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Algebra category theory53 knowls

Abelian category An additive category with kernels and cokernels where exactness behaves like in module categories.
Additive category A preadditive category with a zero object and finite biproducts (so finite products and coproducts agree).
Adjoint functors A pair of functors F ⊣ G equipped with a natural hom-set bijection (equivalently, a unit and counit satisfying the triangle identities).
Automorphism An isomorphism from an object to itself; an invertible endomorphism.
Axioms for an abelian category A convenient list of axioms characterizing abelian categories.
Categorical product An object A×B equipped with projections, universal among cones to A and B.
Category A structure of objects and morphisms with associative composition and identity morphisms.
Category axioms The associativity and identity laws governing composition in a category.
Category Theory Foundations of category theory: categories, functors, natural transformations, limits, adjunctions, and abelian categories.
Coequalizer A universal morphism that forces two parallel morphisms to become equal.
Cokernel (categorical) In a pointed category, the cokernel of f:A→B is the coequalizer of f and the zero morphism A→B.
Colimit A universal cocone from a diagram, generalizing coproducts, pushouts, and coequalizers.
Composition of morphisms The rule that composes morphisms in a category, generalizing function composition.
Contravariant functor A functor that reverses the direction of morphisms; equivalently a functor C^op → D.
Coproduct An object A ⊔ B with injections, universal among cocones from A and B.
Counit of an adjunction For F ⊣ G, the counit ε: F∘G ⇒ Id_D is the natural transformation corresponding to identities under the adjunction bijection.
Distinguished triangle A triangle selected to play the role of an exact sequence in a shifted additive category.
Endomorphism A morphism whose domain and codomain are the same object.
Epimorphism A morphism that is right-cancellative; the categorical analogue of a surjection.
Equalizer A universal solution E → A making two parallel morphisms A ⇉ B equal after composition.
Equivalence of categories A functor that is invertible up to natural isomorphism.
Exact functor A functor between abelian categories that preserves all short exact sequences.
Exact sequence (categorical) In an abelian category, a sequence is exact at an object when the image equals the kernel (equivalently, kernels and cokernels fit together appropriately).
Full Subcategory A subcategory that contains every morphism of the ambient category between its objects.
Functor A map between categories that preserves identities and composition.
Identity morphism A morphism 1_X : X → X acting as a two-sided unit for composition.
Initial object An object admitting a unique morphism to every object of a category.
Isomorphism A morphism that has a two-sided inverse in a category.
Kernel (categorical) In a pointed category, the kernel of f:A→B is the equalizer of f and the zero morphism A→B.
Left exact functor An additive functor that preserves kernels (equivalently, exactness at the left end of short exact sequences).
Limit A universal cone to a diagram, generalizing products, pullbacks, and equalizers.
Monomorphism A morphism that is left-cancellative under composition.
Morphism An arrow between objects in a category.
Natural isomorphism A natural transformation whose components are isomorphisms.
Natural transformation A morphism between functors given by components that commute with all structure maps.
Object An entity of a category; morphisms go between objects.
Octahedral axiom The triangle axiom coherently relating cones of two composable morphisms and their composite.
Opposite Category The category obtained by reversing the direction of every morphism.
Preadditive category A category whose morphism sets are abelian groups and whose composition is bilinear.
Pretriangulated category A shifted preadditive category with distinguished triangles satisfying the first triangle axioms.
Pullback A universal object representing compatible pairs over a cospan.
Pushout A universal object obtained by gluing two objects along a common source.
Representable functor A Set-valued functor naturally isomorphic to a Hom functor.
Right exact functor An additive functor that preserves cokernels (equivalently, exactness at the right end of short exact sequences).
Shift functor An autoequivalence representing an integral degree shift in a category.
Subcategory A category obtained by restricting the objects and morphisms of a given category.
Terminal object An object receiving a unique morphism from every object of a category.
Triangle in a shifted category A cyclic diagram of three objects whose last morphism lands in the shift of the first.
Triangulated category A pretriangulated category satisfying the octahedral axiom.
Unit of an adjunction For F ⊣ G, the unit η: Id_C ⇒ G∘F is the natural transformation corresponding to identities under the adjunction bijection.
Yoneda embedding The fully faithful functor sending an object to its Hom functor (a representable presheaf).
Yoneda lemma Natural transformations from a representable functor correspond to elements of the target functor.
Zero object An object that is simultaneously initial and terminal.

Algebra commutative47 knowls

Algebra: Commutative Algebra Localization, Noetherian rings, primary decomposition, and integral extensions.
Artinian ring A ring in which descending chains of ideals stabilize.
Dedekind domain A Noetherian, integrally closed domain of Krull dimension one; equivalently, a domain with unique factorization of ideals into primes.
Discrete valuation ring A one-dimensional Noetherian local domain with principal maximal ideal; equivalently, a local PID with a unique nonzero prime.
Exactness of localization Localizing a sequence of modules at a multiplicative set preserves exactness.
Extension of scalars Given a ring map R→S, the S-module S⊗_R M obtained from an R-module M by base change.
Going-down theorem For certain integral extensions (e.g. with integrally closed base), prime chains descend inside a fixed prime upstairs.
Going-up theorem Along an integral extension, prime ideals can be lifted to extend prime chains.
Height of a prime The codimension of a prime ideal, measured by the maximum length of chains of primes ending at it.
Hilbert basis corollary Polynomial rings (and finitely generated algebras) over a Noetherian ring are Noetherian.
Integral closure The subring of an overring consisting of all elements integral over a given ring.
Integral element An element b in an R-algebra is integral over R if it satisfies a monic polynomial with coefficients in R.
Integral extension A ring extension A→B is integral if every element of B is integral over A.
Integrally closed domain A domain that already contains every element of its fraction field that is integral over it.
Jacobson radical annihilates simple modules Every element of the Jacobson radical acts as zero on any simple module.
Jacobson radical as intersection of maximal ideals In a commutative ring, the Jacobson radical equals the intersection of all maximal ideals.
Krull dimension The supremum of lengths of chains of prime ideals in a ring (equivalently, the dimension of its prime spectrum).
Krull's principal ideal theorem In a Noetherian ring, any prime ideal minimal over a principal ideal has height at most 1.
Lasker–Noether theorem Every ideal in a Noetherian ring can be written as a finite intersection of primary ideals.
Local ring A commutative ring with exactly one maximal ideal.
Localization at a prime ideal The ring R_p obtained by inverting all elements outside a prime ideal p.
Localization inverts a multiplicative set In the localization S^{-1}R, every element of S becomes a unit, and S^{-1}R is universal with that property.
Localization of a module Given S⊂R multiplicative, the module S^{-1}M obtained by inverting S in an R-module M.
Localization of a ring The ring S^{-1}R obtained from a ring R by inverting a multiplicative set S.
Localization preserves Noetherian rings If a ring is Noetherian, then any localization at a multiplicative set is again Noetherian.
Localization preserves Noetherianity If a ring is Noetherian, then any localization (in particular at a prime) is Noetherian.
Localization preserves prime ideals A prime ideal disjoint from the multiplicative set extends to a prime ideal in the localized ring.
Lying-over theorem In an integral extension, every prime ideal downstairs is the contraction of some prime ideal upstairs.
Maximal ideal of a local ring In a local ring, the unique maximal ideal is exactly the set of nonunits.
Maximal spectrum The set MaxSpec(R) of maximal ideals of a commutative ring, with the induced Zariski topology.
Multiplicative set A subset of a ring closed under multiplication and containing 1, used to form localizations.
Nakayama corollary: generators mod the maximal ideal lift For a finitely generated module over a local ring, generators of M/mM lift to generators of M, and the minimal number of generators is dim(M/mM).
Nakayama's lemma In a local ring, a finitely generated module cannot equal its maximal-ideal multiple unless it is zero.
Noether normalization lemma A finitely generated algebra over a field is integral over a polynomial subalgebra.
Noetherian ring A ring in which ascending chains of ideals stabilize (equivalently, every ideal is finitely generated).
Nullstellensatz corollary: maximal ideals are points Over an algebraically closed field, maximal ideals of a polynomial ring are exactly the ideals of points.
Nullstellensatz: varieties and radical ideals Over an algebraically closed field, Zariski-closed subsets of affine space correspond to radical ideals in a polynomial ring.
Primary decomposition Expressing an ideal as an intersection of primary ideals, with existence guaranteed in Noetherian rings.
Primary decomposition in Noetherian rings In a Noetherian ring, every ideal is a finite intersection of primary ideals.
Prime avoidance lemma If an ideal is contained in a finite union of prime ideals, then it is contained in one of them.
Prime correspondence under localization Prime ideals in a localization S^{-1}R correspond to primes of R disjoint from S via extension and contraction.
Prime spectrum The set Spec(R) of prime ideals of a commutative ring, naturally equipped with the Zariski topology.
Residue field For a local ring (R,m), the field R/m obtained by modding out by the maximal ideal.
Restriction of scalars Given a ring map R→S, any S-module can be regarded as an R-module by forgetting part of the scalar action.
Semisimple Artinian rings decompose as finite products A semisimple Artinian ring is a finite product of simple Artinian rings; if commutative, it is a finite product of fields.
Simple Artinian rings are matrix rings over division rings A simple Artinian ring is isomorphic to a full matrix ring over a division ring.
Zariski topology The natural topology on Spec(R) whose closed sets are defined by vanishing of ideals.

