Average values by mapping them through a continuous strictly monotone generator, taking an arithmetic mean in transformed coordinates, and mapping the result back.
Quasi-category	Higher Category Theory	A simplicial set satisfying every inner horn-filling condition, modeling an infinity-category with composition coherent up to higher homotopy.
Quasi-Finite Field	Field TheoryGalois Theory	A perfect field with procyclic absolute Galois group, equivalently a unique cyclic extension of every finite degree whose union is the separable closure.
Quasi-finite morphism	Algebraic Geometry	A finite-type morphism of schemes whose fibers are zero-dimensional and finite, equivalently one that is locally finite over each image point.
Quasi-Frobenius Lie algebra	Lie Theory	If (\mathfrak{g},[\,\,\,,\,\,\,],\beta ) is a quasi-Frobenius Lie algebra, one can define on \mathfrak{g} another bilinear product \triangleleft by the formula.
Quasi-Hopf algebra	Quantum Algebra	A quasi-bialgebra equipped with an antipode and distinguished elements whose identities replace the strict Hopf antipode laws in the presence of a nontrivial associator.
Quasi-Invariant Measure	Measure TheoryNonsingular Transformations and Group Actions	A measure whose class of null sets, though not necessarily its numerical values, is preserved by every transformation in a specified action, so each pushforward remains equivalent to the original and changes density through a Radon–Nikodym cocycle.
Quasi-Isometry	Geometric Group TheoryMetric Geometry	In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details.
Quasi-Isomorphism	Homological Algebra	A morphism of complexes inducing degreewise isomorphisms on homology or cohomology.
Quasi-Open Map	General Topology	Map every nonempty open subset of a topological domain to a set with nonempty interior, retaining a weakened openness guarantee without requiring the image itself to be open.
Quasi-polynomial	Algebra	In mathematics, a quasi-polynomial (sometimes called pseudo-polynomial) is a generalization of polynomials.
Quasi-projective variety	Algebraic Geometry	A quasi-projective variety is a locally closed subvariety of projective space, equivalently an intersection of a Zariski-open and Zariski-closed subset.
Quasiconvex Function	Convex Analysis	A real-valued function on a convex domain whose sublevel sets are convex, preserving convex feasibility under every threshold without requiring the stronger convex-function inequality.
Quasifield	Nonassociative Algebra	A nonassociative division-like algebra whose additive structure is a group and whose multiplication supports division while satisfying only selected distributive laws.
Quasigroup	Nonassociative Algebra	Equip a set with a closed binary operation for which either missing operand in an equation is uniquely recoverable, equivalently making every left and right translation a bijection, without requiring identity or associativity.
Quasimartingale	Stochastic ProcessesStochastic Integration	An adapted stochastic process of finite mean variation; with càdlàg paths it coincides with a semimartingale.
Quasinormal operator	Operator TheoryFunctional Analysis	A bounded Hilbert-space operator A that commutes with A*A, equivalently one whose partial-isometry and positive factors commute in its polar decomposition.
Quasiperfect number	Number TheoryDivisor Sum Number Classes	Classify a positive integer by the divisor-sum equation sigma of n equals twice n plus one, an unresolved class whose hypothetical members must satisfy strong odd-square and prime-factor restrictions.
Quasiregular Representation	Representation Theory	The unitary representation generated by a group action on an L2 space of a homogeneous or measured space, with a square-root Radon–Nikodym factor correcting any merely quasi-invariant measure.
Quasisymmetric function	Algebraic Combinatorics	A bounded-degree formal power series whose coefficient depends on an exponent composition but not on the particular increasing sequence of variable indices.
Quasisymmetric map	Controlled Metric Homeomorphisms	Control relative metric distortion by requiring every ratio of two distances from a common base point to be bounded through one homeomorphic control function.
Quasitrace	Operator Algebras	A positive tracial functional on a C-star algebra that is homogeneous and additive on commuting positive elements but need not be globally additive.
Quasivariety	Universal Algebra and Model Theory	A class of algebraic structures of a fixed signature axiomatizable by quasi-identities, equivalently closed under isomorphisms, subalgebras, direct products, and ultraproduct or reduced-product conditions under standard formulations.
Quaternary cubic	Algebraic GeometryCubic Forms	A homogeneous polynomial of degree three in four variables, whose projective zero locus is a cubic surface and whose coefficients carry a classical ring of invariants.
Quaternion	AlgebraHypercomplex Numbers	A quaternion is an element a+bi+cj+dk of the four-dimensional real algebra H, where i squared, j squared, and k squared equal minus one and ij=k, jk=i, ki=j with reversed products negated, giving noncommutative multiplication, conjugation, norm, and inversion.
Quaternion-Kähler Manifold	—	A Riemannian manifold of dimension divisible by four whose Levi-Civita holonomy lies in Sp(n)Sp(1), carrying a parallel rank-three quaternionic structure rather than a globally selected complex structure.
Quaternionic discrete series representation	Representation TheoryLie Group Discrete Series	A discrete-series representation of a semisimple real Lie group whose symmetric space carries the quaternionic structure associated with an SU(2) factor in a maximal compact subgroup.
Quaternionic eigenvalue problem	Quaternionic Linear Algebra	The problem of finding left or right eigenvalues and eigenvectors of a matrix with quaternion entries, where noncommutativity makes the side of scalar multiplication constitutive.
Quaternionic representation	Representation Theory	A complex group representation carrying an invariant antilinear equivariant operator whose square is minus the identity, equivalently a representation of quaternionic type.
Queue number	Graph Layouts and Structural Graph Theory	The minimum number of edge queues needed in a vertex ordering of a graph so no two edges in the same queue are properly nested.
Quillen Adjunction	Homotopical Algebra	An adjoint pair between model categories whose left member preserves cofibrations and trivial cofibrations, equivalently whose right member preserves fibrations and trivial fibrations.
Quillen determinant line bundle	Differential GeometryGlobal Analysis	In mathematics, the Quillen determinant line bundle is a line bundle over the space of Cauchy–Riemann operators of a vector bundle over a Riemann surface, introduced by .
Quillen metric	Global Analysis	A Hermitian metric on a determinant line of elliptic operators obtained by correcting an L2 metric with a zeta-regularized analytic torsion factor.
Quillen's Theorem A	Homotopy TheoryAlgebraic K Theory	The two theorems play central roles in Quillen's Q-construction in algebraic K-theory and are named after Daniel Quillen.
Quincunx matrix	Lattice Theory	The two-by-two integer matrix with rows (1, -1) and (1, 1), generating the diagonal same-parity square sublattice.
Quintic function	Algebra	A polynomial function of degree exactly five with nonzero leading coefficient.
Quotient Algebra	Universal Algebra	An algebra of congruence classes whose operations descend from a given algebra independently of representative choice.
Quotient category	Category TheoryCategorical Congruences and Quotients	Keep a category's objects while replacing each hom-set by equivalence classes of morphisms under a composition-compatible congruence, so composition descends and the projection is universal for identifying equivalent arrows.
Quotient Graph	Graph Theory	Collapse each block of a vertex partition into one quotient vertex and push original adjacency through the block map, compressing a graph while recording which aggregate vertices remain connected.
Quotient rule	Calculus	In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions.
Quotient space of an algebraic stack	Algebraic GeometryAlgebraic Stacks and Moduli	Associate an algebraic stack with its underlying Zariski topological space of points or integral substacks, functorially turning stack morphisms into continuous maps while forgetting stabilizer data.
Quotient stack	Algebraic Geometry	An algebraic stack [X/G] that retains stabilizers and families of objects while representing a group action's quotient.
Radial basis function	Applied Mathematics	A function whose value depends only on distance from a center, used as a localized basis for interpolation, approximation, and learning.
Radical of a ring	Ring Theory	An ideal-valued construction that isolates elements regarded as structurally degenerate under a chosen radical theory and yields a semisimple quotient.
Radon Measure	Measure Theory	A compact-finite regular Borel measure on a Hausdorff space, approximable externally by open sets and internally on open sets by compact sets.
Radon transform	Integral Geometry	An integral transform mapping a function to its integrals over hyperplanes, with inversion reconstructing the original under suitable conditions.
Radon–Nikodym theorem	Measure Theory	A measure-theoretic theorem representing a sigma-finite measure absolutely continuous with respect to another as integration against an almost-everywhere unique density.
Rainbow coloring	Graph Theory	An edge coloring of a connected graph in which every pair of vertices is joined by a path whose edges all have distinct colors.
Rainville polynomials	Special Function Theory	A polynomial family defined by the generating function involving the modified Bessel function I0.
Ramanujan's sum	Number TheoryExponential Sums	The finite exponential sum c_q(n) over residues coprime to q, an integer-valued arithmetic function used as a Fourier basis for number-theoretic expansions.
Ramified forcing	Set Theoretic Forcing History	Cohen's original forcing construction, which adds a generic object while building names through a hierarchy ramified by formula complexity and constructibility assumptions.
Ramsey's Theorem	Ramsey TheoryCombinatorics	Every finite coloring of fixed-size subsets forces a homogeneous subset when the finite ground set is sufficiently large, with a distinct infinite homogeneous-set form.
Ran space	Algebraic Geometry	The topological or algebro-geometric space that organizes all nonempty finite subsets of a base space as a single varying configuration object.
Rand index	Cluster Analysis	A pair-counting similarity measure for two partitions that counts element pairs on which both clusterings agree.
Random compact set	Random Set Theory	A measurable random variable whose values are compact subsets of a complete separable metric space equipped with the Hausdorff topology.
Random graph	Graph TheoryProbabilistic Graph Models	A graph-valued random object specified by a probability distribution or stochastic generation rule over vertices and edges.
Random measure	Stochastic Processes	A measure-valued random element or kernel that assigns each outcome a locally finite measure, unifying random point configurations and stochastic mass distributions.
Random Variable	Probability Theory	Translate an uncertain event into a measurable function X: Ω → ℝ on a probability space, so that a number attaches to each outcome and the whole apparatus of expectation, distribution, and convergence becomes computable before any value is observed.
Rank (Linear Algebra)	Linear Algebra	Read the effective dimensionality of a matrix or linear map as one integer — the number of linearly independent columns (equivalently rows) — so that solvability, invertibility, and reachability all reduce to comparing that count against a relevant dimension.
Rank of a partition	Partition TheoryPartition Statistics	Associate an integer partition with Dyson's rank—largest part minus number of parts—while distinguishing the separate Durfee-square rank convention used elsewhere in combinatorics.
Rank-width	Graph TheoryWidth Parameters and Decompositions	A graph width parameter that minimizes, over subcubic decomposition trees, the maximum binary rank of the adjacency matrix crossing any induced vertex cut.
Ratio distribution	Probability Theory	The probability distribution of a random variable formed as the quotient of two random variables.
Ratio Test	Real Analysis	Classify absolute convergence or divergence of a series from the eventual magnitude ratio of successive nonzero terms: below one converges, above one diverges, and equality to one is inconclusive.
Rational dependence	Linear Algebra over Fields	The property that a finite collection of numbers satisfies a nontrivial linear relation with rational coefficients.
Rational homotopy theory	Algebraic Topology	The study of topological spaces after replacing homotopy invariants by rational versions that discard torsion and admit algebraic models.
Rational normal curve	Algebraic Geometry	The degree-n projective curve obtained by mapping a projective line to all degree-n monomials in two homogeneous coordinates.
Rational Normal Scroll	Algebraic Geometry	An irreducible rational ruled projective surface swept by paired directrices and embedded with minimal degree.
Rational sequence topology	General Topology	In mathematics, more specifically general topology, the rational sequence topology is an example of a topology given to the set R of real numbers.
Rationalizable strategy	Game TheorySolution Concepts	A game strategy that is a best response to some belief about opponents' strategies consistent with common knowledge of rationality, equivalently surviving iterated elimination of never-best responses.
Rauzy Fractal	Symbolic DynamicsFractal Geometry	A compact self-similar planar set obtained by projecting prefix-sum paths of the Tribonacci substitution onto the contracting plane of its substitution matrix, with tiling and symbolic-dynamical structure.
Reach (Mathematics)	Geometric Measure Theory	Measure the largest open Euclidean tube around a closed set in which every point has a unique nearest point on the set, exposing the first scale at which metric projection becomes ambiguous.
Real Closed Field	AlgebraOrdered Fields	Identify an ordered field maximal among orderable algebraic extensions, equivalently one where positive elements are squares and every odd-degree polynomial has a root.
Real element	Group Theory	A group element conjugate to its inverse, with strong reality requiring conjugation by an involution.
Real form (Lie theory)	Lie Theory	A real Lie algebra or group whose scalar extension to the complex numbers recovers a specified complex Lie algebra or group.
Real point	Projective GeometryAlgebraic Geometry	In geometry, a real point is a point in the complex projective plane with homogeneous coordinates for which there exists a nonzero complex number such that , , and are all real numbers.
Real Representation	Representation Theory	A representation of a group, algebra, or related structure on a vector space over the real numbers, with equivalence and irreducibility tested over R rather than inferred from a complexification alone.
Real-valued function	—	A function whose codomain is the real numbers.
Reciprocity law	Algebraic Number TheoryReciprocity Theorems	A number-theoretic rule relating splitting behavior of primes in an algebraic extension to congruence or residue information, generalizing quadratic reciprocity.
Reconstruction conjecture	Graph Theory	The conjecture that every finite simple graph with at least three vertices is determined up to isomorphism by the multiset of its vertex-deleted subgraphs.
Recurrence plot	Nonlinear Time Series Analysis	A square matrix visualization marking pairs of observation times whose reconstructed system states are equal or sufficiently close under a declared metric and threshold.
Recurrence relation	Discrete Mathematics	An equation defining terms of a sequence from earlier terms together with enough initial conditions to select a solution.
Recurrent event analysis	Survival Analysis	Statistical analysis of repeated event times experienced by the same observational unit.
Recurrent point	Dynamical SystemsTopological Dynamics	A point of a dynamical system that returns arbitrarily close to itself at arbitrarily late iterates, equivalently belonging to its own omega-limit set.
Recursive tree	CombinatoricsRandom Trees	A rooted labeled non-plane tree whose labels increase strictly along every path away from the root, often generated by attaching each new label to an earlier vertex.
Redheffer Star Product	Operator Theory	An associative partial product on compatibly partitioned linear operators that closes connected input/output ports, eliminates the resulting internal feedback variables, and returns the external scattering operator of the composite system.
Reduced residue system	Number Theory	A complete set of incongruent representatives modulo n chosen from the integer classes coprime to n.
Reduct	Model Theory	Forget selected symbols from a logical or algebraic signature while keeping the same underlying set and exactly the inherited interpretations of every symbol that remains.
Reduction (Computability Theory)	Computability Theory	Compare decision sets by an effective procedure that converts access to a solver for B into a solver for A, with the allowed access defining the reducibility notion.
Ree group	Finite Group Theory	An exceptional twisted group of Lie type in the ²G₂ or ²F₄ Ree families.
Reeb graph	Computational Topology	A quotient graph summarizing how connected components of a real-valued function’s level sets merge and split across values.
Reedy category	Category TheoryHomotopy Theory	In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure.
Rees decomposition	Commutative Algebra	In commutative algebra, a Rees decomposition is a way of writing a ring in terms of polynomial subrings.
Refactorable number	Divisor Function Predicate	Classify a positive integer as refactorable when the number of its positive divisors divides the integer itself.
Refinement (category theory)	Category Theory	A categorical construction that replaces an object's structure through a universal morphism from a chosen class, dual to an envelope construction.
Reflected Brownian motion	Stochastic Processes	A Brownian diffusion constrained to a domain by a regulating process that pushes sample paths inward whenever they reach the boundary.
Reflection principle	Set TheoryReflection	A set-theoretic principle asserting that any specified finite collection of truths about the universe of sets already holds in some set-sized rank-initial structure, with stronger variants serving as large-cardinal axioms.
Reflection principle (Wiener process)	Probability TheoryBrownian Motion and Hitting Times	Reflect a Wiener path after its first hitting time of a level to obtain another process with the same law, converting barrier-crossing events into endpoint-distribution identities.
Reflexive closure	Binary Relations and Formal Semantics	The smallest reflexive binary relation containing a given relation, obtained by adjoining every identity pair on the underlying set.
Reflexive relation	Relation Theory	A binary relation on a set that relates every element of the set to itself.
Regular Category	Category TheoryCategorical Exactness	A finitely complete category in which every morphism has a pullback-stable regular-epimorphism–monomorphism image factorization—equivalently, kernel-pair quotients exist and regular epimorphisms remain regular under pullback.
Regular ideal	Ring TheoryOperator Theory	In operator theory, a right ideal \mathfrak{i} in a (possibly) non-unital ring A is said to be regular (or modular) if there exists an element e in A such that ex - x \in \mathfrak{i} for every x \in A .
