Index
July 2026 operator algebra and geometry additions
Definitions added in the July 2026 expansion of operator algebras, noncompact representation theory, noncommutative geometry, and differential geometry.
Core idea
This permanent provenance and review index records the definitions added in the July 2026 operator-algebra and geometry expansion. Click any term to expand it in place.
Operator algebras
Operator and C*-algebra foundations
- C*-algebra
- C*-algebra C_0(X)
- C*-algebra of bounded operators
- C*-algebra of compact operators
- C*-identity
- C*-subalgebra
- Canonical operator trace
- Commutative C*-algebra
- Concrete C*-algebra
- Even and odd operators on a graded Hilbert space
- Generated C*-subalgebra
- Involutive algebra
- Matrix C*-algebra
- Opposite algebra
- Projection in a C*-algebra
- Unital C*-algebra
- Unitary element of a C*-algebra
Positivity and order
- Order on self-adjoint elements
- Positive and negative parts of a self-adjoint element
- Positive cone of a C*-algebra
- Positive element of a C*-algebra
- Positive square root in a C*-algebra
- Self-adjoint element of a C*-algebra
Maps and morphisms
- *-automorphism
- *-homomorphism
- *-isomorphism
- Completely bounded map
- Completely positive contraction
- Completely positive map
- Completely positive order-zero map
- Conditional expectation of C*-algebras
- Faithful *-homomorphism
- Kadison–Schwarz inequality
- n-positive map
- Nondegenerate *-homomorphism
- Positive linear map
- Stinespring dilation theorem
- Unital *-homomorphism
- Unital completely positive map
Ideals and quotients
- Closed two-sided ideal of a C*-algebra
- Corona C*-algebra
- Essential ideal of a C*-algebra
- Full hereditary C*-subalgebra
- Hereditary C*-subalgebra
- Hull–kernel topology
- Kernel of a *-homomorphism
- Multiplier algebra
- Primitive ideal of a C*-algebra
- Primitive ideal space
- Quotient C*-algebra
- Strict topology on a multiplier algebra
Approximate identities, unitizations, and multipliers
- Approximate identity in a C*-algebra
- Double centralizer of a C*-algebra
- Extension of a nondegenerate homomorphism to multiplier algebras
- Quasicentral approximate identity
- Strictly positive element of a C*-algebra
- Unitization of a C*-algebra
- σ-unital C*-algebra
Spectra and functional calculus
- Character space of a commutative C*-algebra
- Continuous functional calculus
- Gelfand duality
- Gelfand transform
- Spectral mapping theorem
- Spectral permanence
States, characters, and traces
- Cauchy–Schwarz inequality for a positive functional
- Character of a C*-algebra
- Faithful state on a C*-algebra
- Lower semicontinuous trace
- Positive linear functional
- Pure state of a C*-algebra
- Singular state on a von Neumann algebra
- State on a C*-algebra
- State space of a C*-algebra
- Trace on a C*-algebra
- Tracial state
Representations and enveloping von Neumann algebras
- Cyclic representation of a C*-algebra
- Direct sum of C*-representations
- Disjoint representations
- Factorial representation
- Faithful representation of a C*-algebra
- Gelfand–Naimark theorem
- GNS construction
- GNS uniqueness theorem
- Irreducible representation of a C*-algebra
- Nondegenerate representation of a C*-algebra
- Pure states and irreducible GNS representations
- Quasi-equivalent representations
- Representation of a C*-algebra
- States separate positive elements
- Unitary equivalence of C*-representations
- Universal representation of a C*-algebra
- Vector state
- Von Neumann algebra generated by a representation
Type I C*-algebras
- Elementary C*-algebra
- Glimm's type I theorem
- Liminal C*-algebra
- Liminal C*-algebras are type I
- Type I C*-algebra
- Type I composition series
- Type I factor representation
- Type I stability under ideals and quotients
Group C*-algebras
- Amenability and equality of full and reduced group C*-algebras
- Canonical quotient from full to reduced group C*-algebra