Algebra fields galois57 knowls

Algebra: Fields and Galois Theory Field extensions, splitting fields, Galois groups, and the fundamental theorem.
Algebraic closure An algebraic extension of a field that is algebraically closed, unique up to non-canonical isomorphism.
Algebraic element An element α is algebraic over F if it satisfies a nonzero polynomial with coefficients in F.
Algebraic extension An extension E/F in which every element of E is algebraic over F.
Artin's theorem on fixed fields A finite group of field automorphisms yields a finite Galois extension with degree equal to the group order.
Cyclotomic extension An extension obtained by adjoining a primitive n-th root of unity, e.g. Q(ζ_n)/Q.
Cyclotomic polynomial The polynomial Φ_n(x) whose roots are the primitive n-th roots of unity; it factors x^n−1 and is irreducible over Q.
Dedekind independence lemma Distinct K-embeddings of a field are linearly independent as functions.
Degree bounds for splitting fields The splitting field of a separable degree-n polynomial has degree at most n! over the base field.
Degree Equals Galois Group Order For a finite Galois extension L/K, the degree [L:K] equals the size of Gal(L/K).
Degree of a field extension The dimension [E:F] of E as a vector space over F (finite or infinite).
Discriminant (of a field basis) For a finite extension L/K, the discriminant of a K-basis is det(Tr_{L/K}(b_i b_j)).
Existence and uniqueness of finite fields For each prime power q=p^n there is a unique (up to isomorphism) field with q elements.
Existence and uniqueness of splitting fields Every nonconstant polynomial has a splitting field, unique up to K-isomorphism.
Existence of algebraic closures Every field admits an algebraic closure.
Existence of Finite Fields Finite fields exist exactly in prime power cardinalities, and can be constructed from irreducible polynomials.
Field automorphism A bijective field homomorphism; automorphisms fixing a base field form the Galois group.
Field embedding An injective field homomorphism, often required to fix a base field in extension theory.
Field extension An inclusion of fields F ⊆ E (written E/F) and the basic language used to study it.
Field norm For a finite extension L/K, the norm N_{L/K}(α) is the determinant of multiplication-by-α as a K-linear map.
Field trace For a finite extension L/K, the trace Tr_{L/K}(α) is the trace of multiplication-by-α as a K-linear map.
Finite field A field with finitely many elements; necessarily of size p^n and unique up to isomorphism for each p^n.
Finite Field Extensions Are Cyclic Galois The extension 𝔽_{p^n}/𝔽_p is Galois with cyclic Galois group generated by Frobenius.
Finite fields are perfect Every finite field has all algebraic extensions separable (equivalently, Frobenius is an automorphism).
Finite Galois extensions are separable and normal A finite extension is Galois iff it is both separable and normal.
Finitely generated field extension An extension E/F of the form E = F(α1,…,αn) for finitely many generators.
Fixed field The subfield consisting of elements fixed by every automorphism in a given group.
Frobenius endomorphism In characteristic p, the map x ↦ x^p is a ring endomorphism; on finite fields it is an automorphism.
Fundamental theorem of Galois theory For a finite Galois extension L/K, intermediate fields correspond to subgroups of Gal(L/K).
Fundamental theorem of symmetric polynomials A symmetric polynomial can be expressed uniquely in terms of the elementary symmetric polynomials.
Galois Correspondence For a finite Galois extension, intermediate fields correspond bijectively to subgroups of the Galois group.
Galois extension An algebraic field extension that is both normal and separable.
Galois group The group of field automorphisms of an extension that fix the base field pointwise.
Galois group of a finite field extension is cyclic Gal(F_{p^n}/F_p) is cyclic of order n, generated by Frobenius x↦x^p.
Inseparable extension An algebraic extension that is not separable; it contains an element with a repeated-root minimal polynomial.
Intermediate field A subfield K with F ⊆ K ⊆ E inside a given field extension E/F.
Multiplicative Group of a Finite Field Is Cyclic For a finite field 𝔽_q, the group 𝔽_q× of nonzero elements is cyclic of order q−1.
Multiplicative group of a finite field is cyclic For a finite field F_q, the group F_q^× is cyclic of order q−1.
Normal extension An algebraic extension in which every irreducible polynomial having one root splits completely.
Normal extensions and splitting fields An algebraic extension is normal iff it is a splitting field of polynomials over the base field.
Perfect field A field for which every algebraic extension is separable.
Perfect fields and separability of finite extensions Over a perfect field, every algebraic (hence every finite) extension is separable.
Primitive element theorem Every finite separable extension is generated by a single element.
Primitive root of unity An element ζ with ζ^n = 1 whose multiplicative order is exactly n.
Separability in towers Separability is stable under passing up and down a tower of fields.
Separable element An algebraic element whose minimal polynomial has distinct roots in a splitting field.
Separable extension An algebraic extension in which every element is separable over the base field.
Separable polynomials have distinct roots A polynomial is separable iff it has no repeated roots in an algebraic closure (equivalently gcd(f,f')=1).
Simple extension An extension of the form E = F(α), generated by a single element α.
Splitting field The smallest field extension over which a given polynomial factors completely into linear factors.
Tower law In a finite tower K ⊂ L ⊂ M, degrees multiply: [M:K]=[M:L][L:K].
Tower of fields A chain of field extensions F ⊆ K ⊆ E, used to analyze E/F in stages.
Trace and norm in towers In a tower K⊂E⊂L of finite extensions, trace and norm compose multiplicatively/additively.
Transcendental element An element α is transcendental over F if it satisfies no nonzero polynomial in F[x].
Transcendental extension An extension E/F that contains at least one element transcendental over F (i.e. is not algebraic).
Uniqueness of Algebraic Closures Any two algebraic closures of a field are isomorphic over the base field.
Uniqueness of splitting fields Splitting fields are unique up to base-field isomorphism (and unique inside a fixed algebraic closure).

Algebra groups138 knowls

A p-group has nontrivial center A finite group of order p^n always has a center of size divisible by p
Abelian Group A group whose operation is commutative
Abelian implies all subgroups normal In an abelian group every subgroup is normal
Algebra: Groups Group theory through Sylow theorems and structure
Automorphism Group The group of all isomorphisms from a group to itself
Automorphisms of a cyclic group Aut(C_n) is naturally isomorphic to (ℤ/nℤ)×
Burnside's Lemma The number of orbits equals the average number of fixed points
Burnside's p^a q^b Theorem A finite group of order p^a q^b (two primes) is solvable
Cancellation laws Left and right cancellation hold in every group
Cauchy's Theorem (Finite Groups) If a prime p divides |G|, then G contains an element (and subgroup) of order p
Cayley's Theorem Every group embeds into a permutation group via the left regular action
Center is characteristic The center of a group is invariant under all automorphisms
Center of a Group The set of elements commuting with every group element
Central Extension An extension whose kernel lies in the center of the total group
Centralizer The subgroup of elements commuting with a given subset
Characteristic Subgroup A subgroup fixed by every automorphism of the group
Chief series A normal series with no intermediate normal subgroups between successive terms
Class Equation A finite group decomposes into the center plus conjugacy classes of larger size
Class equation decomposition A finite group decomposes into its center and nontrivial conjugacy classes
Class function A function on a group that is constant on conjugacy classes
Classification of Finite Abelian Groups Every finite abelian group is a direct product of cyclic prime-power groups, uniquely up to isomorphism.
Commutator An element measuring the failure of two group elements to commute
Commutator subgroup The subgroup generated by all commutators in a group
Composition series A subnormal series with simple successive quotients
Conjugacy class The set of all conjugates of a given group element
Conjugacy Class Size Lemma The size of a conjugacy class equals the index of the centralizer
Conjugate element Two elements of a group are conjugate if one is obtained from the other by an inner automorphism
Conjugation Action The action of a group on itself (or its subgroups) by conjugation
Conjugation action on itself A group acts on itself by conjugation g·x = gxg^{-1}
Conjugation preserves order Conjugate elements have the same order in a group
Correspondence Theorem (Groups) Subgroups of G containing N correspond to subgroups of the quotient G/N
Coset A left or right translate of a subgroup by a group element
Cosets Partition a Group Left (or right) cosets of a subgroup form a partition of the ambient group
Cyclic Subgroup A subgroup generated by a single element
Derived series The descending series obtained by repeatedly taking commutator subgroups
Direct Product of Groups The product group with componentwise multiplication
Direct Sum of Groups The subgroup of a direct product with finite support
Euler's Theorem If gcd(a,n)=1 then a^{φ(n)} ≡ 1 (mod n).
Exact Sequence of Groups A sequence of homomorphisms where image equals kernel at each stage
Faithful Action An action with trivial kernel, equivalently an injective permutation representation
Fermat's Little Theorem For prime p, a^{p-1} ≡ 1 (mod p) when p ∤ a.
Finite cyclic group is isomorphic to ℤ/nℤ A cyclic group of order n is (canonically) isomorphic to ℤ/nℤ
Finite p-Group Has Nontrivial Center If |G|=p^n with n≥1 then p divides |Z(G)|, so Z(G) is nontrivial.
Finite p-groups have subgroups of all p-power orders If |G|=p^n then for each k there is a subgroup of order p^k
First isomorphism consequence for groups For a homomorphism f, the quotient G/ker(f) is isomorphic to im(f)
First Isomorphism Theorem (Groups) A homomorphism factors through the quotient by its kernel, giving G/ker(f) ≅ im(f)
Fixed-Point Set The subset of points fixed by all group elements in an action
Frattini Argument If N is normal and P is a Sylow p-subgroup of N, then G = N N_G(P)
Free Action An action in which only the identity can fix a point
Free Group The group generated by a set with no relations beyond the group axioms
Fundamental Theorem of Finitely Generated Abelian Groups Every finitely generated abelian group is a direct sum of copies of Z and finite cyclic groups
Generated Subgroup The smallest subgroup containing a given subset
Generating Set A subset whose elements generate the whole group
Group A monoid in which every element has an inverse
Group Action A homomorphism from a group to permutations of a set, equivalently a compatible map G×X→X
Group epimorphism A surjective group homomorphism
Group Extension A group fitting into a short exact sequence 1→N→E→Q→1
Group homomorphism A map between groups that preserves the group operation
Group isomorphism A bijective group homomorphism
Group monomorphism An injective group homomorphism
Group Presentation A description of a group by generators and relations
Groups of order p^2 are abelian Every group of order p^2 (p prime) is abelian
Groups of prime order are cyclic A finite group of prime order is generated by any non-identity element
Hall subgroup A subgroup whose order is coprime to its index in the ambient finite group
Image is a subgroup The image of a group homomorphism is a subgroup of the codomain
Image of a group homomorphism The set of values attained by a group homomorphism
Index of a Subgroup The number of cosets of a subgroup in a group
Inner Automorphism An automorphism given by conjugation by an element
Internal Direct Product A group built from two normal subgroups whose product is the whole group
Internal Semidirect Product A group generated by a normal subgroup and a complementary subgroup with trivial intersection
Intersection of subgroups is a subgroup Any intersection of subgroups of a fixed group is again a subgroup
Jordan-Hölder Uniqueness Any two composition series of a group have the same simple composition factors up to order.
Jordan–Hölder Theorem (Groups) Any two composition series have the same length and the same composition factors up to order
Kernel is normal The kernel of a group homomorphism is a normal subgroup
Kernel of a group homomorphism The set of elements mapped to the identity by a group homomorphism
Kernel of an Action The subgroup acting trivially on every point of the set
Kernels are Normal Subgroups The kernel of a group homomorphism is invariant under conjugation
Krull–Remak–Schmidt Theorem (Groups) Under chain conditions, direct product decompositions into indecomposable normal factors are unique up to order
Lagrange's Theorem In a finite group, the order of a subgroup divides the order of the group
Left multiplication action A group acts on itself by left translation
Loop A quasigroup with an identity element
Lower central series The descending series defined by iterated commutators with the whole group
Magma A set with a binary operation (no other axioms)
Missing Knowls List of referenced knowls that need to be created
Monoid A semigroup with an identity element
Nielsen–Schreier Theorem Every subgroup of a free group is free (with a rank formula in finite index)
Nilpotent Group A group whose lower central series terminates at the trivial subgroup
Normal Closure The smallest normal subgroup containing a given subset
Normal Subgroup A subgroup invariant under conjugation
Normal Subgroup Criterion A subgroup is normal iff it is stable under conjugation by every group element
Normalizer The largest subgroup in which a given subgroup becomes normal
Orbit The set of points reachable from a given point under a group action
Orbit Decomposition Lemma Orbits of a group action form a partition of the underlying set
Orbit–Stabilizer Theorem For a group action, an orbit is in bijection with a coset space G/Stab(x)
Order of Element Divides Order of Group In a finite group, the order of any element divides the order of the group.
Outer Automorphism Group Automorphisms modulo inner automorphisms
p-group A group whose elements have order a power of a fixed prime p
Perfect Group A group equal to its commutator subgroup
Permutation Representation A homomorphism from a group into bijections of a set
Product of normal subgroups is normal If N and M are normal in G then NM is a normal subgroup of G
Proper Subgroup A subgroup that is strictly smaller than the whole group
Quasigroup A magma where division is always possible
Quotient Group The group of cosets of a normal subgroup
Regular Action An action that is both free and transitive
Schreier Refinement Theorem Any two subnormal series admit equivalent refinements with isomorphic factors
Schreier's Lemma A subgroup of a finitely generated group is generated by Schreier generators from a transversal
Schur–Zassenhaus Theorem A normal Hall subgroup has a complement, unique up to conjugacy
Second Isomorphism Theorem (Groups) For H ≤ G and K ⊲ G, there is a natural isomorphism H/(H∩K) ≅ HK/K
Semidirect Product A product of groups twisted by an action by automorphisms
Semidirect product from a splitting exact sequence A split extension yields a semidirect product decomposition
Semigroup A set equipped with an associative binary operation
Simple Group A nontrivial group with no nontrivial normal subgroups
Solvable Group A group whose derived series terminates at the trivial subgroup
Split Extension An extension admitting a homomorphic section, equivalently a semidirect product
Stabilizer The subgroup of elements fixing a point under a group action
Subgroup A subset of a group that is itself a group under the same operation
Subgroup of Index 2 is Normal Any subgroup with exactly two cosets is invariant under conjugation
Subgroup Test (one-step) A nonempty subset of a group is a subgroup iff it is closed under xy^{-1}
Subgroup Test (two-step) A nonempty subset of a group is a subgroup iff it is closed under products and inverses
Subgroups are closed under inverses and products A subgroup contains the identity and is closed under multiplication and inversion
Subgroups of cyclic groups are cyclic Every subgroup of a cyclic group is cyclic, with an explicit generator
Subnormal series A finite chain of subgroups where each is normal in the next
Sylow Congruence The number n_p of Sylow p-subgroups satisfies n_p ≡ 1 (mod p).
Sylow Conjugacy Lemma Every p-subgroup lies in a conjugate of a Sylow p-subgroup
Sylow normality criterion If the Sylow p-subgroup is unique then it is normal
Sylow p-subgroup A maximal p-subgroup of a finite group, of order equal to the largest p-power dividing the group order
Sylow's First Theorem If |G| = p^a m with p ∤ m, then G has a subgroup of order p^a
Sylow's Second Theorem All Sylow p-subgroups are conjugate, and every p-subgroup lies in one
Sylow's Third Theorem The number of Sylow p-subgroups divides the p'-part of |G| and is ≡ 1 mod p
Third Isomorphism Theorem (Groups) If N ⊆ K ⊲ G with N ⊲ G, then (G/N)/(K/N) ≅ G/K
Torsion-free group A group in which the identity is the only element of finite order.
Transitive Action An action with a single orbit
Trivial Subgroup The subgroup consisting only of the identity element
Uniqueness of identity A group has exactly one identity element
Uniqueness of inverses Each element of a group has a unique two-sided inverse
Unital Magma A magma with an identity element
Universal Property of Quotient Groups Homomorphisms out of G that kill N factor uniquely through G/N
Upper central series The ascending series built from successive centers of quotients