Regular Polygon	Euclidean Geometry	A Euclidean polygon whose equally spaced vertices follow one constant edge-connection step, giving a connected rotationally symmetric boundary.
Regular prime	Algebraic Number TheoryCyclotomic Fields	An odd prime p that does not divide the class number of the p-th cyclotomic field, equivalently one that divides none of the relevant Bernoulli-number numerators.
Regular scheme	Algebraic Geometry	A locally Noetherian scheme whose every local ring is regular, so local dimension equals the minimal number of generators of its maximal ideal.
Regular space	Topology	A topological space in which every point can be separated from every disjoint closed set by disjoint open neighborhoods.
Regularization by spectral filtering	—	A regularization family for ill-conditioned inverse and learning problems that applies a bounded filter to an operator's singular or eigenvalue spectrum, suppressing unstable small-mode contributions.
Relative contact homology	Symplectic and Contact Topology	A contact-topological invariant associated with a contact manifold together with a Legendrian submanifold, constructed from Reeb chords and pseudoholomorphic curves within symplectic field theory.
Relative convex hull	Computational GeometryGeodesic Convexity	The smallest geodesically convex set containing given points while constrained to remain inside a surrounding polygon or simple closed region.
Remarkable Cardinal	Set TheoryVirtual Large Cardinal Theory	A virtual large cardinal κ for which arbitrarily high ground-model rank or hereditary-size segments admit, in set-forcing extensions, elementary small embeddings whose critical point is mapped to κ.
Remmert–Stein Theorem	—	Under a strict component-dimension gap, the closure across a lower-dimensional analytic exceptional set of an analytic subset remains analytic.
Renewal theory	Probability TheoryRenewal Processes	A probability framework for processes that restart after independent identically distributed waiting times, studying event counts, ages, residual lives and rewards over repeated cycles.
Representation on coordinate rings	Algebraic Geometry and Representation Theory	A group action on an affine algebraic variety induces a contragredient linear action on its coordinate ring by precomposing regular functions with the inverse geometric action.
Representation ring	Representation Theory	The Grothendieck ring of finite-dimensional group representations, with direct sum as addition and tensor product as multiplication.
Representation theory of the symmetric group	Algebraic Combinatorics	The classification and analysis of symmetric-group actions on vector spaces through partitions, Young diagrams, tableaux, characters and Specht modules.
Residence Time (Statistics)	ProbabilityStochastic Control	The statistical residence time is the expected first exit of a random process from a specified domain, conditional on its starting state.
Residual (numerical analysis)	Numerical Analysis	The discrepancy obtained by substituting an approximate solution into the original equation, usually b−f(x₀) or its signed convention.
Residue (complex analysis)	Complex Analysis	The coefficient of the inverse-linear term in a meromorphic function’s Laurent expansion at an isolated singularity.
Resolution of singularities	Algebraic Geometry	The replacement of a singular algebraic variety by a nonsingular variety connected through a proper birational morphism.
Resolution Theorem (Algebraic K-Theory)	Algebraic K TheoryExact Categories	A resolving exact subcategory with finite resolutions of every ambient object has the same higher K-theory as the ambient category.
Restricted isometry property	Compressed Sensing	A matrix property requiring approximate norm preservation on all sufficiently sparse vectors, thereby controlling the geometry needed for stable sparse recovery.
Restricted Power Series	Topological Algebra	A formal power series over a complete linearly topologized ring whose coefficients tend to zero in the ring topology as degree grows, equivalently an element of the completed polynomial ring.
Restricted product	Topological Groups	The subgroup of a direct product whose coordinates lie in designated compact open subgroups at all but finitely many indices.
Restricted representation	Representation Theory	The representation of a subgroup obtained by retaining the same vector space and limiting a group representation to subgroup elements.
Restricted root system	Lie Theory	The root system obtained by restricting a semisimple Lie algebra’s roots to a maximal abelian subspace in the noncompact part of a symmetric decomposition.
Resultant	Computational Algebra	A polynomial in the coefficients of two univariate polynomials that vanishes exactly when they have a common root over an algebraic closure.
Reuleaux polygon	Convex GeometryCurves of Constant Width	A convex constant-width curve assembled from an odd number of equal-radius circular arcs, with each arc centered at an opposite vertex of its generating polygon.
Reverse mathematics	Mathematical Logic	A program classifying mathematical theorems by the weakest axiomatic subsystems sufficient to prove them.
Reynolds operator	Invariant Theory and Fluid MechanicsAveraging Projections	Project objects onto their invariant part by averaging over a symmetry group or averaging regime, with linearity, idempotence, and compatibility rules governing the result.
Rhombus	Euclidean GeometryQuadrilateral Classification	A nondegenerate Euclidean quadrilateral with four equal sides, forcing a parallelogram whose diagonals bisect at right angles and whose square case adds right-angle symmetry.
Rhumb line	Spherical GeometryNavigationCartography	A path on a sphere or reference ellipsoid that crosses every meridian at the same angle, enabling travel at constant true bearing and appearing as a straight line on a Mercator projection, though usually longer than the great-circle route.
Ribbon category	Category Theory	A rigid braided monoidal category equipped with a twist compatible with braiding and duality.
Ribbon Theory	Differential GeometryKnot Theory	In differential geometry, a ribbon (or strip) is the combination of a smooth space curve and its corresponding normal vector.
Ricci Decomposition	Differential GeometryRiemannian Geometry	The metric-dependent orthogonal splitting of a Riemann curvature tensor into scalar-curvature, traceless-Ricci, and totally trace-free Weyl components.
Ridders' Method	Numerical AnalysisScalar Root Finding	A sign-bracketed scalar root finder that samples the midpoint, corrects a three-point interpolation by an exponential factor, and retains a certified root-containing subinterval.
Ridge function	Approximation TheoryMultivariate Ridge Functions	Factor a multivariate function through one linear or affine projection, so its value varies only along a selected direction and remains constant across every orthogonal affine slice.
Riemann Sphere	Complex Analysis	The one-point compactification of the complex plane, C union {infinity}, equipped with the complex structure of the projective line CP1 and representable geometrically by stereographic projection onto a sphere.
Riemann sum	Real AnalysisRiemann Integration	Approximate a definite integral by partitioning an interval, sampling the function once in each subinterval, multiplying each sampled value by subinterval width, and summing, with mesh refinement controlling convergence.
Riemannian manifold	Differential Geometry	A smooth manifold equipped at every point with a smoothly varying positive-definite inner product on its tangent space.
Riemannian submersion	Riemannian GeometryDifferential Geometry	A smooth surjective submersion between Riemannian manifolds whose differential restricts to an isometry from each horizontal tangent space onto the target tangent space.
Riemann–Hilbert Problem	Complex AnalysisAnalytic Matrix Factorization	A complex-analytic boundary-value problem that reconstructs a piecewise holomorphic scalar or matrix function from prescribed multiplicative jumps across an oriented contour, together with normalization and singularity conditions.
Riemann–Liouville integral	Fractional Calculus	A parameterized fractional-integration operator that extends repeated antiderivation from positive integer orders to noninteger orders.
Riesz potential	Harmonic AnalysisFractional Integration	Apply the convolution kernel proportional to |x|^{α−n} to realize a fractional inverse power of the Laplacian on Euclidean space.
Riesz projector	Spectral Theory	A contour-integral projection onto the invariant spectral subspace associated with an isolated portion of an operator’s spectrum.
Riesz space	Functional Analysis	A real vector space equipped with a lattice order compatible with vector addition and nonnegative scalar multiplication.
Riesz's lemma	Functional Analysis	In mathematics, Riesz's lemma (after Frigyes Riesz) is a lemma in functional analysis.
Riesz–Markov–Kakutani representation theorem	Functional Analysis	A representation theorem identifying continuous linear functionals on suitable spaces of continuous functions with integration against unique regular measures.
Rigged Hilbert Space	Functional Analysis	A dense test space, pivot Hilbert space, and continuous test-space dual linked by compatible embeddings so generalized vectors can act on declared tests.
Right triangle	Euclidean GeometryTriangle Geometry	Specify a triangle with exactly one right angle, making the opposite side the hypotenuse and coupling its sides, acute angles, altitude, circumcircle, and similarity relations through Euclidean constraints.
Rigid category	Category Theory	A monoidal category in which every object has a left and right dual, with evaluation and coevaluation morphisms satisfying triangular identities.
Rigidity Matroid	Combinatorial RigidityMatroid Theory	The edge matroid represented by a framework's rigidity matrix, encoding independent first-order distance constraints and their rank in a fixed Euclidean dimension.
Ring	Abstract Algebra	A set with two operations — an abelian group under addition and an associative multiplication coupled to it by the distributive law — whose axiom tier and ideal structure unlock a single body of theory (quotients, homomorphisms, factorization) across integers, polynomials, and matrices at once.
Ring Homomorphism	Ring Theory	Map one ring to another while preserving addition, multiplication, and—under the declared unital convention—the multiplicative identity, so kernels, images, quotients, and composition retain ring structure.
Ring Ideal	Ring TheoryAbstract Algebra	An additive subgroup of a ring closed under multiplication by arbitrary ambient ring elements on the declared left, right, or both sides, enabling kernel and quotient constructions.
Ring of mixed characteristic	Commutative Algebra	A characteristic-zero commutative ring with a quotient or residue field of positive characteristic, usually considered locally at a prime p.
Ringed Space	Algebraic Geometry and Sheaf Theory	A topological space equipped with a sheaf of rings assigning locally compatible ring-valued algebraic data to every open set and restriction maps to inclusions.
Ringed topos	Category Theory and Algebraic GeometryRinged Topoi	Pair a topos with an internal ring object so generalized spaces carry local algebraic data and morphisms combine geometric inverse-image structure with a compatible map of structure rings.
Rising sun lemma	Real Analysis	A one-dimensional decomposition lemma partitioning the points below a later higher value of a continuous function into disjoint intervals with controlled endpoints.
Risk dominance	Game Theory	An equilibrium-selection criterion favoring the coordination equilibrium with the larger basin of attraction under uncertainty about others’ actions.
Robinson–Schensted–Knuth correspondence	Algebraic Combinatorics	A weight-preserving bijection between nonnegative-integer matrices and pairs of semistandard Young tableaux of equal shape.
Robust Optimization	Optimization	Optimization that selects decisions against a specified set of uncertain parameter values, with feasibility or performance judged across that set.
Roman Dominating Set	Graph DominationGraph Labeling	A graph labeling assigns each vertex 0, 1, or 2 so every zero vertex neighbors a two vertex; the least total label weight is its Roman domination number.
Root System	Lie Theory	Encode reflection symmetry by a finite spanning set of Euclidean vectors closed under root reflections, with integral Cartan pairings in the crystallographic setting.
Root Test	Real Analysis	Classify an infinite series using the limit superior of the nth roots of its term magnitudes: below one gives absolute convergence, above one gives divergence, and one is inconclusive.
Rooted product of graphs	Graph TheoryGraph Products	The rooted product is a subgraph of the cartesian product of the same two graphs.
Rosati involution	Algebraic GeometryAbelian Varieties	The positive involutive anti-automorphism of the rational endomorphism algebra of a polarized abelian variety obtained by taking the dual endomorphism and conjugating through the polarization.
Rotation matrix	Linear Algebra	In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space.
Rotation number	One Dimensional Dynamics	An invariant measuring the average angular displacement of an orientation-preserving circle homeomorphism under repeated iteration, taken modulo one.
Rugosity	Surface Geometry and EcologyHabitat Structural Complexity	A scale-declared ratio of contoured length or surface area to a straight or planar reference that quantifies how much geometric extent roughness adds beyond a smooth baseline.
Rule 184	Cellular Automata	A synchronous binary cellular-automaton rule whose local update moves each unblocked occupied site one step right while conserving occupancy on a ring.
Ruled join	Algebraic Geometry	The projective variety formed by the union of all lines connecting points of two separately embedded projective subvarieties.
Ruled Surface	Differential Geometry	A surface swept by a one-parameter family of straight lines, locally represented as a directrix plus a variable multiple of a ruling direction.
Ruled variety	Algebraic Geometry	An algebraic variety birational to a product with a projective line, so its generic points lie on a rational one-parameter ruling.
Runge–Kutta method (SDE)	Stochastic Numerical Analysis	A family of time-stepping schemes that approximates stochastic differential equations by combining drift and diffusion evaluations with simulated stochastic increments.
Rupture field	Field Theory	A field extension generated by adjoining one root of a polynomial over the base field.
Ruzsa–Szemerédi Problem	Extremal Combinatorics	An extremal-combinatorics problem asking how dense an n-vertex graph can be when every edge lies in exactly one triangle, equivalently how many triples can be chosen with no three supported on six vertices.
S-equivalence	Algebraic Geometry	The equivalence relation identifying semistable vector bundles or sheaves whose Jordan–Hölder graded objects are isomorphic.
S-procedure	Control TheoryConvex OptimizationQuadratic Inequalities	The S-procedure or S-lemma is a mathematical result that gives conditions under which a particular quadratic inequality is a consequence of another quadratic inequality.
S4 (logic)	Modal LogicMathematical Logic	The normal modal logic obtained from K by adding reflexivity and transitivity principles, characterizable by reflexive-transitive Kripke frames.
Sagitta	Circle Geometry	The sagitta is the perpendicular rise from a circular arc's chord midpoint to the arc, linking chord length to radius.
Saint-Venant's Compatibility Condition	Continuum MechanicsElasticity Complex	The differential integrability test for whether a symmetric small-strain field can be the symmetrized gradient of one displacement field.
Sastry Automorphism	Group Theory	A characteristic-two field automorphism admitting a nonzero quadruple that satisfies the Type I or Type II Sastry–Bombieri identity system arising from Ree-group embeddings in F4.
Saturated set (intersection of open sets)	General Topology	A subset of a topological space equal to the intersection of all open sets containing it, equivalently an upper set for the specialization preorder.
Sauer–Shelah lemma	CombinatoricsExtremal Set Systems	An extremal bound stating that a set family of VC dimension d on an n-element ground set has at most the sum of binomial(n,i) for i from zero through d members.
Scalar (Mathematics)	Linear AlgebraModule TheoryAbstract Algebra	In linear algebra, an element of the field (or, for a module, the ring) over which vectors are defined, acting on vectors through scalar multiplication and scaling their algebraic coordinates and combinations.
Scalar multiplication	Linear Algebra	The vector-space or module operation that combines a scalar with a vector to produce another vector.
Scattered order	Order TheoryLinear Orders	In mathematical order theory, a scattered order is a linear order that contains no densely ordered subset with more than one element.
Scattered Space	TopologySpecial Topological Spaces	Classify a topological space by requiring every nonempty subspace to expose an isolated point, equivalently that transfinite removal of isolated points eventually exhausts it.
Schatten norm	Functional AnalysisOperator Ideals	The p-norm of a compact operator's singular-value sequence, generalizing matrix Frobenius and nuclear norms to operators on Hilbert spaces.
Schauder Fixed-Point Theorem	—	A continuous self-map of a nonempty closed convex set has a fixed point when its image is relatively compact in the surrounding locally convex space.
Schneider–Lang Theorem	—	Bounds the complex points where a derivative-stable family of controlled-growth meromorphic functions can simultaneously take values in a number field when the family contains two algebraically independent functions.
Schnirelmann Density	Additive Number Theory	Measure a set of positive integers by the least fraction it occupies in any initial segment, making early gaps permanently visible and enabling quantitative sumset and additive-basis theorems.
Schramm–Loewner evolution	Probability TheoryConformally Invariant Random Curves	Generate conformally invariant random planar curves by driving a normalized Loewner evolution with scaled one-dimensional Brownian motion, with the parameter controlling the curve regime.
Schröder number	Enumerative Combinatorics	A sequence counting diagonal-bounded lattice paths with horizontal, vertical and diagonal steps, equivalently several recursively decomposable combinatorial structures.
Schröder–Bernstein Property	Structural Comparison and Classification	The property of a specified mathematical class, embedding relation, and equivalence notion that mutual embeddability forces equivalence, making the embedding preorder antisymmetric after quotienting by that equivalence.
Schubert polynomial	Algebraic Combinatorics and Schubert Calculus	A polynomial indexed by a permutation that represents the corresponding Schubert variety's cohomology class in a flag variety and forms a basis of the polynomial ring in stable variables.
Schuette–Nesbitt formula	Combinatorics and Actuarial Science	A weighted generalization of inclusion–exclusion that expresses sums over outcomes with exactly or at least a given number of occurring events.
Schur complement method	Numerical Linear Algebra	A nonoverlapping domain-decomposition method eliminating subdomain interiors and solving the remaining interface Schur-complement system.