- Full group C*-algebra
- Group C*-algebras of a discrete group
- Nuclearity of reduced group C*-algebras and amenability for discrete groups
- Reduced group C*-algebra
- Unitary representations and representations of the full group C*-algebra
Crossed products
- Amenable action on a C*-algebra
- C*-dynamical system
- Canonical conditional expectation on a reduced crossed product
- Canonical map from full to reduced crossed product
- Covariant representation of a C*-dynamical system
- Crossed-product convolution *-algebra C_c(G,A)
- Dual action on a crossed product
- Fell bundle
- Full crossed product
- Integrated form of a covariant representation
- Reduced crossed product
- Regular covariant representation
- Takai duality
- Transformation-group C*-algebra
Universal C*-algebras
- C*-algebra by generators and relations
- Enveloping C*-algebra
- Maximal C*-seminorm
- Universal C*-algebra
Exactness and tensor products
- C*-tensor norm
- Completely positive approximation property
- Exact C*-algebra
- Local lifting property
- Local reflexivity for C*-algebras
- Maximal C*-tensor product
- Minimal C*-tensor product
- Nuclear C*-algebra
- Nuclear C*-algebras are exact
- Short exact sequence of C*-algebras
- Weak expectation property
Continuous fields and C*-bundles
- Continuous C_0(X)-algebra
- Continuous field of C*-algebras
- Continuous-trace C*-algebra
- C_0(X)-algebra
- C_0(X)-algebras and upper-semicontinuous C*-bundles
- Dixmier–Douady class
- Fiber of a C_0(X)-algebra
- Section algebra of a C*-bundle
- Upper-semicontinuous C*-bundle
Hilbert modules, correspondences, and Morita theory
- Adjointable operator on a Hilbert C*-module
- Brown-Green-Rieffel stabilization theorem
- C*-correspondence
- Cauchy-Schwarz inequality for Hilbert C*-modules
- Compact operator on a Hilbert C*-module
- Countably generated Hilbert C*-module
- Cuntz-Pimsner algebra
- Exterior tensor product of Hilbert C*-modules
- Fock module of a C*-correspondence
- Green imprimitivity theorem
- Hilbert C*-module
- Imprimitivity bimodule
- Internal tensor product of C*-correspondences
- Kasparov stabilization theorem
- Linking algebra
- Multiplier module
- Rieffel induction
- Standard Hilbert C*-module
- Strong Morita equivalence of C*-algebras
- Toeplitz representation of a C*-correspondence
- Toeplitz-Pimsner algebra
Von Neumann algebra foundations and topologies
- Bicommutant
- Commutant
- Sakai characterization theorem
- Strong operator topology
- Strong-* operator topology
- Ultrastrong operator topology
- Ultrastrong-star topology
- Ultraweak operator topology
- Von Neumann algebra
- Von Neumann bicommutant theorem
- Weak operator topology
Projections and comparison theory
- Abelian projection
- Central support of a projection
- Comparison theorem for projections
- Finite projection
- Infinite projection
- Minimal projection
- Murray–von Neumann equivalence of projections
- Murray–von Neumann subequivalence of projections
- Properly infinite projection
Maps, preduals, and normal states
- Faithful normal state
- Normal *-homomorphism
- Normal completely positive map
- Normal conditional expectation
- Normal functional
- Normal linear map
- Normal positive map
- Normal representation
- Normal state
- Normal unital completely positive map
- Positive cone of the predual
- Predual of a von Neumann algebra
- Support projection of a normal positive functional
- Unique predual theorem
- Vector functional on a von Neumann algebra
Weights, modular theory, and standard form
- Centralizer of a weight
- Connes cocycle derivative
- Cyclic vector
- Cyclic-separating duality
- Extended positive cone of a von Neumann algebra
- Faithful normal semifinite trace
- Faithful weight
- Finite normal trace on a von Neumann algebra
- GNS construction for a weight
- Left ideal of a weight
- Modular automorphism group
- Modular conjugation
- Modular KMS condition
- Modular operator