Algebra homological32 knowls

Chain complex A graded sequence of modules with differentials d lowering degree and satisfying d∘d=0.
Chain homotopy A degree +1 family of maps witnessing that two chain maps differ by a boundary operator.
Chain map A degreewise module homomorphism between chain complexes commuting with differentials.
Cochain complex A graded sequence of modules with differentials d raising degree and satisfying d∘d=0.
Cohomology module The nth cohomology H^n(C) = ker(d^n)/im(d^{n-1}) of a cochain complex of modules.
Connecting homomorphism (boundary map) lemma From a short exact sequence of complexes (or of objects with a left/right exact functor), one constructs natural connecting maps yielding a long exact sequence in homology/cohomology.
Corollary of the five lemma: the short five lemma In a morphism of short exact sequences, isomorphisms on the ends force an isomorphism in the middle.
Derived functor Functors R^nF and L_nF obtained from resolutions, measuring the failure of exactness and yielding Ext and Tor.
Exact complex A chain complex whose homology vanishes in every degree (equivalently, im d = ker d).
Exactness properties of Hom and tensor Hom is left exact and tensor is right exact; flatness, projectivity, and injectivity are exactly the conditions that make these functors exact.
Existence of injective resolutions Every module embeds into an injective module, hence admits an injective resolution.
Existence of projective resolutions Every module admits a projective (in fact free) resolution.
Ext The right derived functors of Hom; measures extension classes and failure of exactness of Hom.
Ext and Tor as derived functors Ext and Tor are the right/left derived functors of Hom and tensor, computed via injective/projective (or flat) resolutions.
Ext¹ classifies extensions Ext¹_R(C,A) is naturally identified with equivalence classes of short exact sequences 0→A→E→C→0.
Five lemma In a morphism of exact sequences, if four vertical maps are isomorphisms (with mild extra hypotheses), then so is the middle map.
Four lemma Diagram-chase criteria ensuring the middle map in a morphism of exact sequences is injective or surjective.
Hom is left exact Hom preserves kernels: Hom_R(M,-) is left exact (covariant) and Hom_R(-,N) is left exact (contravariant); Ext measures the failure of exactness beyond that.
Homological Algebra Chain complexes, derived functors, Ext, Tor, and the fundamental lemmas of homological algebra.
Homology module The nth homology H_n(C) = ker(d_n)/im(d_{n+1}) of a chain complex of modules.
Horseshoe lemma Given a short exact sequence of modules, compatible projective (or injective) resolutions can be spliced to produce a resolution of the middle module.
Injective resolution An exact cochain complex starting at M and continuing with injective modules, used to compute Ext and right derived functors.
Long exact sequence for derived functors A short exact sequence induces a long exact sequence on left/right derived functors via connecting morphisms.
Long exact sequence for Ext The natural long exact sequence in Ext induced by a short exact sequence of modules.
Long exact sequence for Tor The natural long exact sequence in Tor induced by a short exact sequence of modules.
Mapping cone A complex combining the source and target of a chain map and measuring its failure to be an equivalence.
Nine lemma (3×3 lemma) In a commutative 3×3 diagram in an abelian category with exact rows/columns, exactness of one row (or column) follows from the other eight exact sequences.
Projective resolution An exact chain complex of projective modules ending in a given module M, used to compute Tor and Ext.
Snake lemma From a commutative diagram with exact rows, produces an exact sequence of kernels and cokernels with a canonical connecting map.
Snake lemma corollary: long exact sequence in homology A short exact sequence of chain complexes induces a natural long exact sequence in homology.
Tensor product is right exact For fixed N, the functor -⊗_R N preserves cokernels (exactness on the right); its failure to be left exact is measured by Tor.
Tor The left derived functors of tensor product; measures failure of tensor to be left exact (flatness).