Schur decomposition	Linear AlgebraMatrix DecompositionsNumerical Linear Algebra	A unitary similarity that transforms any complex square matrix to upper triangular form, placing its eigenvalues on the diagonal while preserving numerical stability.
Schur functor	Representation Theory	A polynomial functor indexed by a partition that constructs an irreducible polynomial representation from tensor powers using prescribed row symmetries and column antisymmetries.
Schur Orthogonality Relations	Representation Theory	Invariant averaging makes coefficients of inequivalent irreducible group representations orthogonal and normalizes equal-representation coefficients by dimension.
Schur's property	Banach Space Theory	A normed-space property under which every weakly convergent sequence also converges in norm.
Schwartz topological vector space	Functional Analysis	A locally convex topological vector space whose bounded sets are precompact, equivalently whose neighborhoods satisfy a finite-covering condition after suitable shrinking and scaling.
Schwarz triangle function	Complex Analysis	A conformal map from the upper half-plane onto a curvilinear triangle, expressible as a ratio of hypergeometric solutions with angle parameters.
Schützenberger Group	Semigroup Theory	The permutation group induced on a semigroup H-class by its stabilizing translations, even when the H-class is not itself a group.
Seashell surface	Differential GeometryParametric SurfacesMathematical Visualization	A seashell surface is a parametric surface generated by combining axial growth, radial expansion, rotation, and optional ornamentation to model the coiled geometry of a shell.
Secant Line	GeometryDifferential Calculus	A secant line passes through two distinct points of a curve, making a full-line incidence relation that can support chord geometry or average-change reasoning.
Secant Variety	Projective Algebraic GeometryAlgebraic Geometry	The Zariski closure of the union of linear spans of k+1 points of an embedded projective variety; for k=1 it closes all secant lines and their tangent limits.
Second Fundamental Form	Differential Geometry	Encode how an immersed surface or submanifold bends in its ambient space by pairing tangent directions with the normal component of their ambient derivative, yielding normal curvature and the shape operator under explicit sign and normal conventions.
Second-order logic	Mathematical LogicHigher Order Quantificational Logic	Extend first-order languages with quantification over predicates, relations, sets, and functions, while treating full and Henkin semantics as different regimes with different categoricity, completeness, compactness, and axiomatizability behavior.
Second-Order Predicate	Mathematical LogicHigher Order Logic	A predicate whose argument position ranges over a first-order predicate, concept, relation, or set rather than only over individual objects.
Secondary Measure	Orthogonal Polynomials	An auxiliary positive measure derived from an initial moment measure so that its secondary-polynomial sequence becomes orthogonal, with the pair linked by a reciprocal Stieltjes-transform relation.
Secondary Polynomials	Orthogonal Polynomials	Secondary Polynomials is a recurring orthogonal polynomials, analysis identity in which a difference quotient of an orthogonal polynomial is integrated against its density to generate an associated polynomial sequence.
Section (category theory)	Category Theory	In category theory, a branch of mathematics, a section is a right inverse of some morphism.
Segre Class	Algebraic GeometryIntersection Theory	A graded intersection-theoretic cycle class of a cone or closed embedding that records its projective directions and reduces to the inverse Chern class of the normal bundle for a regular embedding.
Seiberg–Witten flow	Differential GeometryGauge TheoryGeometric Flows	In differential geometry, the Seiberg–Witten flow is a gradient flow described by the Seiberg–Witten equations, hence a method to describe a gradient descent of the Seiberg–Witten action functional.
Seifert Surface	Knot Theory and Low Dimensional Topology	Span an oriented knot or link by a compact connected orientable embedded surface, turning a one-dimensional embedding into a two-dimensional carrier from which genus, linking forms, and knot invariants can be derived.
Selberg zeta function	Analytic Number TheorySpectral TheoryHyperbolic Geometry	A zeta function built from primitive closed geodesic lengths of a hyperbolic surface.
Self-complementary graph	Graph Theory	A graph isomorphic to its complement, so some vertex relabeling exchanges edges with nonedges while preserving graph structure.
Self-concordant function	Convex OptimizationInterior Point Methods	A self-concordant function is a function satisfying a certain differential inequality, which makes it particularly easy for optimization using Newton's method A self-concordant barrier is a particular self-concordant function, that is also a barrier function for a particular convex set.
Self-verifying theories	Mathematical LogicIncompleteness and Arithmetic	Weak consistent first-order arithmetical systems constructed to prove an internal statement of their own consistency without containing enough arithmetic for Gödel's second incompleteness theorem to apply in its usual form.
Semantics (logic)	Mathematical LogicModel TheoryFormal Semantics	In logic, the semantics or formal semantics is the study of the meaning and interpretation of formal languages, formal systems, and (idealizations of) natural languages.
Semi-infinite programming	Optimization Theory	Optimization with finitely many decision variables and infinitely many constraints, or dually infinitely many variables and finitely many constraints.
Semi-reflexive space	Functional Analysis	A locally convex topological vector space whose canonical map into its strong bidual is algebraically onto, without necessarily being a topological isomorphism.
Semi-s-cobordism	Geometric Topology	A cobordism whose inclusion of one designated boundary component is a simple homotopy equivalence, with no corresponding requirement on the other boundary.
Semi-symmetric graph	Graph Theory	A regular undirected graph whose automorphism group is transitive on edges but not on vertices.
Semidirect Product	Group TheoryGroup Extensions	A group construction N ⋊φ H on N×H whose multiplication is twisted by an action of H on N, equivalently an internal decomposition with N normal and a complementary subgroup H.
Semigroup with Involution	Semigroup Theory	An associative algebraic system with a self-undoing unary operation that reverses product order.
Semigroupoid	Category TheoryPartial Algebraic Structures	A category-like partial algebra with objects, composable morphisms and associative composition but without requiring an identity morphism at every object.
Semilinear map	Linear Algebra	An additive map between vector spaces whose scalar multiplication is respected after applying a fixed field automorphism.
Seminorm	Functional Analysis	A nonnegative subadditive absolutely homogeneous function on a vector space that may vanish on nonzero vectors.
Seminormal ring	Commutative Algebra	A reduced commutative ring in which compatible square and cube roots already come from one element, preventing certain hidden subintegral identifications.
Semiorder	Order TheoryUtility Representation	A partial order representable by assigning real-valued utilities and a positive discrimination threshold so that one item is preferred to another only when their scores differ by at least that threshold, allowing nontransitive incomparability.
Semiperfect Number	Number TheoryDivisor Sums	A positive integer that equals the sum of a distinct subset of its proper divisors, whether or not the subset uses them all.
Semiprimitive ring	AlgebraRing Theory	A ring with zero Jacobson radical, equivalently one whose simple modules collectively detect every nonzero element.
Semiregular space	General Topology	A topological space whose regular open sets form a base for its topology.
Semisimple module	Abstract AlgebraModule Theory	A module that is a direct sum of simple submodules, equivalently one in which every submodule has a complementary submodule.
Separable polynomial	Field Theory	A polynomial over a field whose roots in an algebraic closure are all distinct.
Separatrix	Dynamical Systems	An invariant state-space boundary separating qualitatively different dynamical trajectories.
Sequent	Mathematical LogicProof Theory	Package antecedent and succedent formula contexts into a two-sided formal judgment whose interpretation and admissible transformations are fixed by a proof calculus.
Sequential Dynamical System	Discrete Dynamical SystemsAsynchronous Graph Dynamics	A finite graph dynamical system whose one-step evolution is the ordered composition of neighborhood-local vertex updates, so later updates in the schedule read changes made earlier in the same sweep.
Sequentially compact space	TopologyCompactness Properties	A topological space in which every sequence has a subsequence converging to a point of the space, coinciding with compactness in metric spaces but not in general.
Serial Module	Module Theory	A module admitting a direct-sum decomposition into uniserial modules, each with submodules linearly ordered by inclusion.
Serre Group	Arithmetic GeometryAlgebraic Number TheoryHodge Theory	An abelian pro-algebraic group, obtained as an inverse limit of algebraic tori associated with number fields, whose representations encode CM motives or polarizable rational Hodge structures with abelian Mumford–Tate group.
Serre's Property FA	Geometric Group TheoryGroup Actions on Trees	A group property requiring every action on a simplicial tree to have a global fixed vertex, equivalently excluding the quotient, amalgam, and ascending-union obstructions identified by Serre.
Seshadri Constant	Algebraic GeometryPositivity	A local positivity invariant of a nef line bundle at a point, computed by the least ratio of curve intersection to curve multiplicity, equivalently by the nef threshold on the blowup.
Sesquilinear form	Linear Algebra	A two-argument form on complex vector spaces that is linear in one argument and conjugate-linear in the other.
Set inversion	Interval Analysis	The problem of characterizing all inputs whose image under a function lies in a specified target set.
Set-Theoretic Code	Set TheoryDescriptive Set Theory	A real coding a hereditarily countable set by a well-founded extensional relation on natural numbers whose Mostowski collapse recovers the set's transitive closure.
Severi–Brauer variety	Algebraic GeometryTwisted Projective Spaces	An algebraic variety over a field that becomes projective space after extending scalars to an algebraic closure, encoding a central simple algebra and its splitting.
Sexy Primes	Analytic and Elementary Number TheoryPrime GapsRecreational Mathematics	A pair of prime numbers separated by exactly six, with triplet and higher variants requiring repeated six-unit prime gaps.
Sheaf	—	A sheaf assigns compatible local data to open subsets of a space through restriction maps and provides a unique gluing rule for recovering sections from locally consistent pieces.
Sheaf of algebras	Algebraic Geometry	A sheaf on a ringed space whose sections form algebras over the structure sheaf compatibly with restriction.
Sheaf of Modules	Algebraic GeometrySheaf Theory	A sheaf whose local sections are modules over a sheaf of rings, with scalar multiplication compatible with restriction and gluing.
Sheaf of spectra	Stable Homotopy and Derived Geometry	A homotopy-coherent assignment of a spectrum to each open set or site object that satisfies descent, so local generalized-cohomological data glue into global spectral data.
Shear mapping	GeometryAffine Transformations	An affine transformation that displaces points parallel to a fixed direction by an amount proportional to signed distance from a fixed parallel hyperplane, preserving volume and collinearity while generally changing lengths and angles.
Sherman–Morrison Formula	Linear AlgebraMatrix Analysis	The Sherman–Morrison formula gives the exact inverse of an invertible matrix after a rank-one outer-product update, provided one scalar denominator is nonzero.
Shift Operator	Functional AnalysisTranslation Operators and Shift Operators on Function and Sequence Spaces	Translate the argument or index of a function, signal, or sequence by a declared displacement, forming a composable family whose algebra, invertibility, norm behavior, and boundary effects depend on the indexed domain and convention.
Shock-Capturing Method	Numerical AnalysisComputational Fluid DynamicsHyperbolic Conservation Laws	A conservative numerical method that captures shocks within a fixed computational grid using flux discretization and controlled dissipation rather than explicitly tracking shock surfaces.
Shrinking Space	Covering Properties	A shrinking space lets every indexed open cover be replaced by a still-covering family whose members' closures fit inside their original sets.
Siegel Disc	Complex DynamicsHolomorphic Iteration	Recognize a periodic Fatou component whose holomorphic first-return dynamics become an irrational rigid rotation after a biholomorphic change of coordinates.
Siegel modular variety	Algebraic GeometryModuli Spaces	In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension.
Siegel upper half-space	Complex GeometrySymmetric Domains and Modular Forms	The complex manifold of symmetric g-by-g matrices with positive-definite imaginary part, a Hermitian symmetric domain on which the real symplectic group acts transitively and the degree-g setting for Siegel modular forms.
Siegel Zero	Analytic Number TheoryZero Free Regions for Dirichlet L Functions	Treat a possible real simple zero of a primitive real Dirichlet L-function exceptionally close to s=1 as the sole allowed intruder in a stated classical zero-free region, with family uniqueness, ineffectivity, and zero-repulsion consequences kept explicit.
Sierpiński Graph	Graph TheoryFractal Graphs	A recursively structured graph whose word-labeled vertices connect by a first-difference and swapped-suffix rule.
Sierpiński Set	Set TheorySet Theory of the Reals	Recognize an uncountable real set whose intersection with every Lebesgue-null set is countable, with existence controlled by additional set-theoretic axioms.
Sign (mathematics)	Mathematical Notation and Order	A positive, negative, or zero classification attached to a real quantity, and by extension a binary orientation or parity factor represented by plus or minus one in typed mathematical structures.
Signed graph	Graph Theory	A graph whose edges carry positive or negative signs, with cycle-sign products determining balance and switching equivalence.
Signed set	CombinatoricsOriented Structures	A set equipped with a positive-or-negative label on every element, equivalently an ordered pair of disjoint positive and negative subsets or a map to a two-element sign set.
Silhouette (clustering)	Cluster Analysis	An internal cluster-validation score comparing each observation’s mean within-cluster dissimilarity with its smallest mean dissimilarity to another cluster.
Silverman's game	Game TheoryZero Sum Games	A two-player zero-sum number-selection game that rewards a moderately larger choice but penalizes a choice whose ratio to the opponent's reaches a fixed threshold.
Simon model	Preferential Attachment and Applied Probability	A stochastic growth model in which new categories enter at a fixed probability while existing categories receive new occurrences in proportion to their current counts, generating a power-law size distribution.
Simple homotopy theory	Algebraic TopologyHomotopy Theory	In mathematics, simple homotopy theory is a homotopy theory (a branch of algebraic topology) that concerns with the simple-homotopy type of a space.
Simple space	Algebraic Topology	A connected topological space whose fundamental group is abelian and acts trivially on every higher homotopy group, usually with a CW-type assumption.
Simplex	GeometryConvex Polytopes	The convex hull of n+1 affinely independent points in n-dimensional space, generalizing a point, segment, triangle and tetrahedron as the simplest full-dimensional polytope.
Simplex category	Category TheorySimplicial Methods	The category Δ of nonempty finite ordinals [n] and order-preserving maps, whose functors into or out of another category define simplicial and cosimplicial objects.
Simplex Graph	Graph Theory	Transform an undirected graph into a bipartite median graph whose vertices are all cliques, including the empty clique, with adjacency given by adding or deleting exactly one original vertex.
Simplicial Group	Algebraic TopologySimplicial Homotopy Theory	A dimension-indexed family of groups whose face and degeneracy homomorphisms obey the simplicial identities, combining algebraic composition with a combinatorial model of homotopy.
Simplicial Localization	Homotopical AlgebraHigher Category TheoryAlgebraic Topology	A localization of a category at chosen weak equivalences into a simplicial category whose mapping-space components recover ordinary localized morphisms while retaining higher homotopy data.
Simplicial Presheaf	Local and Motivic Homotopy Theory	A simplicial presheaf is a contravariant functor from a category to simplicial sets, equivalently a simplicial object in set-valued presheaves, combining sectionwise homotopy data with functorial restriction.
Simplicial set	Algebraic Topology	A contravariant functor from the simplex category to sets, equivalently graded simplices equipped with compatible face and degeneracy maps.
Simplicial space	Homotopy Theory	In mathematics, a simplicial space is a simplicial object in the category of topological spaces.
Simplicial sphere	Combinatorial TopologyPolyhedral Combinatorics	In geometry and combinatorics, a simplicial (or combinatorial) d-sphere is a simplicial complex homeomorphic to the d-dimensional sphere.
Simplicially enriched category	Higher Category TheoryEnriched Categories	A category whose hom-objects are simplicial sets and whose composition and identities are simplicial maps, encoding higher homotopies between morphisms.
Simply connected at infinity	Geometric Topology	A noncompact space property requiring sufficiently remote loops to contract outside any prescribed compact core.
Simply typed lambda calculus	Mathematical LogicType Theory	The term simple type is also used to refer to extensions of the simply typed lambda calculus with constructs such as products, coproducts or natural numbers (System T) or even full recursion (like PCF).
Simulation Decomposition	Uncertainty AnalysisGlobal Sensitivity Visualization	A hybrid uncertainty-and-sensitivity visualization that partitions selected inputs into states, crosses those states into joint scenarios, and decomposes the sampled output distribution into scenario-labeled subdistributions.
Singular function	Real Analysis	A nonconstant continuous real function whose derivative exists and equals zero almost everywhere.
Singular Homology	Algebraic Topology	A functorial topological invariant built from all continuous simplex maps into a space, with homology groups formed as cycles modulo boundaries.
Singular integral operators on closed curves	Complex AnalysisBoundary Singular Integrals	Interpret principal-value Cauchy- and Hilbert-type integrals along a closed curve as boundary operators whose jump relations, projections, and mapping properties encode analytic traces on the curve's two sides.
