Index
July 30 mathematical knowl expansion
Review index for 372 new and 64 substantively refined production knowls in the July 30 mathematical expansion, organized by durable subject and dependency order.
Core idea
This page is a review and navigation index for the July 30 mathematical expansion. It lists every current new production knowl exactly once, followed by the pre-existing knowls selected for substantive integration review. Purely mechanical reference-placement updates elsewhere in the corpus are not repeated in that second list. The lists follow durable mathematical subjects and place foundations before their principal applications.
The atomicity re-audit folded broad survey-style records into subject maps or replaced them with focused definitions, constructions, propositions, and theorems. Folded records are therefore absent from the new-knowl count; their surviving content is reached through the atomic entries below.
New production knowls (372)
Foundational topology and categorical algebra (16)
Shared foundations (1)
Topology (2)
Categorical algebra (6)
- Algebra object
- Core of a category
- Group object
- Groupoid
- Monoidal category
- Symmetric monoidal category
Coalgebras, bialgebras, and Hopf algebras (3)
Modules and algebra constructions (4)
Rings, valuations, ordered algebra, and completions (30)
Commutative algebra (2)
Valuations and fields (4)
- Non-Archimedean absolute value
- Valuation on a field
- Valuation ring of a valued field
- Value group of a valuation
Ordered and projective groups (6)
- Commutative monoid
- Lattice-ordered abelian group
- Ordered abelian group
- PGL–PSL comparison
- Projective general linear group
- Projective special linear group
Rings, semirings, and power series (15)
- Boolean semifield
- Commutative semiring
- Formal inverse function theorem
- Idempotent semifield
- Idempotent semifields and lattice-ordered groups
- Idempotent semiring
- Multivariable formal power series ring
- Natural order of an idempotent semiring
- Ordered semiring
- Parasemifield
- Semifield
- Semiring
- Semiring homomorphism
- Substitution of formal power series
- Tropical semifield
Topological algebra (3)
Hyperstructures, semirings, and blueprint comparisons (34)
- Algebra: Hyperstructures, semirings, and blueprints
- Band
- Canonical hypergroup
- Classification of stringent hyperfields
- Doubly distributive hyperfield
- Doubly distributive hyperfields are stringent
- Field embeddings into semifields and hyperfields
- Fusion band
- Fusion rule for bands
- Hyperfield
- Tract associated with a hyperfield
- Hyperfield of a field quotient
- Hyperoperation
- Hyperring
- Hyperring homomorphism
- Idyll
- Integral hyperring
- Krasner hyperfield
- Nullset of a band
- Partial field
- Tract associated with a partial field
- Partial hyperfield
- Tract associated with a partial hyperfield
- Pasture
- Phase hyperfield
- Quotient hyperring
- Sign hyperfield
- Stringent hyperfield
- Tract
- Ordered blueprint associated with a tract
- Tropical hyperfield
- Tropical semifield and hyperfield tied-sum distinction
- Valuations as tropical hyperfield morphisms
- Valuative hyperfield
Algebraic and blue geometry (58)
- Affine blue scheme
- Algebraic curve
- Algebraic group
- Algebraic space
- Algebraic stack
- Band as an ordered blueprint
- Blue scheme
- Blueprint
- Blueprint as an ordered blueprint
- Borel subgroup
- Coherent sheaf
- Commutative monoid with zero as a blueprint
- D-module
- Derived algebraic stack
- Finite locally free morphism
- Flag variety
- Formal scheme
- Formal spectrum
- Fundamental theorem of projective geometry
- Hyperrings as ordered blueprints
- Idyll as an ordered blue field
- Integrable algebraic connection
- -ideal of a blueprint
- Localization of a blueprint
- Locally blueprinted space
- Locally free sheaf
- Locally topologically ringed space
- Moduli stack of G-bundles on a curve
- Ordered blue field
- Ordered blue scheme
- Ordered blueprint
- Ordered blueprint with unique weak inverses
- Ordered blueprinted space
- Parabolic subgroup
- Unique-weak-inverse reflection of an ordered blueprint
- Pointed algebraic curve
- Pre-addition on a monoid
- Principal G-bundle on a scheme
- Projective geometry
- Projective line
- The projective line as a rank-one flag variety
- Projective morphism
- Projective semilinear group
- Projective transformation
- Punctured algebraic curve
- Quasi-coherent sheaf
- Reductive algebraic group
- Ringed space
- Semiring as a blueprint
- Semiring completion of a blueprint
- Monoid blueprint to semiring blueprint
- Sheaf of differential operators
- Sheaf of modules
- Smooth morphism
- Smooth projective curve
- Spectrum of a blueprint
- Tangent sheaf
- Weil restriction
Formal groups and formal Lie theory (19)
- Formal groups and formal group laws
- Additive and multiplicative formal group laws
