Algebraic Varieties & Topological Invariants

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Abstractions about specific invariants and spaces in algebraic geometry and topology, including varieties and their invariants such as secant variety, degree of a variety and J-multiplicity, cohomology theories and spectral sequences like equivariant cohomology and the Adams spectral sequence, and classifying spaces such as fiber bundles and Stone spaces.

27 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Adams Spectral Sequence — 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.
• Algebraic Surface — A dimension-two algebraic variety over a specified field, defined by polynomial data and studied with explicit affine/projective, singularity, and birational conventions.
• Behrend function — An intrinsic integer-valued constructible weight on a complex scheme whose Euler integral can recover suitable virtual counts.
• Completely Uniformizable Space — A topological space whose topology is induced by at least one complete uniformity, also called Dieudonné complete under conventions that may additionally require Hausdorffness.
• Convex body — A compact convex subset of finite-dimensional Euclidean space with nonempty ambient interior under the standard convention.
• Degree of an algebraic variety — The multiplicity-counted number of intersections between an embedded algebraic variety and a general complementary-dimensional linear subspace.
• Equivariant cohomology — Cohomology of a group's homotopy quotient, recording topology together with an action on a space.
• Euclidean Neighborhood Retract — 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.
• Fiber Bundle — 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.
• Finiteness Properties of Groups — 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.
• J-multiplicity — A local-algebra multiplicity for an ideal obtained from maximal-ideal-supported graded data, extending Hilbert–Samuel multiplicity beyond m-primary ideals.
• Jet Group — The group of fixed-origin invertible Taylor germs truncated at order k, with group law induced by composition of local coordinate changes.
• Microdifferential operator — 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.
• Neutral Geometry — An axiomatic geometry retaining incidence, betweenness, and congruence while withholding any parallel postulate, so its theorems survive in both Euclidean and hyperbolic extensions.
• Polynomially Reflexive Space — 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.
• Rational normal curve — The degree-n projective curve obtained by mapping a projective line to all degree-n monomials in two homogeneous coordinates.
• Rational Normal Scroll — An irreducible rational ruled projective surface swept by paired directrices and embedded with minimal degree.
• Ricci Decomposition — The metric-dependent orthogonal splitting of a Riemann curvature tensor into scalar-curvature, traceless-Ricci, and totally trace-free Weyl components.
• Secant Variety — 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.
• Selberg zeta function — A zeta function built from primitive closed geodesic lengths of a hyperbolic surface.
• Stanley's Reciprocity Theorem — A reciprocity identity turning a rational cone's lattice-point generating function under variable inversion into the signed generating function of its relative interior.
• Stone Space — The compact zero-dimensional Hausdorff space of ultrafilters of a Boolean algebra, with algebra elements represented by membership clopen sets.
• Stunted projective space — A projective-space quotient that collapses a lower skeleton and retains higher cells.
• Subpaving — Nonoverlapping boxes representing or approximating a region in interval analysis.
• Topological Galois Theory — 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.
• Urysohn's lemma — A normal-space theorem that separates any disjoint closed sets by a continuous function taking prescribed values 0 and 1 on them.
• Whitney Sum — The vector bundle over a shared base whose fiber at each point is the direct sum of the corresponding fibers of two input bundles.