Six operations	Homological Algebra	The six-functor formalism relating pullback, pushforward, extraordinary pullback and pushforward, tensor product and internal Hom across geometric categories.
Skeleton (category theory)	Category Theory	A skeleton of a mathematical category is a subcategory that, roughly speaking, does not contain any extraneous isomorphisms.
Skeletonization of fusion categories	Category Theory	Reduction of a fusion category to skeletal simple-object labels, fusion rules, and coherence data.
Skew coordinates	Analytic GeometryTensor Calculus	Skew coordinates are coordinates referred to non-orthogonal basis directions, so coordinate components and metric relations retain cross terms absent from an orthogonal frame.
Skew-symmetric graph	Graph Theory	In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points.
Sklar's theorem	Probability TheoryCopula Theory	Sklar's theorem states that every multivariate distribution can be represented by a copula joining its univariate marginal distributions, with the copula unique when the marginals are continuous.
Skolem Normal Form	Mathematical LogicFirst Order Logic	A first-order formula shape in which a universal prenex prefix governs a quantifier-free matrix and existential witnesses have been encoded by fresh Skolem terms, preserving satisfiability across a signature expansion.
Skyline matrix	Numerical Linear AlgebraSparse Matrix Storage	In scientific computing, skyline matrix storage, or SKS, or a variable band matrix storage, or envelope storage scheme is a form of a sparse matrix storage format matrix that reduces the storage requirement of a matrix more than banded storage.
Slitherlink	Recreational MathematicsLoop Forming Logic Puzzle	Select edges of a clue-labeled planar lattice so local face counts and vertex degrees jointly form exactly one nonbranching closed loop.
Slow Manifold	Geometric Singular Perturbation Theory	An invariant or approximately invariant lower-dimensional manifold in a fast–slow dynamical system on which the reduced long-timescale evolution occurs after nearby fast variables relax toward it.
Slow-Growing Hierarchy	Proof TheoryOrdinal Analysis	Builds an ordinal-indexed family of natural-number functions by successor increments and fundamental-sequence descent at limits.
Small cancellation theory	Geometric Group Theory	The study of group presentations whose relators have sufficiently short mutual overlaps, yielding strong geometric and algorithmic consequences.
Small category	Category TheorySet Theoretic Foundations	A category \mathcal{C} is called small if both \operatorname{ob}(\mathcal{C}) and \operatorname{mor}(\mathcal{C}) are actually sets and not proper classes, and large otherwise.
Small set (category theory)	Category Theory	A set belonging to a fixed foundational universe used to bound categorical size.
Smith space	Functional Analysis	A complete compactly generated locally convex space possessing one compact set that absorbs every compact subset.
Smooth functor	Differential Topology	A functor on finite-dimensional real vector spaces whose action varies smoothly in smoothly parameterized families.
Smooth manifold	Differential GeometryDifferential Topology	A topological manifold equipped with smoothly compatible coordinate charts, enabling coordinate-independent calculus, tangent spaces, and differential structures.
Smooth Number	Number Theory	A positive integer is B-smooth when every prime divisor is at most the declared bound B, making its factorization lie entirely within a small-prime factor base.
Smoothness (probability theory)	Probability and Nonparametric StatisticsDeconvolution and Inverse Problems	Classify an error distribution by the asymptotic decay of its characteristic function, separating polynomially ordinary-smooth laws from exponentially supersmooth laws in nonparametric inverse problems.
Snark (graph theory)	Graph Theory	A connected cubic graph whose edges cannot be properly colored with three colors, usually restricted by girth and nontrivial connectivity conditions.
SO(8)	Lie Groups and Representation Theory	The rank-four, 28-dimensional special orthogonal Lie group of orientation-preserving linear isometries of eight-dimensional Euclidean space, distinguished by triality in its Spin(8) cover.
Sobolev spaces for planar domains	Partial Differential Equations	A Hilbert-space framework that encodes weak derivatives and boundary traces to formulate elliptic boundary-value and eigenvalue problems on bounded planar domains.
Soft Cell	Geometry	Classify a geometric cell by the dimension-specific minimum number of sharp boundary corners, independently of whether it fills space.
Soft set	Soft Set Theory	A parameterized family of subsets used to represent attribute-dependent or uncertain classifications.
Solid of revolution	Solid GeometryIntegral Calculus	In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary).
Solvmanifold	Differential Geometry	A homogeneous manifold of a connected solvable Lie group, with compact lattice quotients forming the special convention central to topology and Lie theory.
Sombrero function	Applied MathematicsSpecial Functions in Imaging	The radial two-dimensional analogue of sinc, commonly defined as 2J1(πρ)/(πρ), and arising as the Fourier transform of a circular aperture.
Somos sequence	Number TheoryRecurrence Relations	In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below.
Sorgenfrey plane	TopologyCounterexample Spaces	The product of two Sorgenfrey lines, a classic separable first-countable space that is not normal and exposes failures of product preservation in topology.
Souček space	Functional AnalysisFunction Spaces	In mathematics, Souček spaces are generalizations of Sobolev spaces, named after the Czech mathematician Jiří Souček.
Space of continuous functions on a compact space	Functional AnalysisBanach Function Algebras	Equip all real- or complex-valued continuous functions on a compact Hausdorff space with pointwise algebra and the supremum norm, obtaining a unital commutative Banach algebra whose structure reflects the underlying space.
Space-Filling Curve	Continuum and Fractal Geometry	A continuous surjection from a one-dimensional interval onto a higher-dimensional region, typically built as the uniform limit of recursively refined approximating paths.
Span (category theory)	Category Theory	A diagram of two morphisms with common domain, used as a generalized relation or correspondence between their codomains.
Sparse polynomial	AlgebraComputational Polynomials	A polynomial represented by relatively few nonzero monomial terms compared with its degree, dimension or dense coefficient array.
Sparsely Totient Number	Number TheoryArithmetic Functions	A natural number n whose Euler totient is a strict suffix minimum: every larger integer m has φ(m) greater than φ(n).
Specialization preorder	General Topology and Domain Theory	The canonical preorder on points of a topological space in which one point is below another when it belongs to the closure of the other, subject to an explicitly stated orientation convention.
Spectral abscissa	Operator Theory	The greatest real part among the eigenvalues or spectral values of a matrix or bounded linear operator.
Spectral Asymmetry	Spectral TheoryIndex Theory	In mathematics and physics, the spectral asymmetry is the asymmetry in the distribution of the spectrum of eigenvalues of an operator.
Spectral band	Spectral AnalysisSpectral Regions and Band Conventions	Bound a contiguous region on a declared spectral coordinate and type it by the physical feature, allocation rule, or instrument response that selects the region, keeping band limits, bandwidth, and overlap conventions explicit.
Spectral Element Method	Numerical AnalysisHigh Order Methods	A high-order PDE discretization that partitions a domain into elements and represents each element with high-degree polynomial bases, combining finite-element geometry with spectral accuracy.
Spectral method	Numerical AnalysisGlobal High Order Methods for Differential Equations	Approximate a differential-equation solution on a usually single global domain by a truncated expansion in smooth global basis functions, then determine its coefficients through Galerkin, tau, collocation, or related residual conditions with convergence tied to regularity and basis fit.
Spectral Submanifold	Nonlinear Modal AnalysisSpectral Submanifold Theory	The distinguished smooth invariant nonlinear continuation of a chosen linear spectral subspace, enabling qualified low-dimensional dynamics on that manifold.
Spectral theory of compact operators	Functional AnalysisSpectral Theory	In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets.
Spectrum (functional analysis)	Functional Analysis and Operator Theory	The set of scalars for which an operator minus that scalar times the identity fails to possess an everywhere-defined bounded inverse.
Spectrum of a C*-Algebra	Operator Algebras	The spectrum of a C*-algebra organizes its nonzero irreducible representations into unitary-equivalence classes.
Sperner property of a partially ordered set	Order Theory	The property that a graded poset’s largest antichain has the same size as its largest rank level.
Sphere packing	Discrete GeometryGeometric Packing	Arrange nonoverlapping equal-radius balls in a specified ambient space to maximize a declared finite or asymptotic density under explicit boundary, periodicity, and congruence conventions.
Spherical 3-manifold	Geometric TopologyThree Manifolds	In mathematics, a spherical 3-manifold M is a 3-manifold of the form.
Spherical Design	Algebraic CombinatoricsDesign Theory	A finite equal-weight point set on a unit sphere whose discrete average exactly matches the sphere average for every polynomial through a declared degree.
Spherical Measure	Geometric Measure Theory	In mathematics — specifically, in geometric measure theory — spherical measure σ n is the "natural" Borel measure on the n-sphere S n .
Spherical trigonometry	Spherical GeometryTrigonometry	Spherical trigonometry is the branch of spherical geometry and trigonometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions.
Spherical variety	Algebraic Geometry	An algebraic variety with an action of a reductive group for which a Borel subgroup has an open dense orbit.
Sphericon	Solid Geometry	Form a developable one-surface roller by bisecting a 90-degree bicone, rotating one half by a quarter turn, and rejoining so its meandering roll brings the whole surface into ground contact.
Spheroid	Solid Geometry	Recognize an ellipsoid of revolution by its two equal transverse axes, and distinguish oblate from prolate by the remaining axis.
Spin Connection	Spin Geometry	A connection on a spinor bundle that lifts an orthonormal-frame or Lorentz connection to spin representations, enabling covariant differentiation and parallel transport of spinor fields and coupling fermions consistently to curved geometry.
Spinc structure	Differential Geometry	A lift of an oriented manifold’s frame bundle to the group Spin-c, generalizing spin structure by coupling spinors to a complex line bundle.
Spitzer's Formula	Random Walk Fluctuation Theory	Recover an i.i.d. random walk's running-maximum laws from positive-part partial-sum laws through Spitzer's time-generating-function identity.
Spline wavelet	Wavelet Analysis	A wavelet whose scaling functions or wavelets are constructed from spline spaces, combining multiresolution structure with piecewise-polynomial regularity.
Split graph	Graph Theory	A graph whose vertices can be partitioned into one clique and one independent set.
Splitting field	Abstract AlgebraField and Galois Theory	Form the field extension generated by all roots of a polynomial so that it factors completely into linear terms, with minimality and uniqueness understood relative to the base field.
Splitting principle	Algebraic Topology and Characteristic Classes	A technique that pulls a vector bundle to a space where it decomposes into line bundles, performs calculations there, and transfers valid identities back through an injective cohomology map.
Springer correspondence	Geometric Representation Theory	Match irreducible Weyl-group representations to eligible nilpotent orbits and centralizer local systems through top cohomology of Springer fibers.
SQ-Universal Group	Combinatorial Group Theory	Require every countable group to embed as a subgroup of some quotient of one host group, preserving the exact quotient-then-subgroup quantifier pattern.
Square number	Elementary Number Theory	An integer equal to the product of some integer with itself.
Square principle	Set Theory	A set-theoretic principle asserting a coherent sequence of short club sets with no single global thread.
Square-Free Element	Commutative AlgebraFactorization Theory	A nonzero element of a unique factorization domain whose irreducible factors all have multiplicity at most one, equivalently one not divisible by the square of any nonunit.
Square-Free Integer	Number Theory	An integer not divisible by any perfect square greater than one, equivalently one whose prime factorization contains no repeated prime factor.
Square-free polynomial	AlgebraPolynomial Algebra	In mathematics, a square-free polynomial is a univariate polynomial (over a field or an integral domain) that has no multiple root in an algebraically closed field containing its coefficients.
Square-free word	Combinatorics on Words	A square-free word is a finite or infinite symbol sequence containing no adjacent repetition XX of any nonempty contiguous factor X.
Squared Triangular Number	Number TheoryFigurate Numbers	In number theory, the sum of the first cubes is the square of the th triangular number.
Squeeze Mapping	Linear AlgebraPlanar Geometry	A planar linear map that stretches one fixed axis by a positive factor and contracts the other by its reciprocal, preserving area and coordinate product.
Stabilizer subgroup	Group TheoryGroup Actions	For a group G acting on a set X, the stabilizer subgroup of a point x is the subgroup of exactly those elements of G that leave x fixed under the action.
Stable manifold	Dynamical SystemsHyperbolic Dynamics	The invariant manifold consisting locally or globally of states whose forward trajectories converge to a hyperbolic fixed point or invariant set, tangent to its stable eigenspace.
Stable manifold theorem	Dynamical SystemsDifferential Equations	In mathematics, especially in the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given hyperbolic fixed point.
Stable Normal Bundle	Differential TopologyBundle Theory	The embedding-independent stable real normal-vector-bundle class of a closed smooth manifold, equal to the stable inverse of its tangent class.
Stable polynomial	Control and Systems Mathematics	A polynomial whose roots all lie in a declared stability region, commonly the open left half-plane for continuous time or open unit disk for discrete time.
Stable Roommates Problem	Matching TheoryCombinatorics	A matching problem on an even-sized, non-bipartitioned set in which every participant strictly ranks every other participant and a solution is a perfect pairing with no blocking pair; unlike stable marriage, a solution need not exist.
Stack (Mathematics)	Category Theory	Organize objects varying over a Grothendieck site as a category fibred in groupoids whose isomorphisms form sheaves and whose compatible local objects glue effectively to a global object.
Staircase paradox	Mathematical Analysis	A sequence of rectilinear curves can converge uniformly to a diagonal while their lengths fail to converge to the diagonal’s length.
Stalk (sheaf)	Sheaf Theory	The local object obtained from a sheaf at one point by identifying sections that agree on some sufficiently small neighborhood of that point.
Standard Borel space	Descriptive Set TheoryMeasurable Spaces	A measurable space isomorphic to the Borel measurable space of a Polish space, providing a regular setting in which measurable bijections and probability constructions behave well.
Standard model (set theory)	Models of Set Theory	Interpret the membership symbol of a set-theoretic structure as actual membership restricted to its domain, separating semantic standardness from transitivity and inner-model conditions.
Standard monomial theory	Algebraic Geometry and Representation Theory	A method constructing explicit bases for coordinate rings and line-bundle sections on flag and Schubert varieties through ordered products satisfying straightening relations.
Standard Part Function	Nonstandard Analysis	Map each finite hyperreal to the unique real number infinitesimally close to it, thereby passing from a nonstandard approximation to its ordinary real shadow.
Stanley symmetric function	Algebraic Combinatorics	A symmetric function indexed by a permutation and generated from its reduced words, encoding reduced-decomposition and Schubert-polynomial combinatorics.
Stanley's Reciprocity Theorem	Enumerative CombinatoricsDiscrete GeometryPolyhedral Geometry	A reciprocity identity turning a rational cone's lattice-point generating function under variable inversion into the signed generating function of its relative interior.
Stanley–Reisner ring	Combinatorial Commutative Algebra	The quotient of a polynomial ring by the squarefree monomial ideal generated by nonfaces of a simplicial complex.
Star domain	Geometry	A set containing a point from which the line segment to every point of the set remains inside the set.
Star polyhedron	Geometry	A nonconvex polyhedron with systematic star-like self-intersection or alternating salient and reentrant geometry.
Starlike tree	Graph Theory	A tree with exactly one vertex of degree greater than two, equivalently several paths joined at one central vertex.
Stationary process	Stochastic Processes	A stochastic process whose probabilistic law is invariant under shifts of its time index, with weaker forms preserving selected moments instead.
Stationary sequence	Stationary Stochastic Processes	A sequence of random variables whose finite-dimensional joint distributions are invariant under shifts of the index origin.
Stationary set	Set TheoryModel Theory	In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets and is analogous to a set of non-zero measure in measure theory.
Statistical manifold	Information Geometry	A differentiable family of probability distributions equipped with information-geometric structures, most canonically the Fisher information metric and compatible affine connections.
Steenrod problem	Algebraic TopologyHomology Theory	In mathematics, and particularly homology theory, Steenrod's Problem (named after mathematician Norman Steenrod) is a problem concerning the realisation of homology classes by singular manifolds.
Steinberg formula	Lie Theory and Representation Theory	A Weyl-group and Kostant-partition-function formula for the multiplicity of an irreducible highest-weight representation inside a tensor product of two irreducible representations of a complex semisimple Lie algebra.
Steinberg group (K-theory)	Algebraic K TheorySteinberg Groups and Low K Groups	Present the universal central extension of the stable elementary linear group of a ring by generators x_ij(a) and Steinberg commutator relations.
Steiner system	Combinatorial Design Theory	An n-point uniform block design S(t,k,n) in which every t-point subset occurs in exactly one k-point block.
Steiner tree problem	Combinatorial and Geometric Optimization	The optimization problem of connecting specified terminal points at minimum total cost while permitting additional intermediate Steiner points or vertices.