Algebra modules87 knowls

Algebra homomorphism A ring homomorphism that respects the chosen base-ring action.
Algebra over a commutative ring A ring equipped with a compatible structure map from a commutative base ring.
Algebra: Module Theory Foundational definitions and theorems in module theory over rings.
Annihilator of a module The ideal of scalars that kill the entire module.
Annihilator of an element The ideal of ring elements that kill a given module element.
Artinian and Noetherian implies finite length A module that is both Artinian and Noetherian has a finite composition series.
Artinian module A module satisfying the descending chain condition on submodules.
Associated graded ring The graded ring gr_F(R)=⊕ F_nR/F_{n-1}R attached to a filtered ring.
Baer's criterion Characterization of injective modules by extension of maps from ideals.
Basis of a free module A set of elements giving unique finite linear combinations in a free module.
Bilinear map A map that is linear in each variable (and balanced over a ring when needed).
Bimodule A module with commuting left and right actions by (possibly different) rings.
Chinese remainder for modules Module quotients by comaximal ideal multiples split as a direct sum of smaller quotients.
Classification of finitely generated abelian groups Every finitely generated abelian group splits as a free part plus finite cyclic invariants.
Cokernel The quotient of the codomain by the image of a module homomorphism.
Composition series A finite chain of submodules with simple successive quotients.
Correspondence theorem for modules Submodules of M containing N correspond to submodules of M/N.
Cyclic module A module generated by a single element.
Diagonalizable operator A linear operator that has a basis of eigenvectors.
Direct product of modules The product of modules: all tuples with coordinatewise operations.
Direct sum of modules The coproduct of modules: tuples with finite support under coordinatewise operations.
Direct sum universal property The direct sum is characterized by a universal mapping property from the summands.
Dual module The Hom module Hom_R(M,R) for a module over a commutative ring.
Elementary divisor theorem Over a PID, a finitely generated module decomposes into primary cyclic summands.
Exact sequence of modules A sequence of module homomorphisms where each image equals the next kernel.
Exactness via kernels and images A sequence is exact at a term precisely when the incoming image equals the outgoing kernel.
Filtered ring A ring equipped with an increasing multiplicative filtration.
Finitely generated module A module generated by finitely many elements.
Finitely generated projectives are locally free Over a commutative ring, finitely generated projective modules become free after localization.
Finitely generated torsion-free over a PID is free Over a PID, finitely generated torsion-free modules are free.
First isomorphism theorem for modules A module homomorphism induces an isomorphism M/ker f ≅ im f.
Flat module A module whose tensor product functor preserves exactness.
Free module A module admitting a basis; equivalently, a direct sum of copies of the ring.
Free module universal property A free module on a set represents functions out of that set by unique linear extension.
Graded module A module decomposed into degrees compatible with a graded ring action.
Graded ring A ring decomposed into homogeneous pieces compatible with multiplication.
Hom module The module (or abelian group) of module homomorphisms between two modules.
Hom turns sums into products Hom out of a direct sum canonically identifies with the product of Homs.
Image of a module homomorphism The submodule consisting of all values attained by a module homomorphism.
Injective module A module with the extension property against injective homomorphisms.
Jordan canonical form from rational canonical form When the relevant polynomials split, rational canonical form refines to Jordan form.
Jordan canonical form theorem Over a splitting field, every linear operator is similar to a direct sum of Jordan blocks.
Kernel and image are submodules For a module homomorphism, both kernel and image are submodules.
Kernel of a module homomorphism The submodule mapped to zero by a module homomorphism.
Kernels are submodules The kernel of a module homomorphism is a submodule of its domain.
Krull–Schmidt–Azumaya theorem Finite-length modules decompose uniquely (up to permutation) into indecomposable summands.
Length of a module The number of simple factors in a composition series (when finite).
Matrix representation A matrix encoding a linear map relative to chosen bases.
Module An abelian group equipped with a compatible scalar action by a ring (left or right).
Module axioms The axioms defining a (left) module over a unital ring.
Module homomorphism A map preserving addition and scalar multiplication between modules.
Noetherian module A module satisfying the ascending chain condition on submodules.
Projective implies flat Every projective module is flat, so tensoring with it preserves exact sequences.
Projective module A module with the lifting property against surjections; equivalently, a direct summand of a free module.
Projective modules are direct summands of free modules A module is projective iff it is a direct summand of a free module.
Projective short exact sequence criterion A module is projective iff every short exact sequence ending in it splits.
Quotient by kernel is isomorphic to image For a homomorphism f, the induced map M/ker(f) → im(f) is an isomorphism.
Quotient module The module obtained by collapsing a submodule to zero.
Rank of a free module The cardinality of a basis of a free module.
Rational canonical form from the structure theorem Rational canonical form arises by viewing (V,T) as a module over F[x] and applying the PID structure theorem.
Rational canonical form theorem Every linear operator is similar to a block diagonal companion-matrix form determined by invariant factors.
Second isomorphism theorem for modules For submodules A,B ≤ M, one has (A+B)/B ≅ A/(A∩B).
Semisimple iff every submodule is a direct summand A module is semisimple exactly when all submodules split off as direct summands.
Semisimple module A module that is a direct sum of simple modules; equivalently, all short exact sequences split.
Short exact sequence An exact sequence 0 → A → B → C → 0 capturing a module extension.
Simple module A nonzero module with no proper nontrivial submodules.
Smith normal form invariants The Smith normal form diagonal entries are canonical invariants and control the cokernel module.
Smith normal form theorem A matrix over a PID can be diagonalized with divisibility conditions on the diagonal.
Split exact sequence A short exact sequence that decomposes as a direct sum.
Splitting lemma A short exact sequence splits iff it has a section or a retraction.
Structure theorem for finitely generated modules over a PID A finitely generated module over a PID splits as a free part plus cyclic torsion factors.
Submodule An additive subgroup closed under the scalar action of a module.
Submodule criterion Closure conditions that characterize when a subset is a submodule.
Tensor commutes with direct limits and sums Tensoring is a left adjoint, hence it preserves direct sums and filtered colimits.
Tensor product of algebras The tensor product A⊗_R B equipped with the induced algebra structure.
Tensor product of modules The universal recipient of balanced bilinear maps from a pair of modules.
Tensor product preserves direct sums Tensoring with a fixed module distributes over arbitrary direct sums.
Tensor product universal property The tensor product represents balanced bilinear maps out of a pair of modules.
Tensor–Hom adjunction The natural identification Hom(M⊗N,P) ≅ Hom(M,Hom(N,P)).
Tensor–Hom adjunction lemma Natural isomorphism between Hom out of a tensor product and Hom into a Hom-module.
Third isomorphism theorem for modules If A ⊆ B ⊆ M then (M/A)/(B/A) ≅ M/B.
Torsion element An element killed by a nonzero scalar in a module over an integral domain.
Torsion module A module in which every element is torsion (over an integral domain).
Torsion-free module A module over an integral domain with no nonzero torsion elements.
Universal property of quotient modules A map that kills a submodule factors uniquely through the quotient.
Universal property of the tensor product Balanced bilinear maps out of M×N correspond to linear maps out of M⊗N.
Vector space axioms The module axioms specialized to scalars in a field.

Algebra representation theory22 knowls

Character of a Direct Sum For complex representations, the character of a direct sum is the sum of the characters.
Character of a representation The class function χ(g)=tr(ρ(g)) attached to a finite-dimensional representation ρ of a finite group.
Character of a Tensor Product For complex representations, the character of a tensor product is the pointwise product of characters.
Character orthogonality Irreducible complex characters are orthonormal under the standard inner product on class functions.
Complete reducibility over ℂ Every finite-dimensional complex representation of a finite group splits as a direct sum of irreducibles.
Completely reducible representation A representation that splits as a direct sum of irreducible subrepresentations.
Group algebra The associative algebra k[G] whose basis is a group G and whose multiplication extends the group law bilinearly.
Group representation A linear action of a group on a vector space, equivalently a homomorphism into a general linear group.
Induced representation A construction Ind_H^G that extends a representation of a subgroup H to a representation of the whole group G.
Irreducible character The character of an irreducible complex representation; these form an orthonormal basis of class functions.
Irreducible representation A nonzero representation with no proper, nontrivial invariant subspaces.
Irreducibles and Conjugacy Classes Over ℂ, the number of irreducible representations equals the number of conjugacy classes of the group.
Maschke corollary (regular representation decomposition) When char(k) does not divide |G|, the group algebra is semisimple and the regular representation splits into irreducibles with multiplicity equal to dimension.
Maschke's theorem If char(k) does not divide |G|, then every finite-dimensional k-representation of a finite group is completely reducible.
Orthonormality of irreducible characters With respect to the standard inner product on class functions, irreducible characters are orthonormal (and over ℂ they form an orthonormal basis).
Regular representation The canonical representation of a group on the vector space with basis the group, via left multiplication.
Representation Theory Representation theory of finite groups: linear actions, characters, Maschke's theorem, and Schur's lemma.
Restricted representation Given a representation of a group and a subgroup, the restriction is the same action viewed only on the subgroup.
Schur corollary: central elements act by scalars In an irreducible representation over an algebraically closed field, every central group element (and more generally every central group-algebra element) acts as a scalar.
Schur's Lemma Intertwiners between irreducible representations are either zero or isomorphisms; equivariant endomorphisms form a division algebra (scalars over ℂ).
Subrepresentation An invariant subspace of a representation, closed under the group action.
Sum of squares of degrees For a finite group, the sum of the squares of the dimensions of its irreducible complex representations equals the group order.