- Natural positive cone
- Normal semifinite faithful weight
- Normal weight
- Operator-valued weight
- Semifinite weight
- Separating vector
- Spatial derivative
- Standard form of a von Neumann algebra
- Tomita operator
- Tomita–Takesaki theorem
- Tracial weight
- Uniqueness of standard form
- Weight on a von Neumann algebra
Factors and type classification
- Center of a von Neumann algebra
- Central decomposition of a von Neumann algebra
- Commutative von Neumann algebra
- Connes classification of type III factors
- Dimension function on projections
- Finite von Neumann algebra
- Flow of weights
- Group von Neumann algebra
- Hyperfinite II₁ factor
- Properly infinite von Neumann algebra
- Semifinite von Neumann algebra
- Type decomposition of von Neumann algebras
- Type I factor
- Type I von Neumann algebra
- Type II von Neumann algebra
- Type III factor
- Type III von Neumann algebra
- Type II₁ factor
- Type II∞ factor
- Type Iₙ factor
- Type I∞ factor
- Von Neumann factor
Affiliated operators and noncommutative integration
- Breuer-Fredholm operator
- Dixmier trace
- Measurable operator affiliated with a von Neumann algebra
- Noncommutative L^p space
- Operator affiliated with a von Neumann algebra
- Spectral characterization of affiliation
- Tau-compact operator
- Tau-measurable operator
- Tracial noncommutative L^p space
- Weak Schatten ideal
K-theory and index-theoretic bridge
Functional analysis
Banach, Hilbert, and bounded-operator foundations
- Adjoint of a bounded operator
- Banach algebra
- Banach–Alaoglu theorem
- Bounded commutator
- Bounded linear operator between normed spaces
- Closed graph theorem
- Commutator of possibly unbounded operators
- Finite-rank operator
- Fredholm operator
- Open mapping theorem
- Partial isometry
- Reflexive Banach space
- Riesz representation theorem for Hilbert spaces
- Schatten-class operator
- Singular values of a compact operator
- Uniform boundedness principle
- Unitary operator
- Z/2-graded Hilbert space
Unbounded operators
- Adjoint of a densely defined operator
- Borel functional calculus for self-adjoint and normal operators
- Closable operator
- Closed linear operator
- Closed quadratic form
- Closure of a closable operator
- Core of a closed operator
- Deficiency indices
- Densely defined operator
- Essentially self-adjoint operator
- Friedrichs extension
- Graph norm
- Graph of a linear operator
- Operator with compact resolvent
- Point, continuous, and residual spectrum
- Projection-valued measure
- Resolvent identity
- Resolvent set of a closed operator
- Self-adjoint extension
- Self-adjoint unbounded operator
- Spectral integral
- Spectral projection of a self-adjoint or normal operator
- Spectral theorem for unbounded self-adjoint operators
- Spectrum of a closed operator
- Symmetric operator
Spectra and functional calculus
- Holomorphic functional calculus in a Banach algebra
- Invertible element in a Banach algebra
- Resolvent of an element in a Banach algebra
- Spectral radius formula for a Banach algebra
- Spectral radius in a Banach algebra
- Spectrum of an element in a Banach algebra
Locally convex spaces
- Barreled space
- Bornological space
- Bounded subset of a topological vector space
- Continuous linear map between topological vector spaces
- Equicontinuous family of continuous linear maps
- Fréchet space
- Inductive limit of locally convex spaces
- LF-space
- Locally convex space
- Montel space
- Nuclear space
- Projective limit of locally convex spaces
- Strong dual of a locally convex space
- Topological dual
- Topological vector space
- Topology generated by a family of seminorms
- Weak topology on a topological vector space
- Weak-star topology
Schwartz spaces and distributions
- Compactly supported distribution
- Convolution of distributions
- Distribution
- Distributional derivative
- Fourier inversion on Schwartz space