Stiefel Manifold	Differential TopologyNumerical Linear Algebra	The space of ordered orthonormal frames in an inner-product space retains a chosen basis while projecting to the Grassmannian of their spans.
Stieltjes moment problem	Analysis and Probability	The problem of deciding whether a sequence is the moment sequence of a positive measure on the nonnegative half-line and whether that measure is unique.
Stieltjes transformation	Analysis and ProbabilityIntegral Transforms and Spectral Measures	Map a measure to an analytic function off its support by integrating the resolvent kernel 1/(t−z), with boundary limits recovering density and encoding moments and spectral information.
Stirling numbers of the second kind	Enumerative CombinatoricsSet Partitions	The numbers S(n,k) counting partitions of an n-element labeled set into exactly k nonempty unlabeled blocks.
Stirling transform	Enumerative Combinatorics	The invertible sequence transform that weights source terms by Stirling numbers of the second kind, with inverse coefficients given by signed first-kind Stirling numbers.
Stochastic drift	Stochastic Processes	The systematic time-directed component of a stochastic process, commonly represented by the conditional mean rate of change apart from random fluctuation.
Stochastic ordering	Probability Ordering	A partial-order comparison of probability distributions stating that one is larger than another according to a declared class of increasing tests or risk criteria.
Stone duality	Categorical Duality and Pointfree Topology	Relate Boolean and related ordered algebraic structures contravariantly to compact topological or ordered spaces so algebraic elements become distinguished subsets and homomorphisms reverse into continuous maps.
Stone Space	General TopologyStone DualityBoolean Algebra	The compact zero-dimensional Hausdorff space of ultrafilters of a Boolean algebra, with algebra elements represented by membership clopen sets.
Stoneham number	Number Theory	In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G.
Stone–Weierstrass Theorem	Mathematical AnalysisUniform Approximation	A density criterion stating that a real subalgebra of continuous functions on a compact Hausdorff space uniformly approximates every continuous function when it contains constants and separates points, with conjugation closure required in the complex case.
Stopping time	Stochastic Processes	A random time whose occurrence can be determined from information available up to that time, without access to future states of the stochastic process.
Strange nonchaotic attractor	Dynamical Systems	An invariant attracting set with geometrically nonsmooth or fractal structure but no positive maximal Lyapunov exponent.
Strategy-stealing argument	Combinatorial Game TheoryPositional Games	A nonconstructive game-theoretic proof that a second-player winning strategy would let the first player make a harmless opening move and then adopt that strategy, producing a contradiction.
Stratified space	TopologySingular Spaces	A topological space decomposed into disjoint manifold-like strata fitted together under frontier and regularity conditions that organize singular behavior by dimension.
Stratifold	Differential Topology	A stratified topological space equipped with a sheaf of smooth functions and manifold strata satisfying controlled local conditions, used as a geometric model for homology theories.
Strength of a graph	Graph TheoryEdge Connectivity Invariants	A graph-connectivity invariant given by the minimum ratio of removed edges to the number of additional components they create, equivalently a minimum partition cut ratio.
Strictification	Category Theory	A coherence-backed replacement of a weak categorical structure by a stricter presentation under a specified structure-preserving equivalence.
Strictly positive measure	Topological Measure Support	Require a measure on a topological measurable space to assign positive measure to every nonempty open set, equivalently giving the measure full topological support under standard regularity conventions.
Strictly simple group	Group TheorySubgroup Properties	A group whose only ascendant subgroups are the identity subgroup and the whole group, coinciding with simplicity for finite groups but stronger in general.
Strictly Singular Operator	Functional Analysis	Identify a bounded linear operator that fails to preserve norm from below on every infinite-dimensional subspace, so no infinite-dimensional restriction is an isomorphic embedding even though finite-dimensional behavior may remain well conditioned.
Strong antichain	Order Theory	An antichain whose distinct elements have no common lower bound, or dually no common upper bound, in the ambient poset.
Strong duality	Optimization Duality	Assert that a typed primal optimization problem and its dual attain equal optimal objective values, while keeping zero gap distinct from feasibility, attainment, and optimality certificates.
Strong measure zero set	Set Theoretic Analysis	A set coverable, for every positive length sequence, by intervals whose respective lengths are bounded by that sequence.
Strong monad	Category Theory	In category theory, a strong monad is a monad on a monoidal category with an additional natural transformation, called the strength, which governs how the monad interacts with the monoidal product.
Strong product of graphs	Algebraic and Structural Graph Theory	A graph product on ordered vertex pairs where two pairs are adjacent when their first coordinates are equal or adjacent and independently their second coordinates are equal or adjacent, excluding equality in both.
Strongly positive bilinear form	Functional Analysis	A bilinear form on a normed vector space that dominates a fixed positive multiple of squared norm on every vector.
Strongly regular graph	Algebraic Graph Theory	A regular graph with fixed numbers of common neighbors for every adjacent pair and for every nonadjacent pair, summarized by parameters (v,k,lambda,mu).
Structure (mathematical logic)	Model Theory	A nonempty carrier together with interpretations of the constants, functions and relations in a formal signature.
Structure Theorem for Finitely Generated Modules over a Principal Ideal Domain	Modules over Principal Ideal Domains	A classification theorem stating that every finitely generated module over a principal ideal domain decomposes into a finite free part and uniquely determined cyclic torsion factors, in invariant-factor or elementary-divisor form.
Stunted projective space	Algebraic TopologyHomotopy Theory	A projective-space quotient that collapses a lower skeleton and retains higher cells.
Størmer number	Number Theory	A positive integer whose squared value plus one has a prime factor at least twice the original integer.
Sub-probability measure	Probability Theory	A nonnegative countably additive measure whose total mass is at most one, allowing missing mass to represent termination, failure or an unmodeled outcome.
Subcategory	Category Theory	A category whose objects and morphisms are selected from a parent category while retaining the same sources, targets, identity morphisms, and composition.
Subderivative	Convex Analysis	A supporting slope or covector that generalizes the derivative of a convex function at a point where an ordinary derivative may not exist.
Subfield of an algebra	Associative AlgebraSubfields and Splitting	An F-subalgebra of an F-algebra that is itself a field, with maximal and strictly maximal variants recording containment and dimension conditions.
Subfunctor	Category Theory	In category theory, a branch of mathematics, a subfunctor is a special type of functor that is an analogue of a subset.
Subharmonic function	Potential Theory	An upper-semicontinuous function whose value at each point is no greater than the average over every sufficiently small surrounding sphere or ball.
Sublime number	Number TheoryDivisor Function Sequences	A positive integer having a perfect number of positive divisors and a divisor sum that is itself a perfect number.
Submersion (mathematics)	Differential Topology	A smooth map between manifolds whose differential is surjective at every point, making each target direction locally attainable.
Subnormal operator	Operator Theory and Functional Analysis	A bounded operator on a Hilbert space that is the restriction of a normal operator to an invariant subspace of a larger Hilbert space.
Subobject	Category Theory	An equivalence class of monomorphisms into an object, abstracting the notion of a subset, subgroup, or subspace inside an arbitrary category.
Subpaving	Interval AnalysisInterval Geometry	Nonoverlapping boxes representing or approximating a region in interval analysis.
Subquotient	Abstract AlgebraSubobjects and Quotients	Obtain an algebraic object by first selecting a subobject and then quotienting it by a compatible normal subobject or congruence.
Subrepresentation	Representation TheoryInvariant Subspaces	An invariant subspace of a representation on which the original group, algebra or category action restricts to a representation in its own right.
Subspace Topology	General Topology	A subset inherits precisely the traces of its ambient space's open sets, making inclusion the defining continuous map.
Subterminal Object	Category TheoryTopos Theory	An object of a category into which every object has at most one morphism; when a terminal object exists, subterminal objects are precisely its subobjects.
Subtraction	ArithmeticElementary Operations	An arithmetic operation that obtains the difference between a minuend and subtrahend, ordinarily defined as addition of the subtrahend's additive inverse where that inverse exists.
Successor Ordinal	Set TheoryOrdinal Numbers	An ordinal of the form α+1: the least ordinal strictly greater than α, represented in the von Neumann model as α united with the singleton containing α.
Sullivan Conjecture	Algebraic TopologyMapping Spaces	The proved Miller theorem that, for a finite group and finite-dimensional CW complex, the based mapping space from the group's classifying space is weakly contractible, equivalently constant maps give a weak equivalence from the target to the unbased mapping space.
Sum of squares function	Number TheoryAdditive Number Theory and Quadratic Forms	Count ordered signed integer k-tuples whose squared coordinates sum to n, yielding the arithmetic function r_k(n) and its divisor-sum, theta-series, and local-obstruction structure.
Sum-Free Sequence	Number TheoryAdditive Combinatorics	A strictly increasing sequence of positive integers in which no term is representable as a sum of a subset of its predecessors, coupling prefix-dependent additive avoidance to sparse-growth and reciprocal-sum questions.
Sum-product number	Recreational Number Theory	Sum-product number denotes a number equal to the product of the sum and product of its digits within recreational number theory.
Sums of three cubes	Diophantine AnalysisNumber Theory	Integer representability by three signed cubes, subject to a modulo-nine obstruction.
Super-Poulet number	Number Theory	A composite base-two pseudoprime for which every positive divisor d also divides two-to-the-d minus two.
Superalgebra	Graded Algebra	A Z₂-graded algebra split into even and odd components whose multiplication adds parity modulo two.
Supercompact cardinal	Set Theory and Large Cardinals	A large cardinal kappa for which, at every target scale lambda, there is an elementary embedding with critical point kappa into an inner model closed under lambda-length sequences.
Supercompact space	General TopologyCompactness and Superextensions	A topological space possessing a subbase for which every subbasic open cover has a subcover of at most two members, a strong cover property implying compactness and preserved under products.
Superconvergence	Numerical Analysis	A higher approximation order at specified points or for recovered quantities than the same method's general error order under stated conditions.
Superegg	Geometry	A superegg is the solid of revolution formed by rotating an elongated superellipse of exponent greater than two about its long axis.
Supermanifold	SupergeometryDifferential Geometry	In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables.
Supermodule	SuperalgebraAbstract Algebra	In mathematics, a supermodule is a Z 2 -graded module over a superring or superalgebra.
Supernatural number	Number Theory	A formal prime product whose exponent at each prime is a natural number or infinity, extending positive integers under divisibility.
Superpartient Ratio	Ancient and Medieval Arithmetic of Ratios	In ancient and medieval ratio theory, a reduced greater-to-lesser ratio (n+a):n with 1<a<n, containing the lesser once plus more than one of its aliquot parts.
Superperfect number	Number Theory	In number theory, a superperfect number is a positive integer that satisfies.
Superpermutation	CombinatoricsShortest Common Superstrings	A string over n symbols that contains every permutation of those symbols as a contiguous substring.
Supersolvable Arrangement	Hyperplane ArrangementsGeometric Lattices	A finite central hyperplane arrangement whose intersection lattice has a full rank-by-rank chain of modular flats.
Supertransitive class	Set Theory	A transitive class A that also contains every subset of each member: whenever x belongs to A, the entire power set P(x) is included in A.
Supnick Matrix	Matrix TheoryMonge Properties	A symmetric square Monge matrix, under a stated diagonal convention, with ordered quadrangle inequalities.
Supplementary Angles	Euclidean Geometry	When summing two angles (either adjacent or separated in space), three special cases are named complementary, supplementary, and explementary angles.
Supporting hyperplane	Convex Analysis	A hyperplane meeting a set while the entire set lies in one of the two closed half-spaces it bounds.
Surface integral	Multivariable CalculusIntegration on Surfaces	An integral of a scalar or vector field over a parameterized surface, combining local field values with induced area or oriented flux elements.
Surgery theory	Geometric TopologyManifold Topology	A theory of controlled manifold modification by cutting out a sphere neighborhood and gluing in a complementary piece.
Surreal number	Number SystemsCombinatorial Game Theory	In mathematics, the surreal number system is a totally ordered proper class containing not only the real numbers but also infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number.
Suspension (Topology)	Algebraic Topology	A space formed by collapsing the two ends of a cylinder on X to distinct points, with a further basepoint-line quotient in the reduced pointed form.
Symbolic Cholesky Decomposition	Numerical AnalysisSparse Direct Methods	Preprocessing that predicts sparse Cholesky-factor nonzeros and elimination dependencies before numerical factorization.
Symbols of Grouping	Mathematical NotationArithmetic	Symbols of grouping are paired mathematical delimiters that bind subexpressions, establish scope, and control the order in which an expression is parsed or evaluated.
Symmetric closure	Relation Theory	The smallest symmetric relation containing a given binary relation R, equal to the union of R with its converse.
Symmetric difference	Set Theory	The set operation retaining elements that belong to exactly one of two sets.
Symmetric group	Permutation Groups	Form the group of every bijection from a set to itself under composition, with finite S_n containing n! permutations and organizing cycle type, parity, actions, and universal embeddings of finite groups.
Symmetric inverse semigroup	Semigroup Theory	The inverse monoid of all partial bijections on a set under composition.
Symmetric polynomial	Algebra	A multivariable polynomial unchanged by every permutation of its variables.
Symmetric relation	Relation Theory	A binary relation in which a related pair remains related when its two elements are reversed.
Symmetric Successive Over-Relaxation	Applied MathematicsNumerical Linear AlgebraIterative Methods	A preconditioner built from matched forward and backward SOR triangular factors around an inverse diagonal, with optional relaxation parameter, for iterative linear-system solution.
Symmetrically continuous function	Real Analysis	A real function whose values at equally spaced points on opposite sides of each point approach one another.
Symmetrization	Representation Theory	Map a multivariable function, tensor, or representation vector to its permutation-invariant component by summing or averaging over a symmetric-group action.
Symplectic category	Symplectic Geometry	In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations, inclusions of Lagrangian submanifolds L into M \times N^{-} , where the superscript minus means minus the given symplectic form (for example, the graph of a symplectomorphism; hence, minus).
Symplectic Integrator	Geometric Numerical IntegrationNumerical Analysis	A numerical time-stepping method for Hamiltonian dynamics whose discrete update preserves the symplectic two-form, yielding bounded long-term energy behavior through the flow of a nearby modified Hamiltonian.
Symplectic Representation	Representation TheoryRepresentations Preserving Bilinear Forms	A group or Lie algebra acts linearly on an even-dimensional vector space while preserving a fixed nondegenerate alternating bilinear form, equivalently factoring through the corresponding symplectic group or Lie algebra.
Symplectic spinor bundle	Differential GeometrySymplectic Spin Geometry	The infinite-rank Hilbert bundle associated to a metaplectic structure on a symplectic manifold through the metaplectic representation.
Symplectization	Symplectic Geometry	The canonical construction that associates a symplectic manifold to a contact manifold by adjoining a nonzero scale coordinate to its contact covectors.
Syndetic set	Ergodic Ramsey Theory	A subset of a semigroup—canonically the natural numbers—with uniformly bounded gaps or finitely many left translates covering the carrier.
Syntax (logic)	Mathematical LogicFormal Languages	The formal symbols, formation rules and derivation or transformation rules that determine which expressions and proofs are well formed independently of their interpretation.
Synthetic differential geometry	Categorical and Synthetic Differential Geometry	A topos-theoretic formalization of differential geometry that encodes smooth infinitesimal behavior synthetically rather than through classical limit analysis.
Synthetic geometry	Foundations of Geometry	A method of developing geometry from primitive objects, incidence or order relations, and axioms, proving results without making coordinates the primary foundation.
T-structure	Homological AlgebraDerived Categories	A pair of subcategories of a triangulated or stable infinity category satisfying shift, orthogonality and truncation axioms, whose intersection forms an abelian heart.
T0 space	General TopologySeparation Axioms	In topology and related branches of mathematics, a topological space X is a T 0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of X, at least one of them has a neighborhood not containing the other.
T4 Space	General TopologySeparation Axioms	In topology and related branches of mathematics, a normal space is a topological space in which any two disjoint closed sets have disjoint open neighborhoods.
Tacnode	Algebraic GeometryCurve Singularities	A plane-curve double singularity where two smooth local branches share the same tangent, canonically modeled by y²=x⁴ and carrying higher contact than an ordinary node.
Tail dependence	ProbabilityAsymptotic Dependence Measures	Measure whether two continuously distributed variables continue to co-exceed matched extreme quantiles by taking an upper- or lower-tail conditional-probability limit determined by their copula rather than their marginal scales.
Tangent bundle	Differential Geometry	The geometric bundle formed by assembling every tangent space of a smooth manifold into one smooth total space over that manifold.
Tangent indicatrix	Differential Geometry	The curve traced on the unit sphere by the unit tangent vector of a regular space curve.