Algebra rings109 knowls

Algebra: Rings Ring theory and ideal structure
Annihilator ideal The set of ring elements that kill a given subset under multiplication.
Artinian semisimple ring A semisimple ring that satisfies the descending chain condition on ideals; equivalently a finite product of matrix algebras over division rings.
Artin–Wedderburn theorem Semisimple Artinian rings are exactly finite products of matrix rings over division rings.
Associated elements Two elements that differ by multiplication by a unit.
Cancellation in integral domains In an integral domain, nonzero elements satisfy left and right cancellation.
Center of a ring The subring of elements that commute with every element of the ring.
Characteristic The additive order of 1 in a unital ring; either 0 or a positive integer.
Characteristic of an integral domain is 0 or prime An integral domain cannot have composite positive characteristic.
Chinese remainder decomposition For comaximal ideals, a quotient ring decomposes as a product of quotients.
Chinese remainder theorem For pairwise comaximal ideals, the quotient by their intersection splits as a product of quotients.
Commutative ring A ring in which multiplication is commutative.
Commutative ring axiom Axiom requiring multiplication in a ring to be commutative.
Content formula Over a UFD, content(fg) is associate to content(f)content(g) for polynomials.
Content of a polynomial The ideal generated by the coefficients of a polynomial.
Correspondence theorem for rings Ideals of a quotient ring correspond to ideals of the original ring containing the kernel.
Division ring A unital ring in which every nonzero element is invertible (not necessarily commutative).
Eisenstein's criterion A sufficient condition (via a prime element) for a polynomial to be irreducible.
Euclidean algorithm yields gcd and Bézout identity In a Euclidean domain, the Euclidean algorithm computes a gcd and expresses it as a linear combination.
Euclidean domain An integral domain admitting division with remainder controlled by a Euclidean function.
Euclidean domain ⇒ PID Every Euclidean domain has all ideals principal.
Every nontrivial commutative ring has a maximal ideal A commutative ring with 1 and 1≠0 contains at least one maximal ideal.
Existence of maximal ideals Every nontrivial unital commutative ring has a maximal ideal (via Zorn's lemma).
Field A commutative unital ring in which every nonzero element is invertible.
Field axioms Axioms defining a field as a commutative unital ring in which every nonzero element is invertible.
Fields and trivial ideals A commutative ring with 1 is a field iff its only ideals are (0) and (1).
Fields are exactly commutative division rings A ring is a field iff it is a commutative division ring.
Finite division rings are commutative By Wedderburn's little theorem, every finite division ring is a field.
Finite integral domains are fields A finite integral domain has multiplicative inverses for all nonzero elements.
First isomorphism theorem for rings A ring homomorphism induces an isomorphism from the quotient by its kernel onto its image.
Formal power series ring The ring R[[x]] of infinite power series with coefficients in R and Cauchy product.
Fraction field The field obtained from an integral domain by adjoining inverses to all nonzero elements.
Gauss lemma (content multiplicativity) In a UFD, the content of a product equals the product of contents up to associates.
Gauss's lemma Over a UFD, primitive polynomials factor over the fraction field exactly when they factor over the ring.
Gauss's theorem (UFD ⇒ polynomial ring is UFD) If R is a UFD, then the polynomial ring R[x] is again a UFD (and likewise in finitely many variables).
Greatest common divisor A divisor d of a and b that is divisible by every common divisor (defined up to associates).
Group of units The multiplicative group consisting of all units in a unital ring.
Hilbert basis theorem If a commutative ring is Noetherian, then its polynomial ring in finitely many variables is Noetherian.
Hilbert's Nullstellensatz (strong) Over an algebraically closed field, the ideal of a variety is the radical of the defining ideal.
Hilbert's Nullstellensatz (weak) Over an algebraically closed field, every proper ideal in a polynomial ring has a common zero.
Ideal An additive subgroup closed under multiplication by ring elements on one side (left or right).
Ideal correspondence for quotients Ideals of R containing I are in bijection with ideals of the quotient ring R/I.
Ideal generated by a subset The smallest ideal containing a given subset, equivalently the set of finite ring combinations of its elements.
Idempotent element An element e satisfying e^2=e.
Idempotents and product decompositions Central idempotents split a ring as a product of two quotient-like pieces.
Image is a subring The image of a ring homomorphism is closed under the ring operations.
Image of a ring homomorphism The subset of the codomain attained by a ring homomorphism.
Integral domain A commutative unital ring with no zero divisors.
Intersection of ideals The set-theoretic intersection of two ideals, which is again an ideal.
Irreducible element A nonzero nonunit that cannot be written as a product of two nonunits.
Irreducible polynomial A nonconstant polynomial that cannot be factored into lower-degree nonunits.
Jacobson radical The intersection of all maximal ideals, equivalently the elements acting trivially on simple modules.
Kernel is an ideal The kernel of a ring homomorphism is a two-sided ideal of the domain.
Kernel of a ring homomorphism The set of elements mapped to zero by a ring homomorphism.
Kernels are two-sided ideals The kernel of a ring homomorphism is always a two-sided ideal.
Laurent polynomial ring The ring of finite sums of a_i x^i allowing negative exponents.
Least common multiple A common multiple m of a and b that divides every other common multiple (defined up to associates).
Matrix ring The ring of n×n matrices over a ring with the usual addition and multiplication.
Maximal ideal A proper ideal maximal under inclusion; in the commutative unital case, equivalently the quotient is a field.
Maximal ideal iff quotient is a field An ideal is maximal exactly when the corresponding quotient ring is a field.
Maximal ideals are prime In a commutative ring, every maximal ideal is a prime ideal.
Minimal polynomial over a field The unique monic irreducible polynomial over K annihilating a given algebraic element.
Nil ideal An ideal all of whose elements are nilpotent.
Nilpotent element An element whose sufficiently high power is zero.
Nilradical The ideal of all nilpotent elements of a commutative ring.
Nilradical equals intersection of prime ideals In a commutative ring, the nilradical is the intersection of all prime ideals.
Opposite ring The ring with the same underlying abelian group but reversed multiplication.
PID ⇒ UFD Every principal ideal domain is a unique factorization domain.
Polynomial ring The ring R[x] of polynomials in an indeterminate x with coefficients in R.
Primary ideal An ideal Q such that ab in Q forces a in Q or a power of b in Q.
Prime element A nonzero nonunit p such that p | ab implies p | a or p | b.
Prime ideal A proper ideal P such that ab in P forces a in P or b in P.
Prime ideal iff quotient is an integral domain An ideal is prime exactly when the corresponding quotient ring has no zero divisors.
Prime ring A ring in which the product of nonzero ideals is never zero.
Prime subfield Every field contains a smallest subfield isomorphic to Q or to F_p.
Primitive polynomial A polynomial whose coefficients generate the unit ideal (content 1).
Principal ideal An ideal generated by a single element.
Principal ideal domain An integral domain in which every ideal is generated by a single element.
Product of ideals The ideal generated by all products of elements from two ideals.
Quotient ring A ring formed from a ring by identifying elements that differ by a two-sided ideal.
Radical of an ideal The set of elements whose some power lies in a given ideal.
Reduced ring A commutative ring with no nonzero nilpotent elements.
Regular element An element that is not a zero divisor (equivalently, multiplication by it is injective).
Ring A set with addition forming an abelian group and multiplication that is associative and distributive over addition.
Ring axioms Axioms defining a ring as an abelian group under addition with associative multiplication distributing over addition.
Ring epimorphism A surjective ring homomorphism.
Ring homomorphism A function between rings preserving addition and multiplication.
Ring homomorphisms preserve structure A ring homomorphism preserves addition and multiplication and sends 0 (and 1 for unital maps) to 0 (and 1).
Ring isomorphism A bijective ring homomorphism with a homomorphic inverse.
Ring monomorphism An injective ring homomorphism.
Second isomorphism theorem for rings A subring modulo its intersection with an ideal is isomorphic to its image in the corresponding quotient.
Semiprime ideal An ideal containing no nonzero nilpotent ideal modulo it; in commutative rings, the same as a radical ideal.
Semisimple ring A ring whose module theory is completely reducible; equivalently a finite product of matrix rings over division rings.
Simple ring A ring with no nontrivial two-sided ideals.
Subring A subset of a ring that is itself a ring under the inherited operations.
Sum of ideals The ideal consisting of all sums of an element from each of two ideals.
Third isomorphism theorem for rings Quotienting by an intermediate ideal is the same as quotienting in one step.
Total ring of fractions Localization of a commutative ring obtained by inverting all regular elements (non-zero-divisors).
Two-sided ideal A subset that is simultaneously a left ideal and a right ideal.
UFD implies GCDs exist In a unique factorization domain, any two elements admit a gcd unique up to associates.
Unique factorization domain An integral domain where every element factors uniquely into irreducibles up to associates and order.
Unique factorization theorem In a UFD, every nonzero nonunit factors uniquely into irreducibles up to associates and order.
Unit An element of a unital ring that has a multiplicative inverse.
Unital ring A ring whose multiplication has an identity element.
Unital ring axiom Axiom asserting existence of a multiplicative identity element in a ring.
Units map to units A unital ring homomorphism sends invertible elements to invertible elements.
Universal property of quotient rings A homomorphism that kills an ideal factors uniquely through the quotient.
Wedderburn's little theorem Every finite division ring is commutative, hence a field.
Zero divisor A nonzero element that multiplies with some nonzero element to give zero.

Algebraic geometry foundations46 knowls

Affine line The affine scheme Spec(k[x]) representing one algebraic coordinate over a base field.
Affine n-space The affine scheme Spec(k[x_1,...,x_n]) representing n algebraic coordinates.
Affine scheme A locally ringed space obtained as the prime spectrum of a commutative ring.
Algebraically closed field A field in which every nonconstant one-variable polynomial has a root.
Base change Pulling an object over a base back along a morphism to a new base.
Closed point A point whose singleton is closed in the underlying topological space of a scheme.
Connected scheme A scheme whose underlying Zariski topological space cannot be split into two nonempty open-and-closed pieces.
Constant finite group scheme The group scheme obtained by placing one copy of the base scheme at each element of a finite group.
Covering family in a site A family of morphisms whose generated sieve is covering in the site's Grothendieck topology.
Diagonal morphism The canonical map from a scheme to its fiber square over the target.
Direct image of a sheaf The sheaf on the target whose sections are sections over inverse images of open sets.
Fiber product of schemes The scheme representing pairs of points or maps with the same image over a base.
Finite Galois algebra A finite étale algebra with a group action satisfying the Galois torsor identity.
Finite morphism A scheme morphism that is affine and is locally induced by a ring map making the target a finite module over the source.
Finite étale algebra A finite algebra whose spectrum is étale over the spectrum of the base ring.
Finite étale morphism A scheme morphism that is both finite and étale.
Flat morphism A scheme morphism whose induced homomorphisms on local rings are flat.
G-torsor on a site A sheaf with a locally trivial simply transitive action of a group sheaf.
Galois extension as an étale torsor A finite Galois field extension gives a connected finite étale torsor on spectra.
Galois tensor-product identity For a finite Galois extension, the self-tensor product splits into one copy for each automorphism.
Generic point A point whose closure is an entire irreducible closed subset.
Grothendieck topology A specification of covering sieves on each object of a category, stable under pullback and satisfying local character.
Group scheme A scheme over a base whose multiplication, identity, and inverse are morphisms of schemes.
Local diffeomorphism A smooth map that restricts near every point to a diffeomorphism onto an open neighborhood.
Locally of finite presentation A scheme morphism that is locally induced by finitely presented algebras.
Locally of finite type A scheme morphism that is locally induced by finitely generated algebras.
Locally ringed space A topological space with a sheaf of rings whose stalks are local rings.
Morphism of locally ringed spaces A continuous map with a compatible sheaf map that is local on every stalk.
Morphism of schemes A continuous map of schemes equipped with a compatible local map of structure sheaves.
Morphism of sheaves Compatible maps between the sections of two sheaves.
Proj of a graded ring The scheme built from homogeneous prime ideals of a graded ring.
Projective space A scheme obtained by gluing affine spaces so that directions at infinity are included.
Relative Kähler differentials The module or sheaf that universally records first-order variation relative to a base.
Scheme A locally ringed space covered by open subsets that are affine schemes.
Scheme over a base A scheme equipped with a specified morphism to a fixed base scheme.
Sheaf A system of local data on open sets that can be uniquely glued when compatible.
Sheaf of groups A sheaf whose sections form groups compatibly with restriction.
Sieve on an object A collection of morphisms into one object that is closed under precomposition.
Site A category equipped with a Grothendieck topology.
Small étale site The site of schemes étale over a fixed scheme, covered by jointly surjective étale families.
Stalk The collection of germs near one point of a sheaf.
Structure sheaf The sheaf of rings that supplies the local algebraic functions on a scheme.
Torsor condition The condition that two points in the same fiber differ by a unique group element.
Unramified morphism A locally finite type scheme morphism whose relative differentials vanish.
Étale morphism A scheme morphism that is flat, unramified, and locally of finite presentation.
Étale topology The Grothendieck topology in which jointly surjective families of étale morphisms are covers.

Analysis2 knowls

Absolute continuity A strong continuity condition on an interval controlling total change over collections of small subintervals
Analysis Real analysis, metric spaces, and function spaces.

Asymptotics6 knowls

Asymptotics Asymptotic methods and approximations
Entropy and multinomial coefficients Approximations and bounds relating multinomial coefficients to Shannon entropy.
Laplace's method Asymptotic evaluation of integrals dominated by a single interior maximizer of the exponent.
Method of types Counting and probability estimates for sequences grouped by their empirical distribution.
Saddle-point method Asymptotic evaluation of contour integrals and coefficient formulas using stationary points of the phase.
Stirling's approximation Asymptotic formulas and bounds for factorials and log-factorials for large n.