- Fourier transform of tempered distributions
- Fourier transform on Schwartz space
- Order of a distribution
- Regular tempered distribution
- Rigged Hilbert space
- Schwartz kernel theorem
- Schwartz space on Euclidean space
- Singular support of a distribution
- Sobolev space
- Support of a distribution
- Tempered distribution
- Test-function space
- Wavefront set
- Weak derivative
Direct integrals
- Decomposable operator
- Direct integral of Hilbert spaces
- Direct integral of operators
- Measurable field of Hilbert spaces
Harmonic analysis
Unitary-representation and character foundations
- Conjugate and contragredient unitary representations
- Continuous unitary character
- Hilbert direct sum of unitary representations
- Tensor product of unitary representations
Haar measure, convolution, and regular representations
- Approximate identity in L1(G)
- Convolution on a locally compact group
- Haar integral
- Haar measure
- Hecke algebra of a locally compact group pair
- Integrated form of a unitary representation
- Involution on a group convolution algebra
- L1 group algebra
- Left and right regular representations of a locally compact group
- Measure algebra of a locally compact group
- Modular function of a locally compact group
- Quasi-regular representation
- Unimodular locally compact group
- Weil integration formula
- Young's convolution inequality
Fourier transform and Plancherel theory
- Dual Haar measure
- Fourier convolution theorem on a locally compact abelian group
- Fourier inversion theorem for locally compact abelian groups
- Fourier transform on a locally compact abelian group
- Hausdorff–Young inequality on a locally compact abelian group
- Plancherel measure for a type I locally compact group
- Plancherel theorem for locally compact abelian groups
- Pontryagin dual
- Pontryagin duality theorem
Positive-definite functions and duality
- Bochner's theorem for locally compact abelian groups
- Fell topology on the unitary dual
- Fourier algebra of a locally compact group
- Fourier–Stieltjes algebra
- GNS construction for a positive-definite function
- Kernel description of Fell topology
- Matrix coefficient of a unitary representation
- Positive-definite function on a group
- Unitary dual of a locally compact group
Amenability
- Amenability via weak containment of the trivial representation
- Amenable locally compact group
- Følner condition
- Invariant mean
- Reiter condition
- Weak containment of unitary representations
Induction and imprimitivity
- Induced unitary representation
- Induction in stages for unitary representations
- Mackey imprimitivity theorem
- Mackey machine for semidirect products
- Quasi-invariant measure under a locally compact group action
- System of imprimitivity
- Unitary Frobenius reciprocity
Direct integrals and nonabelian Plancherel theory
- Direct integral of unitary representations
- Discrete and continuous spectrum of a unitary representation
- Multiplicity function of a representation
- Nonabelian group Fourier transform
- Plancherel theorem for unimodular type I groups
- Tempered dual of a locally compact group
- Tempered unitary representation
Bruhat and Harish-Chandra analysis
- Harish-Chandra Schwartz algebra
- Harish-Chandra Schwartz space
- Harish-Chandra Xi-function
- Restricted tensor product of local test-function spaces
- Schwartz–Bruhat space on a locally compact abelian group
- Schwartz–Bruhat space on the adeles
- Schwartz–Bruhat space over a local field
- Spherical Fourier transform
- Spherical function on a Gelfand pair
Noncommutative geometry
Smooth-algebra bridge
Spectral triples
- Almost-commutative spectral triple
- Canonical spectral triple of a compact spin manifold
- Dirac operator
- Dirac-type operator
- Even spectral triple
- Nonunital spectral triple
- Odd spectral triple
- Product of spectral triples
- Spectral triple
- Unitary equivalence of spectral triples
Real structures