Tangent measure	Geometric Measure Theory	A weak limit of rescaled blow-ups of a Radon measure around a point, capturing its infinitesimal mass geometry.
Tangential quadrilateral	Euclidean Geometry	A convex quadrilateral whose four sides are tangent to one inscribed circle.
Tarski's Plank Problem	Convex Geometry	Relate geometric coverage to directional thickness: any finite family of Euclidean planks covering a convex body must have total plank width at least the body's minimum width.
Tarski's undefinability theorem	Mathematical Logic	A limit theorem stating that sufficiently strong consistent formal systems cannot define within themselves the full truth predicate for their standard arithmetic interpretation.
Tarski–Grothendieck set theory	Axiomatic Set Theory	An axiomatic set theory extending ZFC with an axiom that places every set inside a Grothendieck-style universe, thereby implying unbounded inaccessible cardinals.
Tate topology	Rigid Analytic GeometryAlgebraic Geometry	In mathematics, the Tate topology is a Grothendieck topology of the space of maximal ideals of a k-affinoid algebra, whose open sets are the admissible open subsets and whose coverings are the admissible open coverings.
Tau additivity	Measure Theory	A topological regularity property requiring a measure of any measurable upward-directed union of open sets to equal the supremum of their individual measures.
Tau-leaping	Stochastic Simulation	An approximate stochastic-simulation method that advances a reaction or event system by a finite time step while sampling multiple event counts from Poisson distributions.
Tautological Ring	Algebraic GeometryModuli of Curves	The minimal operation-stable system of natural cycle-class subrings on moduli spaces of stable pointed curves, generated through forgetful and gluing morphisms and carrying the standard psi, kappa, lambda, and boundary constructions.
Tautology (Logic)	Mathematical Logic	A formula that evaluates as true under every admissible valuation in a specified logic, with propositional tautology distinguished from broader first-order logical validity and from merely repetitive language.
Taxicab number	Recreational Number Theory	The smallest positive integer expressible as a sum of two positive cubes in a specified number of distinct unordered ways.
Taylor series	Analysis	An infinite power series whose coefficients are a function’s derivatives at one expansion point.
Telephone number (mathematics)	Enumerative CombinatoricsInteger Sequences	In the mathematics of chess, the telephone numbers count the number of ways to place rooks on an chessboard in such a way that no two rooks attack each other (the so-called eight rooks puzzle), and in such a way that the configuration of the rooks is symmetric under a diagonal reflection of the board.
Telescoping Markov chain	Probability TheoryStochastic Processes	In probability theory, a telescoping Markov chain (TMC) is a vector-valued stochastic process that satisfies a Markov property and admits a hierarchical format through a network of transition matrices with cascading dependence.
Tensor	Linear Algebra Multilinear Algebra	A multilinear map over a vector space, represented in coordinates by an indexed array whose components co-vary contravariantly and covariantly under change of basis so the underlying geometric object stays invariant — the transformation law being what makes it a tensor rather than a mere array.
Tensor field	Differential Geometry and Mathematical Physics	A smoothly or otherwise regularly varying assignment of a tensor of fixed type to every point of a manifold or region.
Tensor product of fields	Field TheoryCommutative Algebra	In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield.
Tensor Rank Decomposition	Multilinear AlgebraTensor Decomposition	Represent a multiway tensor as a sum of rank-one outer products, coupling one factor vector from every mode in each component.
Tensor representation	Representations of General Linear and Classical Groups	A representation of a general linear or matrix group obtained from finite tensor products of a fundamental representation and its dual, including their irreducible factors.
Tensor Sketch	Randomized Numerical Linear AlgebraSketching Algorithms	A randomized linear embedding for tensor-product features that combines factorwise CountSketches by convolution, approximating inner products or norms without materializing the full Kronecker vector.
Term algebra	Universal Algebra	The freely generated algebra of formal terms built from variables and operation symbols in a signature, initial among algebras receiving those generators.
Terminal singularity	Birational GeometryAlgebraic Geometry	In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group.
Ternary cubic	Algebraic Geometry	A homogeneous polynomial of degree three in three variables, studied through plane cubic curves and invariant theory.
Ternary Operation	Universal AlgebraFinitary Operations	A total three-input function—internally T:A³→A when defined on a set—that maps each ordered triple to one output, with any further identities stated separately.
Ternary Quartic	Algebraic Geometry	A homogeneous polynomial form of total degree four in exactly three variables, carrying a fifteen-coefficient parameter space whose positivity, zero locus, and linear-change invariants support distinct algebraic and geometric analyses.
Tertiary ideal	Noncommutative Ring Theory	A two-sided ideal in a possibly noncommutative ring satisfying the tertiary irreducibility condition used to obtain decompositions where ordinary primary decomposition may fail.
Test ideal	Commutative AlgebraPositive Characteristic Algebra	A test ideal is a positive characteristic analog of a multiplier ideal in, say, the field of complex numbers.
Tetracategory	Higher Category Theory	A weak four-dimensional categorical structure in which composition and coherence extend tricategorical cells by one dimension rather than holding strictly.
Tetrahedron packing	Discrete GeometryPacking Problems	The geometric optimization problem of arranging congruent regular tetrahedra without overlapping so as to maximize the fraction of three-dimensional space they occupy.
Thabit number	Number TheoryInteger Sequences	An integer of the form 3·2^n−1 for a nonnegative integer n, historically linked through special primality conditions to constructions of amicable numbers.
Theta representation	Representation TheoryHeisenberg Group	A representation of the real Heisenberg group on a function space whose lattice-compatible vectors and transformation laws generate Jacobi theta functions.
Thickness (graph theory)	Topological Graph Theory	The minimum number of planar spanning subgraphs needed to partition a graph's edge set.
Thompson factorization	Finite Group TheorySubgroup Factorizations	A factorization of certain finite groups as a product of two structurally selected subgroups, commonly normalizers or centralizers of p-subgroups.
Three-dimensional space	GeometryDimensional Spaces	A geometric space in which a point requires three independent coordinates, most commonly three-dimensional Euclidean space but also including general three-manifolds.
Three-valued logic	Mathematical LogicMany Valued Logic	A three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false, and some third value.
Thurston Elliptization Conjecture	—	A proved three-manifold classification theorem: a closed three-manifold has finite fundamental group exactly when it admits a spherical metric of constant positive sectional curvature.
Tightness of measures	Measure Theory	The property that probability mass can be captured uniformly well inside compact subsets.
Tilting theory	Representation Theory and Homological Algebra	Use a self-orthogonal finite-projective-dimension module or analogous tilting object to construct an endomorphism algebra and transport controlled homological information between representation categories.
Time-series segmentation	Time Series AnalysisStructural Segmentation	The partition of an ordered signal into contiguous intervals whose observations are internally coherent according to a selected model, feature or regime.
Toeplitz operator	Operator Theory	The compression to Hardy space of multiplication by a bounded function on the unit circle, yielding an operator with constant matrix diagonals.
Tomita–Takesaki theory	Operator Algebras	The modular theory deriving a one-parameter automorphism group and commutant duality from a von Neumann algebra with a cyclic separating vector or faithful normal weight.
Toothpick sequence	Integer SequencesGeometric Cellular Growth	Count the total line segments in a deterministic planar growth process that starts from one segment and at each stage adds a perpendicular segment at every exposed endpoint.
Topological Algebra	Functional AnalysisGeneral Topology	In mathematics, a topological algebra A is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
Topological Boundary	General Topology	In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of .
Topological category (enriched category theory)	Category TheoryEnriched Categories	A category whose hom-sets carry topological-space structure and whose identity and composition maps are continuous, usually formalized as enrichment over compactly generated Hausdorff spaces.
Topological Closure	General Topology	Enlarge a subset to the least closed set containing it in a specified topology, equivalently including every point whose neighborhoods meet the subset.
Topological Degree Theory	Topological Methods in Analysis	Assign an integer-valued signed preimage count that survives admissible homotopy, so a nonzero degree certifies that a continuous map hits the target even when individual solutions cannot be found or tracked.
Topological Dynamical System	Topological DynamicsErgodic Theory	A topological space equipped with a continuous discrete-time map or continuous group/semigroup action for qualitative study of orbits and recurrence.
Topological Galois Theory	Topological Galois TheoryComplex Analysis	A theory using the topology and monodromy of branched coverings defined by multivalued analytic functions to derive obstructions to solving equations or representing functions by specified classes of explicit formulas.
Topological game	Set Theoretic Topology	An infinite perfect-information game on a topological space whose moves are points, sets or covers and whose winning condition encodes a topological property.
Topological homomorphism	Functional Analysis	A continuous linear map between topological vector spaces that induces a topological isomorphism from the quotient by its kernel onto its image.
Topological Space	General Topology	Capture the minimum data continuity needs by pairing a set with a collection of its subsets — the open sets, closed under arbitrary unions and finite intersections — so that continuity, compactness, and connectedness can be defined with no reference to distance.
Topological Surface	Topology	A connected two-dimensional topological manifold whose points have planar neighborhoods, with half-plane neighborhoods allowed at a boundary.
Topos	Category TheoryTopos Theory	A category with finite limits, exponentials and a subobject classifier, providing a categorical universe in which objects, maps and internal predicates can be studied.
Toric manifold	Toric TopologyTorus Manifolds	A smooth compact even-dimensional manifold with an effective locally standard action of a half-dimensional torus and a simple convex polytope as orbit space.
Toric variety	Algebraic Geometry	An algebraic variety containing an algebraic torus as a dense open subset whose self-action extends to the entire variety.
Torsion sheaf	Sheaf Theory	A sheaf of abelian groups whose every local section is annihilated by some nonzero integer.
Torsion-Free Abelian Group	Abelian Group Theory	A commutative group in which nx = 0 for a positive integer n implies x = 0, equivalently the identity is its only finite-order element.
Torus action	Algebraic and Differential Geometry	An algebraic or smooth group action of a torus on a variety or manifold, organizing points into orbits and exposing weights, fixed points, quotients, and combinatorial structure.
Torus knot	Knot Theory	A knot isotopic to a closed curve winding p and q times around the two generating directions of an unknotted torus, with coprime p and q.
Total algebra	Algebra	An algebra of all coefficient functions on a suitably finite-factorization monoid, with convolution multiplication extending the finite-support monoid algebra to infinite formal sums.
Total graph	Graph Theory	A graph encoding both vertices and edges of a source graph, with adjacency for adjacency or incidence in the source.
Total ring of fractions	Commutative AlgebraLocalization and Quotient Rings	Localize a commutative ring at all of its non-zero-divisors, embedding it injectively into the largest localization that makes every regular element invertible without forcing zero divisors to invert.
Total subset	Functional AnalysisDense Spanning Sets	A subset of a topological vector space whose linear span is dense in the entire space.
Total variation	Analysis	A supremum-based measure of the total accumulated magnitude of change in a function, path, signed measure, or related object.
Totally disconnected space	TopologyConnectedness Properties	A topological space whose only connected subspaces are single points and the empty set.
Tower of objects	Category Theory	An inverse sequence in a category: objects indexed by nonnegative integers with compatible maps from every later stage to each earlier stage.
Traced monoidal category	Category Theory and Compositional Systems	A monoidal category equipped with a trace operation that feeds an output object back into a matching input while satisfying naturality, dinaturality, vanishing, superposing and yanking axioms.
Train track (mathematics)	Low Dimensional Topology	A smoothly branched graph embedded in a surface that carries measured laminations through compatible nonnegative branch weights.
Transfer (group theory)	Group TheoryFinite Group Homomorphisms	A homomorphism from a group to the abelianization of a finite-index subgroup, constructed by multiplying subgroup residues across coset representatives and used in finite-group structure theorems.
Transfer matrix	Wavelet Theory	The block-Toeplitz linear operator induced by a refinement mask whose eigenstructure characterizes refinable functions and their regularity.
Transfer principle	Model Theory	A theorem or schema carrying all sentences of a specified logical language that hold in one structure to another related structure.
Transfinite number	Set Theory	An ordinal or cardinal number larger than every finite number, used to order or measure infinite sets.
Transition-rate matrix	Stochastic Processes	The infinitesimal generator of a finite-state continuous-time Markov chain, with nonnegative off-diagonal jump rates and rows summing to zero.
Transitive reduction	Graph Theory	A smallest-edge directed graph preserving exactly the reachability relation of a given directed graph.
Transitive Set	—	A set that contains every member of each of its members, equivalently a set T satisfying union(T) subseteq T.
Transitively normal subgroup	Group TheoryNormality Conditions	A subgroup H of G such that every subgroup normal in H is also normal in G, making normality transitive through H.
Translation plane	Finite and Incidence Geometry	A projective plane containing a line whose elation group acts transitively on the affine points of each line parallel to a fixed direction, yielding an affine translation structure.
Translation surface	Flat and Teichmuller Geometry	A surface assembled from Euclidean polygons by identifying parallel boundary edges through translations, yielding a flat metric away from finitely many conical singularities.
Transport of Structure	Structural Mathematics	Define operations, relations, or other structure on one mathematical carrier through a chosen equivalence so that the equivalence becomes structure-preserving by construction.
Transpose of a linear map	Linear Algebra	The induced linear map between dual spaces obtained by precomposing functionals with the original map.
Transpositions matrix	Matrix TheoryXor Structured Matrices	A power-of-two square matrix generated from one vector by indexing entries with the bitwise XOR of row and column indices, so every row and column is a permutation of the vector.
Transseries	Asymptotic AnalysisGeneralized Series	Formal generalized series built from nested powers, exponentials and logarithms and ordered by asymptotic dominance, supporting algebra, differentiation and solution of differential equations beyond ordinary power series.
Transversal (Combinatorics)	—	A collision-free assignment that chooses one member from each indexed set, equivalently a matching that saturates the family side of its incidence graph.
Traveler's Dilemma	Game TheoryIterated Dominance and Equilibrium Refinement	Isolate recursion depth as the variable governing whether an iterated-dominance equilibrium predicts behavior — using the same weak-dominance move as the Prisoner's Dilemma chained 98 times, so the $2 prediction evaporates while behavior tracks an incentive gradient the concept is blind to.
Tree (Graph Theory)	Graph Theory	Characterize a connected acyclic graph — equivalently, one with a unique path between every pair of vertices, or exactly n−1 edges on n vertices — so that recognizing the structure by any one handle imports the bridge-everywhere, fundamental-cycle, and cheapest-connector consequences for free.
Tree (Set Theory)	Set Theory and Infinitary Combinatorics	Organize a partially ordered set so every node's strict predecessors form a well-order, turning ancestry into ordinal height while allowing branching and transfinite limit levels.
Tree Decomposition	Graph TheoryTreewidth	In graph theory, a tree decomposition is a mapping of a graph into a tree that can be used to define the treewidth of the graph and speed up solving certain computational problems on the graph.
Tree of primitive Pythagorean triples	Number Theory	A rooted ternary tree that generates every primitive positive integer solution of the Pythagorean equation exactly once by fixed linear transformations.
Treewidth	Structural Graph Theory and Algorithms	The minimum, over all tree decompositions of a graph, of the largest bag size minus one, measuring how closely the graph can be organized around tree-like separators.
Trembling-Hand Perfect Equilibrium	Game TheoryEquilibrium Refinement	Filter the Nash equilibria by giving every action a vanishing positive probability of being played by mistake and keeping only those that survive as the tremble shrinks to zero, discarding equilibria propped up by threats that never have to be carried out.
Trend-stationary process	Time Series Analysis	A nonstationary time series that becomes stationary after subtracting a deterministic time trend.
Trial Division	Number TheoryComputational Algebra	Test a complete, carrier-bounded cover of candidate divisors by exact division to find a factor or rule out reducibility.
Triangle group	Geometric Group TheoryReflection Groups	A reflection group generated by the sides of a spherical, Euclidean or hyperbolic triangle with angles π/l, π/m and π/n, acting on the corresponding tessellation.
Triangle inequality	Metric GeometryDistance Axioms	The distance or norm axiom stating that a direct separation is no greater than the length of any two-step path, d(x,z)≤d(x,y)+d(y,z).
Triangle-free graph	Graph Theory	An undirected graph containing no cycle of length three, equivalently no three-vertex clique.
Triangulated category	Category Theory	An additive category equipped with an autoequivalence and distinguished exact triangles satisfying axioms that abstract exact sequences and homotopy fiber-cofiber sequences.
Triangulation (topology)	Geometric Topology	A homeomorphic representation of a topological space by the geometric realization of a simplicial complex.
Trigonometric integral	Special Functions and Asymptotic Analysis	A family of special functions defined by nonelementary integrals involving sine or cosine divided by the integration variable, including the sine and cosine integrals.
Trilateration	Applied MathematicsRange Based Positioning	Locate an unknown point by intersecting distance constraints from known reference points, using redundant ranges and an uncertainty model when real measurements do not meet at one exact solution.