- Affine formal groups and complete Hopf algebras
- BCH group of a complete filtered Lie algebra
- Coordinate Hopf algebra of an affine formal group
- Distribution algebra of a formal group
- Distribution algebra and universal enveloping algebra
- Formal affine space
- Formal completion of a group at the identity
- Formal group
- Formal group law
- Morphism of formal group laws
- Formal group laws as coordinates on formal groups
- Formal group logarithm
- Height of a one-dimensional formal group law
- Equivalence between Lie algebras and formal groups
- One-dimensional commutative formal group law
- Failure of tangent classification in positive characteristic
- Tangent Lie algebra of a formal group
Complex analysis and Möbius geometry (53)
- Complex Analysis
- Analytic continuation
- Anti-Möbius transformation
- Argument principle
- Automorphisms of the Riemann sphere
- Casorati–Weierstrass theorem
- Cauchy integral formula
- Cauchy integral theorem
- Cauchy–Riemann criterion
- Cauchy–Riemann equations
- Complex contour integral
- Complex derivative
- Complex domain
- Complex projective structure
- Cross-ratio
- Cross-ratio invariance under Möbius transformations
- Cross-ratio-preserving bijections are Möbius
- Entire function
- Equal Schwarzians differ by Möbius postcomposition
- Fundamental theorem of algebra (complex analysis)
- Generalized circle
- Great Picard theorem
- Holomorphic functions are analytic
- Holomorphic germ
- Identity theorem
- Classification of isolated singularities
- Laurent series
- Liouville's theorem in complex analysis
- Little Picard theorem
- Logarithmic derivative
- Maximum modulus principle
- Meromorphic function
- Möbius characterization by the Schwarzian
- Möbius transformation
- Möbius transformation group
- Monodromy theorem
- Montel theorem
- Normal family
- Open mapping theorem for holomorphic functions
- Order of a zero or pole
- Holomorphic projective connection
- Projective connections form an affine space
- Rational function
- Residue
- Residue theorem
- Riemann mapping theorem
- Riemann sphere
- Rouché's theorem
- Schwarzian derivative and second-order linear ODEs
- Schwarzian chain rule
- Schwarzian derivative
- Sharp three-transitivity of the Möbius group
- Winding number
Smooth, complex, symplectic, Kähler, and spin geometry (56)
Differential geometry (52)
- Almost-complex map
- Associated graded Clifford algebra
- Biholomorphism group
- C-infinity ring
- Category of complex manifolds
- Category of symplectic manifolds and symplectic maps
- Cauchy hypersurface
- Causal curve
- Causal diamond
- Celestial sphere
- Chirality operator
- Chronological and causal future
- Clifford algebra as a super-enveloping quotient
- Classification of complex Clifford modules
- Complex projective space
- Conformal map
- Connection Laplacian
- Diffeomorphism groupoid of smooth manifolds
- Dirac spinor
- Flux group
- Flux homomorphism
- Global hyperbolicity and Cauchy hypersurfaces
- Globally hyperbolic spacetime
- Hamiltonian isotopy
- Holomorphic isometric immersion
- Hyperbolic three-space
- Hyperkähler isometric immersion
- Hyperkähler isometry
- Laplace–Beltrami operator
- Local C-infinity-ringed space
- Lorentzian Dirac operator
- Lorentzian manifold
- Lorentzian spin structure
- Lorentzian spinor bundle
- Majorana spinor
- Majorana–Weyl spinor
- Newlander–Nirenberg theorem
- Nijenhuis tensor of an almost-complex structure
- Pseudo-Riemannian manifold
- Real projective space
- Riemannian isometric immersion
- Rotating hyperkähler isometry
- Scalar curvature
- Full faithfulness of the smooth-function functor
- Smooth splitting of globally hyperbolic spacetimes
- Spinor module
- Strong causality
- Symplectic isotopy
- Symplectomorphism group
- Time orientation
- Triholomorphic map
- Weyl spinor
Bundles, sections, and local systems (4)
- Category of vector bundles over a manifold
- Local system
- Sheaf of smooth sections
- Vector bundles and locally free sheaves
Lie groups and linear models (25)
Lie groups and Lie algebras (19)
- Celestial sphere and Möbius action
- Complete filtered Lie algebra
- Complex Lie group
- Complexification of a real Lie algebra
- Hermitian matrix model of Minkowski space
- Indefinite pin group
- Indefinite spin group
- Lie functor commutes with realification
- Projective general linear Lie group
- Projective special linear Lie group
- Proper orthochronous Lorentz group
- PSL(2,C)
- PSL(2,C) action on hyperbolic three-space
- PSL(2,C)–proper Lorentz isomorphism
- Restricted spin group
- SL(2,C) as a real and complex Lie group
- SL(2,C) spin cover of the Lorentz group
- Underlying real Lie algebra
- Underlying real Lie group
Linear algebra (6)
- Hermitian matrix
- Minkowski vector space
- Quadratic form
- Realification of a complex vector space
- Semilinear map