Convex analysis164 knowls

A nonnegative real below every epsilon is zero If ℓ≥0 and ℓ<ε for all ε>0, then ℓ=0
Affine Hull and Affine Combination The smallest affine set containing Ω, and linear combinations with coefficients summing to 1.
Affine images and preimages of convex sets are convex Affine maps preserve convexity under both images and inverse images
Affine mapping A map of the form x↦Ax+b, i.e., linear plus a translation
Affine Set A set containing the entire line through any two of its points.
Affine Sets are Translates of Subspaces Ω is affine iff Ω−ω is a linear subspace (equivalently, Ω=ω+L).
Algebra of limits in normed spaces Limits commute with addition and scalar multiplication
Algebraic Interior (Core) The algebraic analogue of interior for subsets of vector spaces
Auxiliary Separation Lemma Disjoint convex sets are separable if one has nonempty core and the sets are disjoint.
Balanced and absorbing sets Two scaling properties of subsets in a vector space
Bases are maximal linearly independent sets A nonempty set is a basis iff it is linearly independent and maximal for inclusion
Basic properties of closed sets Intersections of closed sets are closed; finite unions of closed sets are closed
Basic properties of closure Monotonicity, idempotence, and compatibility with finite unions
Basic properties of interior Monotonicity, idempotence, and compatibility with finite intersections
Basic properties of open sets Unions of open sets are open; finite intersections of open sets are open
Basis and dimension A Hamel basis is a linearly independent set that spans the whole vector space
Biconjugate The conjugate of the conjugate, which produces a canonical closed convex minorant of a function.
Bounded Linear Functional and Its Norm A linear functional is bounded iff it is continuous; its operator norm is sup_{||x||≤1}|f(x)|.
Bounded sets and sequences A set is bounded if it lies in some ball; a sequence is bounded if its range is bounded
Cartesian product of convex sets is convex The product Ω1×Ω2 is convex when each factor is convex
Cauchy sequence with a convergent subsequence converges A Cauchy sequence converges if one of its subsequences converges
Cauchy sequences are bounded A Cauchy sequence must lie in some ball
Characterization of affine mappings Affine maps are exactly those that preserve two-point convex combinations
Characterization of direct sums A sum is direct iff every element has a unique decomposition into components
Closed balls are closed In any metric space, every closed ball is a closed set
Closed convex function A convex function whose epigraph is closed, equivalently a lower semicontinuous convex function.
Closed set A set whose complement is open
Closed sets via sequences (proof I) A set is closed iff it contains limits of all convergent sequences from it
Closed sets via sequences (proof II) A set is closed iff it contains limits of all convergent sequences from it
Closure The smallest closed set containing a given set
Closure of intersections under an interior-point condition If convex sets have intersecting interiors, closure distributes over their intersection
Closure via balls A point is in the closure iff every ball around it meets the set
Closure via sequences In metric spaces, a point is in the closure iff it is a limit of a sequence from the set
Codimension The dimension of the quotient space X/L for a subspace L⊂X.
Codimension-One Subspaces Give Direct Sum Decompositions If codim(L)=1 and x0∉L, then X=L⊕span{x0}.
Complete metric space and complete subset A metric space is complete if every Cauchy sequence converges (in the space)
Completeness and closedness Complete subsets are closed; closed subsets of complete spaces are complete
Completeness of R^k Every Cauchy sequence in Euclidean space converges
Complex Separation Theorem (Real Parts) In complex vector spaces, separation holds via the real part of a complex linear functional.
Continuity and Level Sets of the Minkowski Gauge If 0 lies in the interior of a convex set, its gauge is continuous and recovers int(Ω) and cl(Ω).
Continuity of Linear Functionals via Closed Level Sets A linear functional on a normed space is continuous iff one of its level sets is closed.
Convergence implies convergence of norms If x_n→x, then ||x_n||→||x||
Convergence in normed spaces A sequence converges if the norm of its difference to the limit goes to zero
Convergence of a sequence in a metric space A sequence converges if points eventually lie arbitrarily close to the limit
Convergent sequences are bounded A convergent sequence in a metric space must lie in some ball
Convergent sequences are Cauchy Convergence implies the Cauchy property in any metric space
Convex Analysis Convex sets, convex functions, separation theorems, and the Hahn-Banach theorem
Convex combination A weighted average of finitely many points with nonnegative weights summing to one
Convex duality: primal and dual problems Primal and dual convex optimization problems and the relationship between their optimal values.
Convex function via epigraph A function is convex if and only if its epigraph is a convex set
Convex hull The smallest convex set containing a given set
Convex hull is the smallest convex set containing Ω co(Ω) is convex, contains Ω, and lies in every convex superset of Ω
Convex hull via convex combinations The convex hull equals the set of all finite convex combinations of points in Ω
Convex set A set is convex if it contains the line segment between any two of its points
Convex sets via convex combinations A set is convex iff it contains convex combinations of its points
Convexity characterized by monotonicity of the derivative A differentiable function on an interval is convex iff its derivative is nondecreasing
Convexity characterized by positive semidefinite Hessian A C^2 function on an open convex set is convex iff its Hessian is positive semidefinite everywhere
Convexity of the Marginal (Optimal Value) Function Under convexity of the objective and the set-valued map, the value function is convex
Convexity on a convex subset via extension Define convexity on Ω by extending f to X with value ∞ outside Ω
Convexity Preserved Under Affine Composition Precomposition of a convex function with an affine map preserves convexity
Convexity Preserved Under Monotone Convex Composition If f is convex and φ is convex and nondecreasing, then φ∘f is convex
Convexity via nonnegative second derivative A twice differentiable function is convex iff f''≥0 on the interval
Core Characterized by Absorbing Translations A point lies in core(Ω) iff translating Ω by that point makes it absorbing
Core Equals Interior for Convex Sets in Normed Spaces For convex sets with nonempty interior, algebraic and topological interiors coincide.
Core of a Convex Set is Convex Taking algebraic interior preserves convexity
Direct sum of subspaces A sum of subspaces with trivial intersection
Distance function to a set d_Ω(x)=inf{||x−w||: w∈Ω} in a normed space
Domain of a convex function is convex The effective domain dom(f) of a convex function is a convex set
Domain, epigraph, and proper function dom(f) is where f is finite; epi(f) is the set above the graph; proper means dom(f)≠∅
Dual Space and Duality Pairing The continuous dual X and the pairing ⟨x,x⟩=x*(x).
Equivalent characterizations of convex functions Convexity via epigraph is equivalent to Jensen and extended Jensen inequalities
Existence of a basis Every nonzero vector space admits a Hamel basis
Existence of a Norming Functional For any nonzero z0, there is a bounded functional f with ||f||=1 and f(z0)=||z0||.
Extended real number system and conventions Conventions for inf/sup and extended-real-valued functions used in convex analysis
Extension of a linearly independent set to a basis Any nonempty linearly independent set sits inside some Hamel basis
Fenchel conjugate The convex conjugate of an extended-real-valued function, defined by a supremum of affine functionals.
Fenchel-Moreau theorem A closed proper convex function equals its Fenchel biconjugate.
Fenchel-Young inequality An inequality relating a function and its Fenchel conjugate via the dual pairing.
Hahn–Banach Extension Dominated by a Seminorm (Real Case) A real linear functional bounded by a seminorm extends with the same bound.
Hahn–Banach Theorem (Complex Vector Spaces) Complex linear functionals dominated by a seminorm extend to the whole space.
Hahn–Banach Theorem (Real Vector Spaces) A linear functional dominated by a sublinear function extends to the whole space.
Hahn–Banach Theorem in Normed Spaces A bounded linear functional on a subspace extends to the whole space without increasing its norm.
Hyperplane An affine set whose direction subspace has codimension one.
Hyperplanes as Level Sets of Linear Functionals In real vector spaces, Ω is a hyperplane iff Ω={x : f(x)=α} for some f≠0.
Hölder inequality (finite sums) ∑|x_i y_i| is bounded by the product of ℓ^p and ℓ^q norms for conjugate exponents
Hölder inequality (integrals) ∫|fg| ≤ (∫|f|^p)^(1/p)(∫|g|^q)^(1/q) for conjugate exponents
Idempotence of the Core Operator Taking the core twice gives the same set: core(core(Ω))=core(Ω).
Image, kernel, and linear isomorphism The image and kernel of a linear operator; bijective linear maps are isomorphisms
Images and preimages of subspaces under linear maps Linear maps send subspaces to subspaces and pull back subspaces to subspaces
Index bound for subsequences If n1<n2<… are positive integers, then nk≥k
Indicator function of a set The extended-real function that is 0 on Ω and ∞ outside Ω
Interior The largest open set contained in a given set
Interior and closure of a convex set are convex In a normed space, convexity is preserved under interior and closure
Interior and closure relations for convex sets with nonempty interior For convex sets with nonempty interior: cl(int Ω)=cl Ω and int(cl Ω)=int Ω
Interior via balls A point lies in the interior iff a ball around it is contained in the set
Intersections of convex sets are convex Any intersection of convex sets is convex
Intersections of subspaces The intersection of any family of linear subspaces is a linear subspace
Isomorphism theorem for linear operators The image of a linear map is isomorphic to the quotient by its kernel
Kernel of a Nonzero Functional Has Codimension One If f≠0 is linear, then codim(ker f)=1.
Legendre transform A smooth, strict-convex special case of convex conjugation defined via the gradient map.
Legendre–Fenchel transform The general convex-conjugation transform defined by a supremum pairing, without smoothness assumptions.
Line Connecting Two Points The affine line through a and b: {λa+(1−λ)b : λ∈R}.
Line segments in a vector space Segments are sets of convex combinations of two points
Linear Closure The algebraic analogue of closure for subsets of vector spaces
Linear Closure Equals Topological Closure for Solid Convex Sets For convex sets with nonempty interior in a normed space, lin(Ω)=cl(Ω).
Linear Closure of a Convex Set is Convex The set lin(Ω) is convex whenever Ω is convex.
Linear combination A finite sum of scalar multiples of vectors
Linear independence and dependence A set is linearly independent if only the trivial finite linear combination equals zero
Linear operator A map between vector spaces preserving addition and scalar multiplication
Linear subspace A subset closed under addition and scalar multiplication, forming a vector space in its own right
Marginal (Optimal Value) Function The infimum of an objective over a set-valued constraint mapping
Metric and metric space A distance function satisfying positivity, symmetry, and the triangle inequality
Minkowski Function (Gauge) A set-generated sublinear functional pΩ(x)=inf{t≥0 : x∈tΩ}.
Modern Analysis: Lecture Notes and Further Reading Materials Lecture notes on vector spaces, metric spaces, normed vector spaces, convex sets, convex functions, and convex separation
Nonnegative (positive-semidefinite) operator A self-adjoint operator A is nonnegative if ⟨Ax,x⟩≥0 for all x
Norm and normed vector space A norm assigns lengths to vectors and induces a metric
Norm induces a metric (and conversely) A norm defines a metric by d(x,y)=||x−y||; conversely, certain metrics come from norms
Open and closed balls Basic neighborhoods defined by a metric
Open balls are open In any metric space, every open ball is an open set
Open set A set that contains a small open ball around each of its points
Operations on subsets of a vector space Set addition, scalar multiplication, and difference inside a vector space
Operations Preserving Convexity Nonnegative scaling, finite sums, and finite maxima preserve convexity
Parallel Affine Set An affine set Ω is parallel to a subspace L if Ω=ω+L for some ω∈Ω.
Parallel Subspace to an Affine Set is Ω−Ω Every nonempty affine set is parallel to a unique subspace L=Ω−Ω.
Piecewise-polynomial function A function represented by polynomial formulas on a finite semialgebraic cover.
Product space A Cartesian product of vector spaces with componentwise operations
Properties of Affine Sets and Affine Hulls Characterizations and closure properties of affine sets; representation of aff(Ω).
Properties of the Minkowski Gauge of a Convex Set For absorbing convex Ω, pΩ is sublinear and its level sets describe core(Ω) and lin(Ω).
Quasiconvex function A function with f(λx+(1−λ)y)≤max{f(x),f(y)}
Quasiconvexity via convex sublevel sets f is quasiconvex iff all sublevel sets {x: f(x)≤α} are convex
Quotient vector space and codimension A vector space of cosets modulo a subspace; its dimension defines codimension
Segments from Core Points Stay in the Core If a is in core(Ω) and b in Ω, then points on [a,b) remain in core(Ω).
Segments from interior points stay in the interior From an interior point, the segment to any other point stays interior except possibly at the endpoint
Self-adjoint linear operator An operator A with ⟨Ax,y⟩=⟨x,Ay⟩ on an inner product space
Semialgebraic set A subset of real affine space described by finitely many polynomial equalities and inequalities.
Seminorm A subadditive, absolutely homogeneous function p(λx)=|λ|p(x).
Separating a Point from a Convex Set via the Core If x0 is outside core(Ω) and core(Ω)≠∅, then Ω and {x0} are separable by a hyperplane.
Separation by a Closed Hyperplane Separation using a nonzero continuous functional in the dual space.
Separation by a Hyperplane Two sets are separable if a nonzero linear functional orders them.
Separation by Closed Hyperplane Under an Interior Condition If int(Ω1)≠∅ and int(Ω1)∩Ω2=∅, then Ω1 and Ω2 are separable by a continuous functional.
Separation of a Point and a Subspace If a point has positive distance to a subspace, a bounded functional separates them.
Separation of Two Convex Sets via the Core Condition If core(Ω1)≠∅ and core(Ω1) is disjoint from Ω2, then Ω1 and Ω2 are separable by a hyperplane.
Separation via Sup/Inf Inequality Hyperplane separation is equivalent to sup_{Ω1}f ≤ inf_{Ω2}f for some f≠0.
Set-valued mapping (multifunction), domain, graph, and convexity A set-valued map assigns sets to points; convexity is defined via its graph
Slope inequalities for convex functions Secant slopes of a convex function are ordered
Span The smallest linear subspace containing a given set
Span equals finite linear combinations The span of a set consists exactly of its finite linear combinations
Strict Separation by a Closed Hyperplane Strict separation means there is a positive gap between the two sets under a continuous functional.
Strict Separation of Compact and Closed Convex Sets Disjoint compact convex and closed convex sets in a normed space admit strict separation by a continuous functional.
Strict Separation When One Set is Open An open convex set can be separated from a convex set with a strict inequality gap.
Strictly convex function A convex function with strict inequality for distinct points
Subadditive, Positively Homogeneous, and Sublinear Functions Key algebraic properties for gauges and Hahn–Banach domination.
Subdifferential The set of all subgradients of a convex function at a point, defined by global supporting inequalities.
Subgradient A vector that defines an affine global lower bound to a convex function at a point.
Subsequences of convergent sequences converge to the same limit Any subsequence of a convergent sequence converges to the same limit
Subspace test A nonempty subset is a subspace iff it is closed under addition and scalar multiplication
Sum of subspaces and span of the union The sum of two subspaces is a subspace and equals the span of their union
Sums and scalar multiples of convex sets are convex Minkowski sums and dilations preserve convexity
Supporting hyperplane of a convex function An affine function whose graph supports the epigraph of a convex function.
Supremum of Convex Functions The pointwise supremum of any family of convex functions is convex
Uniqueness of limits A sequence in a metric space has at most one limit
Uniqueness of limits and boundedness in normed spaces Limits are unique, and every convergent sequence is bounded
Weighted arithmetic–geometric mean inequality For a,b≥0 and θ∈(0,1): a^θ b^(1−θ) ≤ θa+(1−θ)b
Young's Inequality A conjugate-exponent bound: |xy| is controlled by |x|^p/p + |y|^q/q