Trilinear Interpolation	Numerical Analysis	Estimate a value inside an axis-aligned rectangular grid cell by tensor-product linear interpolation of the eight corner values along three coordinates.
Triple system	Nonassociative AlgebraTernary Algebraic Structures	Equip a vector space with a trilinear product returning to that space, creating a generic ternary algebra whose added identities specialize it into Lie, Jordan, and geometry-bearing triple systems.
Trivial measure	Measure TheoryMeasures	The zero measure on a measurable space, assigning measure zero to every measurable set and serving as the least element under pointwise measure comparison.
Trivial Representation	Representation Theory	A representation whose entire acting group operates as the identity, or whose acting Lie algebra operates as zero, making every vector invariant and exposing invariant multiplicity.
Tropical Projective Space	Tropical Geometry	Tropical projective space identifies tropical coordinate tuples that differ by a common additive shift.
Truncated Distribution	Probability TheoryStatistics	A probability law conditioned on a positive-probability retained region and renormalized over that region.
Truncated normal distribution	Probability Theory	A normal probability law conditioned to lie within a specified lower, upper, or two-sided interval.
Truncation error	Numerical AnalysisApproximation Error	The discrepancy introduced when an exact infinite, limiting or continuous mathematical process is replaced by a finite approximation, distinct from finite-precision roundoff and analyzable through omitted terms or local discretization expansions.
Tsirelson space	Functional AnalysisBanach Space Geometry	The first reflexive Banach-space construction containing no subspace isomorphic to any classical ℓp space or c0, built through an implicit norm that recursively controls separated block sequences.
Tunnell's theorem	Number TheoryCongruent Numbers and Elliptic Curves	Test a square-free integer for the congruent-number property through equalities among counts of representations by four ternary quadratic forms—necessary unconditionally and sufficient conditional on Birch–Swinnerton-Dyer.
Turán Graph	Extremal Graph Theory	The balanced complete r-partite graph T(n,r), formed by making nearly equal vertex blocks independent and joining every cross-block pair, uniquely maximizing edges among n-vertex graphs with no K_(r+1).
Turán number	Extremal Combinatorics	The minimum number of r-element blocks on n vertices needed so every k-element vertex subset contains at least one block.
Twin-width	Graph TheoryStructural Graph Theory	A graph parameter equal to the least possible maximum red degree over all complete vertex-contraction sequences that mark adjacency disagreements.
Twisted diagonal (category theory)	Category Theory	A category whose objects are arrows of a category and whose morphisms are oppositely directed domain-codomain squares.
Twisted K-theory	Algebraic Topology	A generalized cohomology theory in which K-theory classes are modified by a background twist, often a degree-three integral cohomology class or bundle of operator algebras.
Twisted sheaf	Algebraic GeometryGerbes and Sheaves	A sheaf-like object whose local pieces glue only up to multiplication by a prescribed gerbe or multiplicative two-cocycle, encoding sheaves on a twisted geometric background.
Two-Element Boolean Algebra	Abstract AlgebraBoolean Algebra	In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose underlying set (or universe or carrier) B is the Boolean domain.
Two-Terminal Series–Parallel Graph	Graph TheoryNetwork Decomposition	A source–sink edge network built recursively from single edges by terminal-preserving series and parallel joins.
Type and Cotype of a Banach Space	Functional AnalysisBanach Space Theory	In functional analysis, the type and cotype of a Banach space are a classification of Banach spaces through probability theory and a measure for how far a Banach space is away from being a Hilbert space.
Type constructor	Type TheoryType Formation	A type-level operator that builds a new type from zero or more argument types, such as function, product, list or parameterized generic constructors.
Type variable	Type TheoryPolymorphism	A formal variable ranging over types, enabling polymorphic expressions and quantified type schemes without denoting a mutable runtime storage location.
Typed lambda calculus	Type Theory	A lambda-calculus formalism assigning types to variables and terms and restricting abstraction and application through typing rules.
Typing rule	Type TheoryTyping Judgments	A formal inference rule specifying how types of component expressions and contextual assumptions justify a type judgment for a larger syntactic construction.
Typographical Number Theory	Mathematical LogicFormal Arithmetic	Typographical Number Theory (TNT) is a formal axiomatic system describing the natural numbers that appears in Douglas Hofstadter's book Gödel, Escher, Bach.
U-invariant	Quadratic Form Theory	The supremum of dimensions of anisotropic quadratic forms over a field, equivalently the threshold above which every form is isotropic when finite.
Ulam number	Additive Number Theory	Generate a seed-dependent increasing integer sequence by repeatedly choosing the least larger integer having exactly one representation as a sum of two distinct earlier terms.
Ulam's packing conjecture	Discrete Geometry	Ulam's packing conjecture, named for Stanisław Ulam, is a conjecture about the highest possible packing density of identical convex solids in three-dimensional Euclidean space.
Ultrametric space	Metric Geometry	A metric space satisfying the strong triangle inequality, so every triangle is isosceles with its two largest distances equal and balls form a nested hierarchy.
Ultrastrong topology	Operator Algebras	A locally convex operator topology on bounded operators generated by seminorms obtained from countable square-summable families of Hilbert-space vectors or positive normal functionals.
Ultraviolet divergence	Quantum Field TheoryRenormalization and Scale Behavior	Identify a field-theoretic integral whose high-energy or short-distance contribution fails to converge as the ultraviolet cutoff is removed.
Ultraweak Topology	Operator Algebras	Topologize a von Neumann algebra as the weak-* dual of its predual, so a net converges exactly when every normal linear functional converges on it.
Unate function	Boolean Function TheorySwitching Functions	A Boolean function that is monotone in each variable after independently choosing whether that variable is interpreted positively or negatively.
Unavoidable Pattern	Combinatorics on Words	Require that, over every finite alphabet, some alphabet-dependent length threshold forces every longer word to contain a contiguous nonerasing morphic instance of the pattern.
Unbounded operator	Functional Analysis	A linear operator whose domain is typically a proper dense subspace of a normed space and which is not required to satisfy a global boundedness estimate.
Uncorrelatedness	Probability and Statistics	A second-moment relation in which two random variables or vectors have zero covariance, excluding linear association without generally implying independence.
Unibranch local ring	Algebraic GeometryCommutative Algebra	In algebraic geometry, a local ring A is said to be unibranch if the reduced ring A red (obtained by quotienting A by its nilradical) is an integral domain, and the integral closure B of A red is also a local ring.
Uniform module	AlgebraModule Theory	A nonzero module in which every two nonzero submodules intersect nontrivially, equivalently every nonzero submodule is essential.
Uniform norm	Functional Analysis	The supremum of pointwise magnitudes of a bounded function, inducing the metric of uniform convergence and the maximum-coordinate norm in finite dimensions.
Uniform polyhedron	GeometrySymmetric Polyhedra	A polyhedron with regular polygonal faces and a symmetry group transitive on vertices, including convex, star and self-intersecting examples.
Uniform space	General Topology and Uniform Structures	A set equipped with a uniform structure that formalizes relative closeness and supports uniform continuity, convergence, and completeness without a chosen metric.
Uniformizable space	General Topology	A topological space whose topology is induced by at least one uniform structure, equivalently a completely regular space under the stated separation convention.
Uniformly disconnected space	Metric Geometry	A metric space with one scale-independent constant preventing any two distinct points from being joined by a chain of sufficiently small relative steps.
Uniformly distributed measure	Geometric Measure Theory	In mathematics — specifically, in geometric measure theory — a uniformly distributed measure on a metric space is one for which the measure of an open ball depends only on its radius and not on its centre.
Unimodality	Probability and Statistics	Unimodality is the property of a distribution or other mathematical object having one mode or single highest region under a stated definition.
Unimodular matrix	Linear Algebra	A square integer matrix with determinant plus or minus one, equivalently an integer matrix invertible over the integers.
Uniqueness quantification	Mathematical Logic	The logical assertion that exactly one object in a domain satisfies a specified predicate.
Unisolvent functions	Approximation Theory and Interpolation	A finite-dimensional function family for which interpolation values at any admissible set of distinct nodes determine one and only one function, equivalently yielding a nonsingular evaluation matrix.
Unisolvent Point Set	Multivariate Interpolation	A finite sampling set for which evaluation on a declared finite-dimensional function space is injective, equivalently making interpolation uniquely solvable when the dimensions match.
Unit disk graph	Geometric Graph Theory	The intersection graph of equal-radius disks in the Euclidean plane, equivalently a graph connecting points whose pairwise distance is at most a fixed threshold after scaling.
Unit measure	Probability Axioms	The probability axiom requiring the measure of the entire sample space to equal one.
Unit sphere	Geometry and Functional Analysis	The set of all vectors or points at norm exactly one from an origin in a specified normed space.
Unit type	Type Theory	A type with exactly one inhabitant up to equality, carrying no information beyond successful presence or completion.
Unit-Quaternion Rotation Representation	Rotation RepresentationsQuaternion Algebra	A two-to-one representation of proper three-dimensional rotations by unit quaternions, where q and -q denote the same rotation and multiplication composes rotations under a declared convention.
Unital (geometry)	Finite GeometryBlock Designs and Projective Planes	A 2-(n³+1,n+1,1) block design in which every pair of points lies on exactly one block, with embedded unitals meeting each projective-plane line in one or n+1 points.
Unitary modular tensor category	Category Theory	A modular tensor category equipped with compatible Hilbert-space and dagger structure making braiding, duality and fusion unitary.
Unitary operator	Functional Analysis	A surjective linear operator on a Hilbert space that preserves inner products, equivalently one whose adjoint is its inverse.
Universal C*-algebra	Operator Algebras	In mathematics, a universal C*-algebra is a C*-algebra described in terms of generators and relations.
Universal generalization	Predicate LogicMathematical Logic	In predicate logic, generalization (also universal generalization, universal introduction, GEN, UG) is a valid inference rule.
Universal property	Category Theory	Define a mathematical object not by its internal construction but by the unique pattern of maps it sustains with every other object in a class — a commuting-diagram condition plus a unique-mediating-morphism clause that pins the object down up to unique isomorphism.
Universal quadratic form	Quadratic Form Theory	A quadratic form over a declared ring that represents every element of that ring, or every element of a specified target subset under a qualified convention.
Universal quantification	Mathematical Logic	The logical operation asserting that a predicate holds for every member of a declared domain of discourse.
Universal set	Set Theory and Foundations	A set intended to contain every object admitted by a theory, including every set in the relevant universe and potentially itself.
Universally measurable set	Descriptive Set Theory	A subset of a Polish space measurable in the completion of every finite Borel measure on that space.
Universe (mathematics)	Foundations of Mathematics	A contextually fixed collection large enough to contain every object and construction under consideration while controlling size, paradox, and quantification in mathematical foundations.
Unstructured grid	Numerical Methods	A mesh that tessellates a domain with irregularly connected cells such as triangles or tetrahedra, allowing local refinement around complex geometry.
Untouchable Number	Number Theory	A positive integer lying outside the image of the aliquot-sum function, classified by the nonexistence of any positive integer whose proper divisors sum to it.
Unusual number	Number Theory	A natural number whose largest prime factor is strictly greater than its square root.
Upper Half-Plane	Complex AnalysisHyperbolic Geometry	The open set H={z∈C: Im(z)>0}, serving as a canonical domain in complex analysis and, with metric ds²=(dx²+dy²)/y², as the upper-half-plane model of hyperbolic geometry.
Urn problem	Probability Models	A probability model representing random sampling from a finite population by drawing colored or labeled balls with a specified replacement and reinforcement rule.
Urysohn's lemma	General Topology	A normal-space theorem that separates any disjoint closed sets by a continuous function taking prescribed values 0 and 1 on them.
Utility graph	Graph TheoryPlanarity	The complete bipartite graph K3,3, whose two sets of three vertices and nine cross-edges encode the nonplanar three-utilities puzzle.
V-Ring (Ring Theory)	Ring Theory	Classify a ring by requiring every simple module on a specified side to be injective, equivalently forcing radical and maximal-ideal intersection properties across all modules or ideals.
V-topology	Algebraic Geometry	A very fine Grothendieck topology in algebraic geometry whose covers are universally subtrusive and can be tested by lifting valuation-ring maps.
Vague set	Fuzzy Mathematics	A set-valued uncertainty model assigning separate lower evidence for membership and lower evidence against membership, leaving an explicit hesitation interval.
Vague topology	Measure TheoryTopologies on Measure Spaces	A topology on Radon measures defined by convergence of integrals against a declared class of continuous test functions, making local mass behavior observable while allowing mass to escape to infinity.
Valuation (geometry)	Convex GeometryIntegral Geometry	In geometry, a valuation is a finitely additive function from a collection of subsets of a set X to an abelian semigroup.
Valuation (logic)	Mathematical LogicModel Theory	In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema.
Van den Berg–Kesten inequality	Probability Theory	A product-measure inequality bounding the probability of disjoint occurrence of two events by the product of their individual probabilities.
Vanish at infinity	Functional AnalysisTopology	A function property requiring values to become arbitrarily small outside sufficiently large norm balls or, more generally, outside a compact set for every positive tolerance.
Variance-gamma distribution	Probability TheoryLevy Distributions	A continuous heavy-tailed probability family obtained by evaluating Brownian motion with drift at an independent gamma-distributed random time, equivalently a normal variance-mean gamma mixture.
Variation diminishing property	Approximation TheoryTotal Positivity and Geometric Design	A transformation is variation diminishing when it cannot increase sign changes, oscillations, or intersection counts, making its output no more geometrically or algebraically variable than its input control data.
Varifold	Geometric Measure Theory	A Radon measure on position–tangent-plane space representing a generalized surface with mass and orientation-free tangent information.
Vector logic	Mathematical Logic and Computing	An algebraic representation of logical truth values and connectives as vectors and matrices acting on a finite-dimensional space.
Vector measure	Measure Theory	A finitely or countably additive set function taking values in a vector space, typically a Banach space.
Vector-valued differential form	Differential Geometry	In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V.
Venn Diagram	Set Theory and Logic	Represent a finite family of sets by closed contours that realize every possible inside/outside membership zone, then mark those zones to express intersections, unions, complements, emptiness, or occupancy.
Verdier duality	Algebraic Topology	A derived-sheaf duality that exchanges proper direct image with exceptional inverse image and extends Poincaré duality to singular spaces and maps.
Verma module	Lie Representation Theory	A highest-weight module induced from a one-dimensional Borel representation, universal among highest-weight modules of the same weight.
Vertex (curve)	Differential GeometryPlane Curves	In the geometry of plane curves, a vertex is a point of where the first derivative of curvature is zero.
Vertex Cover in Hypergraphs	Hypergraph TheoryCombinatorics	A vertex subset that intersects every hyperedge, with feasibility distinct from inclusion-minimality or minimum size.
Vertex enumeration problem	Computational Geometry	The computational problem of listing all vertices of a polyhedral or discrete-geometric object from an implicit constraint representation.
Vicious circle principle	Foundations of MathematicsPredicative Definition	A predicativist restriction forbidding definition of an entity by quantification over a totality that already contains the entity being defined.
Virtual fundamental class	Enumerative GeometryModuli and Obstruction Theory	Replace the missing ordinary fundamental class of an obstructed moduli space with a cycle of expected dimension derived from a perfect obstruction theory, enabling deformation-invariant enumerative integrals.
Virtual knot	Knot Theory	An equivalence class of knot diagrams with classical and virtual crossings under classical Reidemeister moves and virtual detour moves, equivalently knots in thickened surfaces up to stabilization.
Virtual Valuation	Game Theory	An auction-theory transform that converts an agent value and its prior distribution into marginal revenue, enabling expected-revenue analysis through virtual surplus under stated incentive and regularity conditions.
Visibility (geometry)	Computational Geometry	A geometric relation in which two points see one another when the line segment joining them remains inside free space and avoids declared obstacles.
Vitali covering lemma	Measure Theory	A geometric selection lemma extracting pairwise disjoint balls from a family so that a fixed enlargement of the selected balls covers the original union or set of centers.
Vitali set	Measure Theory and Set Theory	A choice-dependent subset containing one representative from each rational-translation equivalence class in an interval, yielding a canonical example of a non-Lebesgue-measurable set.
Vogel Plane	Lie TheoryRepresentation Theory	A projective parameter space that represents a simple Lie algebra by three Casimir eigenvalues on the nontrivial components of its symmetric square, modulo scaling and permutation.
Volume Element	Differential Geometry	A local top-dimensional density or form that converts coordinate cells into invariant geometric volume, transforming by a Jacobian and arising from coordinate, metric, Gram-determinant, or orientation data.