- First-order condition for a real spectral triple
- KO-dimension sign table for a real spectral triple
- Order-zero condition for a real spectral triple
- Real spectral triple
- Real structure on a spectral triple
Regularity, summability, and dimension spectrum
- Derivation delta(T)=[|D|,T]
- Dimension spectrum of a spectral triple
- Finitely summable spectral triple
- Metric dimension of a spectral triple
- Noncommutative integral of a spectral triple
- p-summable spectral triple
- Regular spectral triple
- Smooth domain of the Dirac derivation
- Spectral zeta function of a spectral triple
- Theta-summable spectral triple
Metric noncommutative geometry
K-homology and index pairing
- Analytic K-homology
- Bounded transform of a spectral triple
- Even Fredholm module
- Even K-theory/K-homology index pairing
- Fredholm module
- Odd Fredholm module
- Odd K-theory/K-homology index pairing
- Spectral flow
- Unbounded Kasparov module
Cyclic homology and local index machinery
- Chern character of a Fredholm module
- Cyclic cohomology
- Hochschild chain complex
- Hochschild cycle
- Jaffe-Lesniewski-Osterwalder cocycle
- Local index formula in noncommutative geometry
Reconstruction and orientability
- Commutative reconstruction theorem for spectral triples
- Finiteness and absolute-continuity axiom for a spectral triple
- Hochschild orientability axiom for a spectral triple
Semifinite noncommutative geometry
- Semifinite K-theory index pairing
- Semifinite local index formula
- Semifinite spectral triple
- Tau-summable semifinite spectral triple
Lie groups and noncompact representation theory
Continuous and unitary representations
- Cyclic vector and cyclic unitary representation
- Intertwining operator between unitary representations
- Invariant closed subspace of a unitary representation
- Irreducible unitary representation
- Projective unitary representation
- Schur's lemma for unitary representations
- Strongly continuous unitary representation
- Unitary equivalence of representations
- Weak and strong continuity for unitary representations
- Weakly continuous unitary representation
Smooth vectors and derived representations
- Analytic vector of a Lie-group representation
- Derived representation on smooth vectors
- Distribution vectors of a Lie-group representation
- Dixmier–Malliavin factorization theorem
- Essential skew-adjointness of derived representation operators
- Gårding density theorem
- Gårding subspace
- Infinitesimal generator of a unitary representation
- Nelson Laplacian
- Nelson's analytic vector theorem
- Poincaré–Birkhoff–Witt theorem
- Smooth vector of a Lie-group representation
- Stone's theorem for one-parameter unitary groups
- Universal enveloping algebra action on smooth vectors
- Universal enveloping algebra of a Lie algebra
Admissibility and Harish-Chandra modules
- (g,K)-module
- Admissible representation of a real reductive group
- Casselman–Wallach globalization theorem
- Casselman–Wallach representation
- Harish–Chandra admissibility theorem
- Harish–Chandra module
- Harish–Chandra module of K-finite vectors
- Infinitesimal equivalence of admissible representations
- K-finite vector
- K-type
Real reductive structure
- Cartan decomposition of a real reductive Lie algebra
- Cartan involution of a real Lie algebra
- Furstenberg boundary of a semisimple Lie group
- Global Cartan decomposition of a real reductive group
- Iwasawa decomposition
- Langlands decomposition of a parabolic subgroup
- Maximal compact subgroup of a real reductive group
- Minimal parabolic subgroup
- Parabolic subgroup of a real reductive group
- Real rank of a reductive group
- Real reductive Lie group
- Restricted root system
- Restricted Weyl group
- Riemannian symmetric space of noncompact type
Noncompact representation series
- Casselman subrepresentation theorem
- Complementary series representation
- Discrete series representation