Differential geometry8 knowls

Almost-complex structure A smooth tangent-bundle endomorphism whose square is minus the identity.
Complex manifold A space locally modeled on complex Euclidean space with holomorphic transition maps.
Differential Geometry Definitions and results in differential geometry, including manifolds, tangent spaces, and related structures.
Holomorphic map A complex-differentiable map between open subsets or complex manifolds.
Integrable almost-complex structure An almost-complex structure induced locally by holomorphic coordinate charts.
Stokes' theorem Generalization of the fundamental theorem of calculus to differential forms on oriented manifolds with boundary.
Symplectic manifold A smooth manifold equipped with a closed, nondegenerate 2-form.
Tangent Space The vector space of tangent vectors at a point, defined intrinsically using derivations or curves.

Discrete structures7 knowls

Boundary of a finite region Standard notions of boundary for a finite subset of a graph or lattice.
Discrete Structures Graphs and lattice structures for statistical mechanics
Finite box in the lattice A finite cube-shaped subset of the integer lattice used as a finite region.
Finite graph A graph with finitely many vertices (and edges).
Graph: vertices and edges Defines vertices and edges in a graph, along with incidence and adjacency.
Integer lattice Z^d The set of all d-dimensional vectors with integer coordinates.
Nearest-neighbor adjacency on Z^d The standard notion of adjacency on the integer lattice where points differ by 1 in one coordinate.