- Signature of a symmetric bilinear form
Supergeometry and supersymmetry (27)
- Supergeometry
- Batchelor theorem
- Category of super vector spaces
- Functor of points of a supermanifold
- Koszul sign rule
- Lie superalgebra
- Lie superalgebra of a Lie supergroup
- Lie supergroup
- Equivalence of Lie supergroups and super Harish–Chandra pairs
- Parity shift
- Representation of a Lie superalgebra
- Split supermanifold
- Super Harish–Chandra pair
- Super internal Hom
- Super-Minkowski space
- Super PBW theorem
- Super vector space
- Superalgebra
- Supercommutative algebra
- Supercommutator
- Superdomain
- Supermanifold
- Supermodule
- Superspace
- Supertranslation distribution
- Symmetric algebra of a super vector space
- Universal enveloping algebra of a Lie superalgebra
Mathematical physics and hyperbolic equations (21)
- Mathematical physics
- Advanced and retarded Green operators
- Cauchy problem for normally hyperbolic operators
- Cauchy problem for the Lorentzian Dirac operator
- Clifford slash notation
- Conformal coupling of a scalar field
- Connection form of a normally hyperbolic operator
- d’Alembert operator
- Dirac equation
- Dirac–Klein–Gordon factorization
- Existence of advanced and retarded Green operators
- Gamma matrices
- Klein–Gordon equation
- Klein–Gordon operator
- Minkowski Dirac operator
- Minkowski spacetime
- Normally hyperbolic operator
- Poincaré algebra
- Super-Poincaré algebra
- Supertranslation algebra
- Wave equation
Geometric Langlands and its categorical dependencies (33)
- Geometric Langlands
- Affine Grassmannian
- Affine Schubert variety
- Convolution of sheaves on the affine Grassmannian
- Dominant coweight
- G-local system
- Categorical geometric Langlands theorem
- Geometric Satake equivalence
- Global nilpotent cone for local systems
- Hecke correspondence
- Hecke eigensheaf
- Geometric Hecke functor
- Hecke modification
- Ind-coherent sheaf
- Ind-coherent sheaves with nilpotent singular support
- Ind-scheme
- Intersection-cohomology complex
- Irregular-singular connection
- Iwahori level structure
- Level structure on a G-bundle
- Loop group
- Moduli stack of G-local systems
- Parahoric level structure
- Perverse sheaf
- Positive loop group
- Ramification of a G-local system
- Ramified geometric Langlands
- Regular-singular connection
- Riemann–Hilbert correspondence
- Singular support of a coherent sheaf
- Stokes data
- Tame ramification
- Wild ramification
Pre-existing knowls substantively refined for integration review (64)
These records existed on develop and were expanded, corrected, cross-linked, or updated as section maps during the July 30 integration. This is the curated comparison set used by the old/new review page; it excludes corpus-wide reference-placement cleanup.
Algebra and algebraic-geometry records (7)
Categorical algebra (1)
Commutative algebra (1)
Valuations and fields (1)
Modules and algebra constructions (1)
Rings, semirings, and power series (2)
Algebraic-geometry foundations (1)
Differential geometry and bundles (25)
Differential geometry (17)
- Differential Geometry
- Algebra of smooth functions
- Almost-complex structure
- Category of smooth manifolds
- Clifford algebra
- Clifford module
- Complex manifold
- Hamiltonian diffeomorphism
- Holomorphic map
- Hyperkähler manifold
- Integrable almost-complex structure
- Irreducible holomorphic symplectic manifold
- Kähler manifold
- Sheaf of smooth functions
- Spinor bundle
- Symplectic manifold
- Symplectomorphism
Bundles, sections, and local systems (8)
- Fiber Bundles
- Module of smooth sections
- Reconstructing a vector bundle from a projective module
- Section module is finitely generated projective
- Vector bundle from an idempotent matrix
- Serre–Swan theorem
- Universal principal bundle EG→BG
- Vector bundle morphism
Analysis, noncommutative geometry, and operator algebras (20)
Functional analysis (3)
- Decomposable operator
- Fourier transform on Schwartz space
- Fourier transform of tempered distributions
Harmonic analysis (3)
- Quasi-invariant measure under a locally compact group action
- System of imprimitivity
- Weil integration formula
Noncommutative geometry (2)
Operator algebras (11)
- C*-algebra of compact operators
- Continuous core of a von Neumann algebra
- Factorial representation
- Faithful normal semifinite trace
- Modular automorphism group
- Normal completely positive map
- Nuclear C*-algebra
- Nuclear C*-algebras are exact
- Quasi-equivalent representations
- Tomita–Takesaki theorem
- Type I∞ factor
Real analysis (1)
Langlands, Lie theory, and linear algebra (11)
Langlands foundations (2)
Lie groups and Lie algebras (8)
- Lie Groups and Lie Algebras
- Baker–Campbell–Hausdorff formula
- Lie algebra
- Lorentz group
- Nelson Laplacian
- Special linear group
- Special linear Lie algebra
- Spin group