- Harish–Chandra's discrete-series criterion
- Knapp–Stein intertwining operator
- Langlands classification for real reductive groups
- Langlands quotient
- Limit of discrete series representation
- Principal series representation
- Spherical principal series representation
- Square-integrable representation modulo the center
- Standard module of a real reductive group
Infinitesimal characters and distribution characters
- Casimir element and Casimir operator
- Center of the universal enveloping algebra
- Global character of an admissible representation
- Harish–Chandra isomorphism
- Harish–Chandra regularity theorem
- Infinitesimal character
Induction, direct integrals, and Plancherel theory
Spherical representation theory
Orbit method
- Induced representation attached to a polarization
- Kirillov correspondence for nilpotent Lie groups
- Orbit method
- Polarization at a coadjoint functional
Examples and model classifications
Quaternionic Lie groups
Differential geometry
Smooth manifolds, maps, and calculus
- Algebra of smooth functions
- Ambient isotopy
- Category of smooth manifolds
- Constant rank theorem
- Critical point of a smooth map
- Critical value of a smooth map
- Diffeomorphism group
- Disjoint union of smooth manifolds
- Inverse function theorem for manifolds
- Isotopy extension theorem
- Iterated tangent bundle
- Jet of a smooth map
- Mapping class group of a smooth manifold
- Product manifold
- Proper smooth map
- Rank of a smooth map
- Sard's theorem
- Smooth homotopy
- Smooth isotopy
- Space of smooth maps
- Tangent functor
- Whitney C-infinity topology
Submanifolds and transversality
- Clean intersection
- Diagonal transversality criterion
- Embedded submanifold
- Immersed submanifold
- Map transverse to a submanifold
- Neat embedding
- Neat submanifold
- Normal bundle
- Parametric transversality theorem
- Proper injective immersion is an embedding
- Regular level set
- Smooth fiber product
- Submanifold chart
- Thom transversality theorem
- Transverse intersection theorem
- Transverse preimage theorem
- Transverse smooth maps
- Transverse submanifolds
- Tubular neighborhood
- Tubular neighborhood theorem
- Whitney embedding theorem
Boundaries, corners, and cobordisms
- Bordism group
- Boundary and interior of a manifold with boundary
- Boundary chart
- Boundary defining function
- Boundary face of a manifold with corners
- Cobordism
- Cobordism category
- Cobordism equivalence relation
- Collar neighborhood theorem
- Corner stratification
- Depth of a point in a manifold with corners
- Double of a manifold with boundary
- Oriented cobordism
- Smooth manifold with boundary
- Smooth manifold with corners
- Smooth map between manifolds with boundary
- Smooth map between manifolds with corners
- Symmetric monoidal structure on the cobordism category
Partitions of unity, sheaves, and differential forms
- Boundary orientation
- Bump function
- Compactly supported de Rham cohomology
- Compactly supported differential form
- de Rham complex
- de Rham homotopy operator
- de Rham theorem
- Degree of a smooth map
- Fine sheaf
- Homotopy invariance of de Rham cohomology
- Integration of differential forms
- Mayer-Vietoris sequence for de Rham cohomology
- Orientability of a smooth manifold
- Orientation of a smooth manifold
- Poincaré lemma
- Sheaf of smooth functions
- Smooth exhaustion function
- Smooth Urysohn lemma
Complex manifolds and holomorphic geometry
- Biholomorphism
- Canonical bundle of a complex manifold
- Complex atlas
- Complex coordinate chart
- Complex submanifold
- Differential form of type (p,q)
- Dolbeault cohomology
- Dolbeault complex
- Dolbeault operators
- Holomorphic cotangent bundle
- Holomorphic line bundle
- Holomorphic section
- Holomorphic tangent bundle
- Holomorphic vector bundle
- Holomorphic vector-bundle morphism
- Riemann surface
- Sheaf of holomorphic functions