Volunteer's Dilemma	Game TheoryCoordination Games and Public Goods	A game where a shared good is produced if any single member pays a private cost to provide it, so each prefers someone else volunteer — and, counterintuitively, larger groups grow no more (often less) likely to produce it.
Von Neumann algebra	Functional AnalysisOperator Algebras	A unital star-algebra of bounded operators on a Hilbert space closed in the weak operator topology, equivalently equal to its double commutant.
Von Neumann Paradox	Paradoxical Decompositions	A planar paradoxical equidecomposition in which finitely many nonmeasurable pieces are rearranged by area-preserving affine transformations to duplicate a bounded figure or equidecompose bounded sets with nonempty interior.
Von Neumann stability analysis	Numerical Pde	A Fourier-mode method for testing linear finite-difference schemes by requiring their amplification factors not to grow beyond the stability bound.
Von Neumann's Closed-Operator Theorem	Functional AnalysisUnbounded Operator Theory	A closed densely defined Hilbert-space operator has a densely defined positive self-adjoint adjoint-product on its correctly restricted composition domain.
Von Neumann–Bernays–Gödel set theory	Foundations of Mathematics	A finitely axiomatizable two-sorted set theory with sets and classes that conservatively extends ZFC for statements about sets.
W-algebra	Conformal Field TheoryChiral and Vertex Algebras	Extend the Virasoro chiral symmetry algebra by finitely many generating fields of additional conformal weights whose modes close through generally nonlinear operator-product or commutation relations.
W-curve	Projective Differential GeometryHomogeneous Curves	A curve in projective space invariant under a one-parameter subgroup of projective transformations, so its entire path is an orbit and its projective differential invariants remain constant.
Wagstaff Prime	Number Theory	A prime number of the form (2^p + 1)/3 for an odd prime exponent p.
Waldhausen category	Algebraic K TheoryCategories with Cofibrations	A category equipped with designated cofibrations and weak equivalences satisfying gluing axioms so its algebraic K-theory spectrum can be constructed by the S-construction.
Wallman compactification	General TopologyCompactifications from Closed Set Lattices	Embed a T1 space densely into a compact space whose points are maximal centered families of closed sets and whose closed subbasis records membership in each original closed set.
Wall–Sun–Sun prime	Number TheoryFibonacci Congruences	A conjectural prime p for which p² divides the Fibonacci number indexed by p's Pisano period, equivalently a Fibonacci–Wieferich prime under standard formulations.
Wandering set	Ergodic TheoryDynamical Systems	A measurable set in a dynamical system whose distinct nonidentity translates are pairwise disjoint up to measure zero, thereby witnessing dissipative rather than recurrent behavior.
Wang tile	Tiling TheoryMathematical Logic	Wang tiles (or Wang dominoes), first proposed by mathematician, logician, and philosopher Hao Wang in 1961, is a class of formal systems.
Waraszkiewicz spiral	Continuum Theory	A member of an uncountable family of planar continua constructed so distinct members are incomparable under continuous surjections.
Wave Equation	Hyperbolic Partial Differential Equations	A second-order hyperbolic field equation that equates temporal acceleration with spatial curvature, encoding finite-speed propagation, traveling and standing modes, energy transport, and initial-boundary-value dynamics.
Wave function renormalization	Quantum Field Theory	The rescaling of a quantum field that normalizes its propagator residue and absorbs interaction-dependent field-strength corrections.
Wavelet	Harmonic Analysis	A localized, oscillating template function whose scaled and translated copies form a basis that resolves a signal simultaneously in position and scale, producing a sparse two-dimensional coefficient plane where features localize in both axes at once — the precondition being a sampled signal with scale-localized, transient structure.
Weak derivative	Functional Analysis	A weak derivative extends differentiation to integrable functions by transferring the derivative to test functions through integration by parts.
Weak dimension	Homological Algebra	The least upper bound of the flat dimensions of all modules over a ring, measuring how far the ring is from making every module flat.
Weak n-category	Higher Category TheoryWeak Higher Categories	Organize cells through dimension n with composition and units that satisfy associativity and unitality up to coherently related higher cells rather than by strict equality.
Weak Trace-Class Operator	Operator Ideals	A compact Hilbert-space operator whose singular values decay at least at harmonic order, placing it in the weak Schatten ideal where ordinary trace summability can fail but singular traces become available.
Weakly chained diagonally dominant matrix	Matrix Analysis	A weakly diagonally dominant matrix in which every non-strict row can reach a strictly dominant row through a directed chain of nonzero off-diagonal entries.
Weakly Inaccessible Cardinal	Set TheoryLarge Cardinals	An uncountable regular limit cardinal: a cardinal greater than aleph-zero whose cofinality equals itself and which is not the successor of any smaller cardinal, without requiring the strong-limit property.
Weakly measurable function	Functional AnalysisVector Valued Measure and Integration	Call a Banach-space-valued function weakly measurable when every scalarization by a continuous linear functional is measurable, and use essential separable-valuedness to determine when this implies strong measurability.
Weakly o-minimal structure	Model TheoryO Minimality	In model theory, a weakly o-minimal structure is a model-theoretic structure whose definable sets in the domain are just finite unions of convex sets.
Weakly symmetric space	Differential Geometry	A complete Riemannian homogeneous space in which an isometry can exchange any chosen pair of points.
Webbed space	Functional Analysis	A topological vector space equipped with a web structure that supports generalized closed-graph and open-mapping theorems.
Weierstrass function	Real Analysis	A classical infinite trigonometric series that is continuous everywhere and differentiable nowhere under suitable parameters.
Weierstrass M-Test	Real AnalysisSeries Convergence	In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely.
Weierstrass–Mandelbrot function	Fractal Modeling	A multiscale fractal function formed by summing frequency-scaled oscillatory components to model rough self-affine surfaces and signals.
Weight function	Mathematical Analysis	A function assigning unequal influence to elements in a sum, integral, average, norm or approximation.
Weinstein Conjecture	Contact Geometry	Every Reeb vector field determined by a contact form on a closed contact manifold has at least one closed periodic orbit, a theorem in dimension three and an open assertion in full higher-dimensional generality.
Weitzenböck identity	Differential Geometry	An identity expressing one Laplace-type operator as a rough Laplacian plus a curvature-dependent lower-order term.
Well-colored graph	Graph Coloring	A graph for which greedy vertex coloring uses the chromatic number of colors under every vertex ordering.
Well-Formed Formula	—	In mathematical logic, propositional logic, and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet, constructed following the defined grammar of a formal language.
Well-founded set	Set TheoryOrder Theory	A binary relation on a set or class for which every nonempty subset has a minimal element, equivalently under suitable choice principles one admitting no infinite descending chain.
Well-Pointed Category	Category TheoryTopos Theory	In category theory, a category with a terminal object 1 is well-pointed if for every pair of arrows f,g:A\to B such that f\neq g , there is an arrow p:1\to A such that f\circ p\neq g\circ p .
Well-posed problem	Hadamard Well Posedness	Classify a mathematical problem as well posed relative to declared data and solution spaces when a solution exists, is unique, and depends continuously on the data.
Well-quasi-ordering	Order TheoryWell Quasi Orders	A quasi-order in which every infinite sequence contains an earlier element below a later one, equivalently having neither infinite descending chains nor infinite antichains.
Weyl Algebra	Noncommutative AlgebraD Module Theory	The noncommutative algebra of polynomial-coordinate and derivative generators satisfying canonical commutator relations.
Weyl law	Spectral Geometry	An asymptotic formula linking the high-eigenvalue counting function of a Laplace-type operator to geometric volume and dimension.
Weyl's Theorem on Complete Reducibility	Lie Algebra Representations	Every finite-dimensional module over a finite-dimensional semisimple Lie algebra in characteristic zero splits into irreducible submodules.
Weyr canonical form	Linear AlgebraCanonical Matrix Forms	A canonical matrix form obtained by regrouping Jordan chains by level, yielding a block structure particularly suited to describing matrices that commute with a given operator.
Whitehead problem	Set Theoretic Algebra	The question whether every abelian group A with Ext-one of A and the integers equal to zero must be free, a statement independent of ZFC.
Whitehead Theorem	Algebraic Topology	A weak homotopy equivalence between CW complexes is a homotopy equivalence, so component and homotopy-group data detect the full homotopy type inside the CW setting.
Whitehead's point-free geometry	Point Free GeometryMereotopology	A region-primitive geometry that axiomatizes inclusion or connection among extended areas and derives point-like locations and topology rather than assuming points as basic entities.
Whitney Sum	Vector Bundle TheoryDifferential Topology	The vector bundle over a shared base whose fiber at each point is the direct sum of the corresponding fibers of two input bundles.
Width of a hypergraph	Hypergraph TheoryGraph Theory	In graph theory, there are two related properties of a hypergraph that are called its "width".
Wigner semicircle distribution	Random Matrix Theory	A compactly supported probability distribution whose density is proportional to a semicircle and which governs limiting eigenvalue spectra of many random symmetric matrices.
Wilf Equivalence	—	Two permutation classes are Wilf equivalent when they contain the same number of permutations at every length, equivalently when their ordinary generating functions coincide.
Wilkie's theorem	Model TheoryO Minimality	In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties.
Wilson quotient	Number Theory	For a prime p, the integer ((p−1)!+1)/p, whose residues encode refinements of Wilson's theorem and define Wilson primes when divisible by p.
Wine/water mixing problem	Recreational MathematicsConservation Puzzles	A conservation puzzle showing that after equal-volume containers exchange material and return to equal volume, the amount of wine in the water container equals the amount of water in the wine container, regardless of stirring or transfer size.
Winged Edge	Computational GeometryPolygon Mesh and Boundary Representation	An edge-centered polygon-mesh representation whose endpoint, incident-face, and four neighboring-edge references make local vertex and face incidence directly navigable.
Wirtinger Derivatives	Complex Analysis	Decompose the real differential of a function on complex coordinates into formal derivatives with respect to each variable and its conjugate, exposing holomorphic and antiholomorphic change without pretending nonholomorphic functions are complex-differentiable.
Wittgenstein's Rod	Geometric Loci	A geometric locus construction in which one endpoint of a fixed-length directed segment moves on a circle while the segment remains aligned with a fixed external point, producing a parameter-dependent figure-eight or degenerate one-lobed curve.
Woodall number	Number Theory	A natural number of the form n times two to the n minus one.
Word metric	Geometric Group Theory	A left-invariant distance on a generated group equal to the shortest length of a generator word representing one element's difference from another.
Word problem for groups	Combinatorial Group TheoryAlgorithmic Group Problems	Decide whether two finite words in a group's generators represent the same element, equivalently whether their quotient word represents the identity; finitely presented groups can make this problem undecidable.
X-Ray Transform	Integral Geometry	Map a spatial function or field to its integrals over every line, producing projection data from which the source may be reconstructed under stated geometric and analytic conditions.
XYZ inequality	Order Theory	A correlation inequality constraining relative ordering probabilities for three incomparable elements in a finite partially ordered set.
Yang–Mills flow	Differential Geometry	The negative gradient flow of the Yang–Mills energy on connections, evolving curvature toward Yang–Mills critical connections.
Yau's conjecture	Differential Geometry	The statement that every closed Riemannian three-manifold contains infinitely many smooth closed immersed minimal surfaces, now a theorem.
Yau's conjecture on the first eigenvalue	Differential Geometry	The conjecture that every closed embedded minimal hypersurface of the unit sphere S to the n plus one has first Laplace–Beltrami eigenvalue n.
Yetter–Drinfeld category	Hopf AlgebraBraided Categories	The braided monoidal category of modules and comodules over a Hopf algebra whose action and coaction satisfy the Yetter–Drinfeld compatibility condition.
Yoneda Extension	Category TheoryPresheaves	For a functor from a small category into a cocomplete category, the essentially unique colimit-preserving functor on the presheaf category whose restriction along the Yoneda embedding recovers the original functor.
Z-matrix (mathematics)	Linear Algebra	A real square matrix whose every off-diagonal entry is nonpositive.
Zak transform	Harmonic Analysis	A quasi-periodic time–frequency transform that maps a function on the real line to a function on a two-dimensional fundamental cell indexed by position and frequency phase.
Zappa–Szép product	Group Theory	A group factorization in which every element has a unique product from two subgroups, with each subgroup acting on the other rather than either necessarily being normal.
Zariski Tangent Space	Algebraic GeometryScheme Theory	The residue-field dual of a point's local maximal ideal modulo its square, recording first-order directions of an algebraic variety or scheme.
Zermelo set theory	Foundations of Mathematics	The original axiomatic set theory built from extensionality, elementary sets, separation, power set, union, choice, and infinity without the later replacement axiom.
Zero Divisor	Ring TheoryAbstract Algebra	In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective.
Zero-Divisor Graph	Commutative Ring TheoryAlgebraic Graph Theory	A zero-divisor graph connects distinct nonzero zero divisors of a commutative ring when their product is zero, exposing annihilation structure while forgetting other ring data.
Zero-Sum Problem	Zero Sum Theory in Finite Abelian Groups	Determine how long a sequence over a finite abelian group must be before a subsequence with zero group-sum and prescribed length properties is unavoidable.
Zero-symmetric graph	Graph Theory	A connected cubic graph whose automorphism group acts regularly on vertices but is not transitive on edges.
Zeros and Poles	Complex AnalysisMeromorphic Functions	This induces a duality between zeros and poles, that is fundamental for the study of meromorphic functions.
Zeta distribution	Probability TheoryDiscrete Power Law Distributions	A discrete power-law distribution on positive integers with probability proportional to k^−s and normalized by the Riemann zeta function for s>1.
Zeta Function	Mathematical AnalysisAnalytic Number TheorySpectral Geometry	A zeta function is a complex-valued function, typically defined initially by a Dirichlet series, Euler product, determinant, or trace-derived generating expression, that packages an indexed family of arithmetic, geometric, spectral, topological, or dynamical invariants and admits analytic study or continuation.
Zeta function regularization	Spectral Analysis	An analytic-continuation method that assigns finite determinants, products, or sums to divergent spectral expressions through an associated zeta function.
Zig-zag lemma	Homological Algebra	The homological-algebra result that a short exact sequence of chain complexes induces a natural long exact sequence in homology.
Ziggurat Algorithm	Random Variate GenerationRejection Sampling	A table-driven rejection sampler that partitions a monotone density into equal-area horizontal layers, making most draws a fast interior test while routing overhang and tail cases to exact fallback tests.
Zolotarev polynomials	Approximation TheoryExtremal Polynomials	Extremal polynomials with prescribed leading coefficients that minimize uniform deviation on an interval, generalizing Chebyshev polynomials in approximation theory.
Zugzwang	Endgame Theory	A turn-based game condition in which the obligation to move worsens a player's position because every legal move is worse than a pass or an alternate-turn counterfactual.
Étale Algebra	Commutative Algebra and Galois Theory	A finite-dimensional commutative algebra over a field that is a finite product of finite separable field extensions, and therefore becomes a finite product of copies of the base after separable closure.
Étale morphism	Algebraic GeometryScheme Theory	An étale morphism is a scheme morphism that is smooth of relative dimension zero, equivalently locally of finite presentation, flat, and unramified, providing the algebraic analogue of a local isomorphism.
Γ-convergence	Calculus of Variations	A variational convergence of functionals defined by a liminf inequality and recovery sequences, designed so minimizers and minimum values behave stably under suitable compactness.
Γ-space	Selection Principles in Topology	A topological space in which every open omega-cover contains a gamma-cover whose members contain each point all but finitely often.
Η set	Set Theoretic Order TheorySaturated Linear Orders	An eta-alpha set is a dense linear order in which every two less-than-aleph-alpha-sized subsets separated left from right have an interpolating element.
Σ-Algebra of τ-past	Probability TheoryStochastic Processes	The stopped sigma-algebra Fτ containing exactly those events whose truth is knowable by a stopping time τ, defined by compatibility of each event with {τ≤t} and the filtration Ft.
Ω-complete theory	Mathematical Logic	A first-order arithmetic theory that proves a universal formula whenever it proves every numeral instance of that formula.
Ω-consistent theory	Mathematical Logic	A consistent arithmetic theory that never proves every standard numeral instance of a formula while also proving that some natural number is a counterexample.
Ω-logic	Set Theory	An infinitary set-theoretic deductive system whose validity is defined through universally Baire sets and generic extensions under large-cardinal assumptions.
∞-Chern–Simons theory	Higher Geometry	A higher-categorical generalization of Chern–Simons gauge theory formulated with higher bundles, connections and characteristic maps in a cohesive infinity-topos.
∞-Chern–Weil theory	Higher Differential Geometry	A higher-geometric extension of Chern–Weil theory that constructs differential characteristic classes and cocycles for higher principal bundles and infinity-groupoids.