- Type decomposition of the complexified tangent bundle
Hermitian and Kähler geometry
- Almost-Hermitian manifold
- Complex torus
- Fubini–Study metric
- Fundamental 2-form of an almost-Hermitian manifold
- Hermitian manifold
- Kähler class
- Kähler form
- Kähler manifold
- Kähler metric
- Local Kähler potential
- Ricci form
- The d-c operator
- The ∂∂̄-lemma
Hodge theory
- Codifferential
- Einstein manifold
- Formal adjoint of a differential operator
- Frölicher spectral sequence
- Hard Lefschetz theorem
- Harmonic differential form
- Hodge decomposition of a compact Kähler manifold
- Hodge decomposition on a compact Riemannian manifold
- Hodge filtration
- Hodge Laplacian
- Hodge numbers
- Hodge theorem
- Hodge–Riemann bilinear relations
- Kähler identities
- Lefschetz operator
- Polarized Hodge structure
- Primitive cohomology
- Pure Hodge structure
- Ricci curvature
- Riemann curvature tensor
- Variation of Hodge structure
Special holonomy, quaternionic, and hyperkähler geometry
- Almost-quaternionic manifold
- Calabi–Yau manifold
- Calabi–Yau theorem
- Holomorphic symplectic form
- Holomorphic symplectic manifold
- Hyper-Hermitian manifold
- Hypercomplex manifold
- Hyperkähler manifold
- Irreducible holomorphic symplectic manifold
- Obata connection
- Quaternion-Kähler manifold
- Quaternion-Kähler manifolds are Einstein
- Quaternionic manifold
- Quaternionic-Hermitian manifold
- Twistor space of a quaternionic manifold
Grassmannians and linear symplectic geometry
- Coisotropic subspace
- Compatible almost-complex structure
- Grassmannian
- Isotropic Grassmannian
- Isotropic subspace
- Lagrangian Grassmannian
- Lagrangian submanifold
- Lagrangian subspace
- Linear coisotropic reduction
- Linear symplectomorphism
- Oriented Grassmannian
- Symplectic basis
- Symplectic basis theorem
- Symplectic linear map
- Symplectic orthogonal complement
- Symplectic subspace
- Symplectic vector space
- Symplectomorphism
- Tangent space of a Grassmannian
- Tautological bundle on a Grassmannian
Symplectic manifolds and symplectomorphisms
- Almost symplectic manifold
- Canonical symplectic form on a cotangent bundle
- Characteristic distribution of a coisotropic submanifold
- Characteristic foliation
- Coisotropic submanifold
- Conormal bundle is Lagrangian
- Cotangent lift
- Darboux theorem for symplectic manifolds
- Exact symplectic manifold
- Isotropic submanifold
- Lagrangian neighborhood theorem
- Liouville vector field
- Moser stability theorem
- Symplectic embedding
- Symplectic map
- Symplectic submanifold
- Tautological one-form on a cotangent bundle
Hamiltonian mechanics
- Action-angle coordinates
- Classical phase space
- Completely integrable Hamiltonian system
- First integral
- Functions in involution
- Hamilton's equations
- Hamiltonian diffeomorphism
- Hamiltonian flow
- Hamiltonian function
- Hamiltonian system
- Hamiltonian vector field
- Liouville–Arnold theorem
- Poisson algebra of smooth functions
- Poisson bracket on a symplectic manifold
- Symplectic vector field
- Time-dependent Hamiltonian system
Hamiltonian actions and reduction
- Atiyah–Guillemin–Sternberg convexity theorem
- Canonical moment map for a cotangent-lifted action
- Coadjoint orbit
- Coisotropic reduction
- Comoment map
- Equivariant moment map
- Hamiltonian Lie group action
- Hamiltonian torus action
- Infinitesimal generator of a Lie group action
- Kirillov–Kostant–Souriau symplectic form
- Marsden–Weinstein–Meyer reduction theorem
- Moment maps as Poisson maps
- Moment polytope
- Moment-map component
- Shifting trick
- Singular symplectic reduction
- Symplectic Lie group action
- Symplectic quotient
- Symplectic reduction at nonzero momentum
- Uniqueness and ambiguity of moment maps
Poisson geometry
Maslov theory
- Lagrangian Gauss map
- Maslov class of a Lagrangian submanifold
- Maslov class of the Lagrangian Grassmannian
- Maslov cycle
- Maslov index
- Relative Maslov index
Spectral and gauge-analytic geometry
- Clifford algebra
- Clifford module
- Differential operator between vector bundles
- Elliptic differential operator
- Hodge star operator
- Principal symbol of a differential operator
- Riemannian manifold
- Spinor bundle
Fiber bundles and gauge theory
Vector bundles, sections, and bundle operations
- Compactly supported section
- Kernel and image bundles of a constant-rank morphism
- Line bundle
- Module of smooth sections
- Nowhere-vanishing section
- Quotient vector bundle
- Section of a fiber bundle
- Section of a vector bundle
- Short exact sequence of vector bundles
- Sphere bundle
- Splitting theorem for vector bundles
- Support of a section
- Vector subbundle
- Zero section
Connections and curvature
- Chern connection
- Connection on a Hom bundle
- Connection on exterior and symmetric powers
- Dual connection
- Hermitian connection
- Pullback connection on a vector bundle
- Tensor product connection
- Vector-bundle-valued differential form
Reductions and geometric structures
- Connection compatible with a reduction
- Curvature of the Chern connection
- G-structure
- Intrinsic torsion of a G-structure
- Quaternionic vector bundle
- Spin connection
- Spin structure
Characteristic classes
- Characteristic number
- Chern number
- Chern roots
- Chern-Gauss-Bonnet theorem
- Chern-Weil homomorphism
- Euler class as pullback of the Thom class
- Euler class obstruction to a nowhere-zero section
- Invariant polynomial on a Lie algebra
- Pontryagin number
- Splitting principle
- Stiefel-Whitney class
- Thom class
- Thom isomorphism theorem
- Universal characteristic class
- Whitney sum formula
Serre-Swan duality
- Reconstructing a vector bundle from a projective module
- Section module is finitely generated projective
- Serre–Swan theorem
- Vector bundle from an idempotent matrix
Gauge actions and quotients
- Based gauge group
- Coulomb gauge
- Gauge theory
- Gauge-fixing condition
- Irreducible connection
- Moduli stack of connections
- Reducible connection
- Slice theorem for the gauge action
- Sobolev completion of connections and gauge transformations
- Stabilizer of a connection
- Wilson loop
Analytic and complex gauge theory
- Covariant Hodge Laplacian
- Donaldson-Uhlenbeck-Yau theorem
- Formal adjoint of the covariant exterior derivative
- Hermitian Yang-Mills connection
- L2 inner product on bundle-valued forms
Yang-Mills theory and instantons
- Anti-self-dual moduli space
- ASD deformation complex
- Bogomolny monopole equation
- Instanton number
- Regular ASD connection
- Self-dual and anti-self-dual connection
- Uhlenbeck compactness theorem
- Uhlenbeck removable singularity theorem
- Yang-Mills energy identity
- Yang-Mills instanton
- Yang-Mills moduli space
Flat connections and moduli
- Atiyah-Bott symplectic form
- Character variety
- Curvature as a moment map
- Deformation complex of a flat connection
- Holonomy correspondence for flat connections
- Moduli space of flat connections
- Representation variety
- Tangent space to flat-connection moduli
Cross-cutting foundations: topology
Characteristic-class and gauge-moduli foundations
Cross-cutting foundations: linear algebra
Quaternionic and Hermitian linear algebra
Audit-driven foundational closure
These entries close prerequisite gaps found by the final semantic-dependency audit of the original expansion.
Operator-algebra closure
- Graded C*-algebra
- Order-unit space
- Bounded star-representation
- Regular operator on a Hilbert C*-module
- Entire analytic element for a one-parameter automorphism group
- Von Neumann crossed product
- Dual action on a von Neumann crossed product
- Continuous core of a von Neumann algebra
- Marcinkiewicz–Macaev ideal
Noncommutative-geometry closure
Functional- and harmonic-analysis closure
- Hilbert–Schmidt operator
- Locally compact homogeneous space
- Integrated operator of a continuous representation