Fiber bundles256 knowls

Adjoint bundle The associated bundle with fiber G where the structure group acts on G by conjugation, yielding a bundle of groups over the base.
Adjoint bundle Ad(P) The bundle of groups associated to a principal G-bundle via the conjugation action of G on itself.
Adjoint Lie algebra bundle ad(P) The Lie algebra bundle associated to a principal G-bundle via the adjoint representation on the Lie algebra.
Ambrose–Singer curvature span The theorem that the holonomy algebra is generated by curvature values transported back to a basepoint.
Ambrose–Singer holonomy theorem The Lie algebra of the restricted holonomy group is generated by parallel transports of curvature.
Associated bundle A fiber bundle built from a principal bundle and a left group action on a model fiber by taking a quotient of the product.
Associated bundle from a principal bundle and a left G-space Construction of the fiber bundle P×_G F associated to a principal G-bundle and a left G-space.
Associated connection theorem A principal connection induces a compatible connection on every associated bundle and every associated vector bundle.
Associated vector bundle A vector bundle obtained from a principal bundle and a linear representation of its structure group.
Atiyah algebroid of a principal bundle The quotient TP/G with its natural Lie algebroid structure induced by G-invariant vector fields on the total space.
Atiyah algebroid TP/G and its anchor Construction of the quotient bundle TP/G as a Lie algebroid over M with anchor induced by the projection to TM.
Atiyah sequence The short exact sequence 0 to ad(P) to TP/G to TM to 0 associated to a principal bundle.
Basic differential form on a principal bundle A differential form on a principal bundle that is horizontal and invariant, hence the pullback of a unique form on the base.
Basic forms theorem Characterizes which differential forms on a principal bundle descend to the base manifold.
Bianchi identity The covariant exterior derivative of the curvature form of a connection vanishes.
Bundle atlas A collection of compatible local trivializations covering the base of a fiber bundle.
Bundle isomorphism An invertible bundle morphism whose total-space map and base map are diffeomorphisms.
Bundle map A morphism of fibered manifolds, i.e. a smooth map of total spaces compatible with the projections.
Bundle metric A smoothly varying inner product on the fibers of a real vector bundle.
Bundle morphism A map of fiber bundles compatible with the projections and covering a specified base map.
Bundle of connections An affine bundle over a manifold whose sections are connections on a fixed bundle.
Bundle of orbits The quotient of a product P × F by the diagonal action of the structure group, yielding the associated bundle.
Cartan connection A g-valued 1-form on a principal H-bundle that models the geometry of a manifold on a homogeneous space G/H.
Cartan's first structure equation (torsion) in the frame bundle On the frame bundle, the torsion form equals the exterior derivative of the solder form plus the connection form acting on it.
Cartan's second structure equation (curvature) in the frame bundle On the frame bundle, the curvature form is given by d omega plus one half the bracket of omega with itself.
Change of connection formula for Chern Weil characteristic forms Exact formula relating characteristic forms computed from two different principal connections
Characteristic class A de Rham cohomology class of a principal bundle defined from curvature via the Chern–Weil construction.
Chern character via Chern–Weil theory A characteristic class of complex vector bundles defined as the trace of the exponential of curvature; it is additive under direct sum.
Chern class via Chern–Weil theory Characteristic cohomology classes of a complex vector bundle defined from curvature using invariant polynomials.
Chern–Simons form A differential form whose exterior derivative is the difference of two Chern Weil forms.
Chern–Weil classes are independent of the connection Characteristic classes obtained from invariant polynomials in curvature do not depend on the chosen principal connection.
Chern–Weil form A differential form built from the curvature of a principal connection using an invariant polynomial.
Chern–Weil theorem Invariant polynomials in curvature yield closed forms whose cohomology class does not depend on the connection.
Classification of principal G-bundles by homotopy classes of maps into BG Principal G bundles over a paracompact manifold are classified up to isomorphism by homotopy classes of maps into the classifying space BG.
Classifying map of a principal bundle A map from the base into BG whose pullback of EG reproduces a given principal G-bundle.
Classifying space BG A space whose homotopy classes of maps from a base classify principal G-bundles up to isomorphism.
Closed differential form A differential form with vanishing exterior derivative: =0.
Clutching function A map on an overlap used to glue trivial bundles into a global bundle.
Coadjoint action of a Lie group The induced action of a Lie group on the dual of its Lie algebra obtained by dualizing the adjoint action.
Cocycle condition for transition functions The compatibility identities on double and triple overlaps needed to glue a fiber bundle.
Complex vector bundle A smooth vector bundle whose fibers are complex vector spaces and whose transition functions are complex linear.
Conjugation action of a Lie group on itself The smooth action of a Lie group on itself given by sending an element to its conjugate by another element.
Connection 1-form on a principal bundle Definition of a principal connection 1-form and the horizontal distribution it determines.
Connection on a vector bundle A rule for differentiating sections along vector fields, linear over constants and satisfying a Leibniz rule.
Connections on vector bundles via frame bundles Equivalence between covariant derivatives on a rank-n vector bundle and principal connections on its frame bundle.
Construction: Connection on Fr(E) induced by a vector bundle connection (and conversely) Equivalence between covariant derivatives on a vector bundle and principal connections on its frame bundle.
Construction: Frame bundle Fr(E) of a vector bundle E Define the principal GL(n)-bundle of frames of a rank-n vector bundle.
Construction: local trivialization from a local section A local section of a principal bundle determines a canonical local trivialization by multiplying by group elements.
Construction: pullback principal bundle Given a principal bundle P over M and a smooth map f from N to M, the pullback f-star P is a principal bundle over N.
Construction: quotient manifold P/G for a free proper action If a Lie group acts freely and properly on a smooth manifold P, the orbit space P/G is a smooth manifold and the projection is a submersion.
Construction: Splitting of the Atiyah sequence from a principal connection How a principal connection produces a canonical splitting of the Atiyah sequence of a principal bundle.
Convention: Ad(P) uses the conjugation action on G Notation convention that the adjoint bundle is formed using the conjugation action of the structure group on itself
Convention: associated bundles use a left action on the fiber Convention for forming an associated bundle from a right principal action and a left action on the typical fiber
Convention: fundamental vector field uses the right action The fundamental vector field X-sharp is defined by differentiating the right action p·exp(tX).
Convention: local curvature is F = dA + A wedge A The sign and bracket convention relating a local connection 1 form to its local curvature 2 form.
Convention: manifolds are smooth, Hausdorff, and second countable Throughout, a manifold means a smooth Hausdorff second-countable manifold (unless explicitly stated otherwise).
Convention: principal bundles use a right G-action on P A principal G-bundle is written with a right action of G on the total space, matching standard connection and equivariance formulas.
Cotangent bundle The smooth vector bundle whose fiber at each point is the dual of the tangent space.
Covariant derivative of a section The derivative of a vector bundle section along a vector field as defined by a connection.
Covariant exterior derivative on ad(P)-valued forms The exterior derivative on differential forms with values in the adjoint bundle, defined using a principal connection.
Covariant exterior derivative preserves tensoriality For a principal connection, the covariant exterior derivative sends tensorial forms to tensorial forms.
Curvature A measure of the failure of parallel transport to be path-independent, or equivalently, the non-integrability of horizontal distributions.
Curvature 2-form in a frame The matrix of 2-forms computed from a local connection 1-form by dA plus A wedge A.
Curvature 2-form of a principal connection A Lie-algebra-valued 2-form measuring the non-integrability of the horizontal distribution of a principal connection.
Curvature of a vector bundle connection The obstruction to commuting covariant derivatives, yielding an endomorphism-valued 2-form.
Curvature of an induced associated connection via a representation How principal curvature induces curvature on an associated vector bundle through the Lie algebra representation.
de Rham cohomology group The quotient of closed forms by exact forms, measuring global obstructions to solving =.
Diffeomorphism A bijective smooth map with smooth inverse; an isomorphism of smooth manifolds.
Difference of two principal connections is tensorial The difference of two principal connection 1-forms is a tensorial one-form with values in the Lie algebra.
Differential (pushforward) of a smooth map The linear map on tangent spaces induced by a smooth map, satisfying the chain rule.
Differential k-form A smooth alternating covariant k-tensor field; equivalently, a smooth section of the kth exterior power of the cotangent bundle.
Differential of a smooth map The linear map between tangent spaces induced by a smooth map, also called the pushforward.
Dirac monopole connection on the Hopf bundle A principal U(1) connection on the Hopf bundle whose curvature is a nonzero two-form on the 2-sphere.
Direct sum vector bundle (Whitney sum) The bundle over a common base whose fiber is the direct sum of the fibers of two bundles.
Dual vector bundle The vector bundle whose fiber over each point is the dual space of the original fiber.
Ehresmann connection A choice of horizontal subspaces complementary to the vertical tangent spaces of a fibered manifold.
Equivalence of cocycles Two transition function cocycles are equivalent if they differ by a change of local trivializations
Equivalent bundle atlases Two atlases are equivalent if they define the same smooth bundle structure via compatible trivialisations.
Equivalent conditions for reduction of structure group Reduction of a principal G bundle to a subgroup H is equivalent to an H subbundle, H valued transition functions, or a section of the G mod H bundle.
Equivalent conditions for triviality of a principal bundle A principal G bundle is trivial exactly when it has a global section, or equivalently when its transition cocycle is cohomologous to the identity.
Equivalent descriptions of a principal connection A principal connection can be specified by a horizontal distribution, a splitting of the tangent sequence, or a connection one-form.
Equivariant cohomology (Cartan model) A cohomology theory for manifolds with a Lie group action, computed by the Cartan complex of equivariant differential forms.
Equivariant local trivialization A local trivialization of a principal bundle that intertwines the right group action with right multiplication on the model fiber.
Equivariant map A smooth map between G-manifolds that intertwines the group actions.
Equivariant map associated to a section of an associated bundle How a section of an associated bundle corresponds to an equivariant map from the principal bundle to the fiber
Euler class via Chern–Weil theory The top-degree characteristic class of an oriented even-rank real vector bundle defined from curvature using the Pfaffian.
Every principal bundle admits a connection Any principal bundle over a smooth manifold admits at least one principal connection.
Every vector bundle admits a connection Any smooth vector bundle over a smooth manifold admits at least one covariant derivative.
Exact differential form A differential form that is the exterior derivative of another form: =d.
Existence of partitions of unity on paracompact manifolds On a paracompact smooth manifold, every open cover admits a smooth partition of unity subordinate to it.
Exponential map The map from a Lie algebra to its Lie group defined by flowing left-invariant vector fields for unit time.
Extension of structure group A construction that turns a principal G-bundle into a principal H-bundle using a homomorphism from G to H.
Exterior covariant derivative A differential operator on tensorial forms on a principal bundle obtained by differentiating and projecting to horizontal directions.
Exterior derivative The differential operator on differential forms satisfying ^2=0 and the graded Leibniz rule.
Exterior power bundle The vector bundle whose fiber at each point is the k-th exterior power of the original fiber.
Fiber Bundles Differential geometry of fiber bundles, principal bundles, and connections.
Fiber of a map The subset of the domain mapping to a fixed point in the codomain, also called a preimage fiber.
Fiber of a map The preimage of a point under a map, viewed as a subset of the domain.
Fiber-preserving map A smooth map between total spaces that sends fibers to fibers over a base map.
Fibered manifold A smooth manifold E equipped with a surjective submersion onto a base manifold M.
Flat connection on a trivial bundle The product connection on a trivial bundle whose curvature and holonomy are trivial.
Flat principal connection A principal connection whose curvature 2-form vanishes identically.
Flat vector bundle connection A vector bundle connection with zero curvature, admitting local parallel frames and homotopy-invariant transport.
Flatness implies holonomy depends only on homotopy class of loops For a flat connection, holonomy around a loop depends only on the loop's based homotopy class.
Flatness implies path-independence on simply connected domains On a simply connected region where the curvature vanishes, parallel transport depends only on the endpoints.
Frame bundle of a manifold Principal GL(n) bundle of ordered tangent frames on a smooth n-manifold.
Frame bundle of a rank-n vector bundle The principal bundle whose fiber consists of ordered bases of the fibers of a rank-n vector bundle.
Gauge equivalence classes of connections form an orbit space The space of connections modulo gauge transformations is the set of orbits for the gauge group action
Gauge group The gauge group of a principal G bundle is the group of principal bundle automorphisms that cover the identity map of the base.
Gauge group action on connections by pullback Gauge transformations act on principal connections by pulling back the connection one-form.
Gauge transform of a local connection form How a local connection 1-form changes under a change of local section by a G-valued gauge function.
Gauge transformation A principal bundle automorphism that covers the identity map on the base manifold.
Gauge transformation behavior of Chern–Simons forms Under a gauge transformation, a Chern–Simons form changes by an exact term plus a group term, yielding a functional well-defined modulo integers in integral normalizations.
Gauge transformation of local bundle data How transition functions and local connection forms change under a change of local sections.
Good cover An open cover whose nonempty finite intersections are contractible.
Hermitian metric A smoothly varying Hermitian inner product on the fibers of a complex vector bundle.
Holonomy algebra The Lie algebra generated by parallel transport around loops for a given connection.
Holonomy element from parallel transport around a loop Definition of the holonomy element in G obtained by transporting a point around a based loop.
Holonomy group The subgroup of the structure group obtained by parallel transport around loops based at a point.
Holonomy reduction principle If the holonomy of a connection lies in a subgroup H, the principal bundle admits an H-reduction preserved by the connection.
Holonomy representation For a flat connection, the induced representation of the fundamental group into the structure group via parallel transport.
Homotopy class [M,BG] The set of homotopy classes of continuous maps from a manifold M to the classifying space BG.
Hopf fibration as a principal U(1)-bundle The classic circle bundle with total space the 3-sphere and base the 2-sphere.
Horizontal differential form on a principal bundle A differential form on a principal bundle that vanishes whenever any input vector is vertical
Horizontal distribution A smooth choice of horizontal tangent subspaces complementing the vertical spaces in a fiber bundle.
Horizontal lift of a curve A curve in the total space projecting to a base curve and whose velocity is everywhere horizontal.
Horizontal lift of a tangent vector The unique horizontal vector at a point in the total space that projects to a given base tangent vector.
Horizontal lift of a vector field The unique horizontal vector field on the total space that projects to a given vector field on the base.
Horizontal lift of curves and uniqueness Existence and uniqueness of the horizontal lift of a base curve for a given starting point in the total space.
Horizontal subbundle A subbundle of the tangent bundle of a total space that complements the vertical tangent bundle.
Induced connection on an associated bundle via horizontals Construction of an Ehresmann connection on an associated bundle from a principal connection on P.
Induced covariant derivative on sections of an associated vector bundle How a principal connection induces a covariant derivative on sections of an associated vector bundle.
Induced map on associated bundles How a principal bundle morphism induces a map between associated bundles.
Integrable horizontal distribution A horizontal distribution closed under Lie brackets, equivalently tangent to a foliation transverse to the fibers.
Integrality of Chern classes Chern–Weil forms representing Chern classes have integral periods and come from integral cohomology classes.
Interior product The contraction of a differential form with a vector field, lowering degree by one.
Interior product (contraction) ι_X Insertion of a vector field into a differential form, producing a form of one lower degree.
Invariant differential form A differential form preserved by pullback under a Lie group action.
Invariant function A smooth function constant along the orbits of a Lie group action.
Isomorphic principal bundles have the same Chern–Weil classes Chern–Weil characteristic classes agree for isomorphic principal bundles.
Jet bundle (first jets of sections) A bundle whose points record the value and first derivative of a local section at a basepoint.
Left Translation on a Lie Group For g G, the diffeomorphism L_g:G G, L_g(h)=gh, used to transport geometric data by left multiplication.
Leibniz rule for a connection The product rule relating differentiation of a scaled section to derivatives of the function and the section.
Leibniz rule for induced connections on associated bundles The induced covariant derivative on an associated vector bundle is a derivation with respect to multiplying sections by functions.
Lemma: Chern–Weil forms are basic Applying an invariant polynomial to the curvature of a principal connection produces a basic differential form.
Lemma: local curvature transforms by conjugation Under a gauge transformation, the local curvature 2-form is conjugated by the gauge function
Levi–Civita connection as a principal O(n)-connection The unique torsion-free metric-compatible connection on a Riemannian manifold, viewed on the orthonormal frame bundle.
Lie bracket A bilinear alternating operation satisfying the Jacobi identity; for vector fields it is the commutator.
Lie derivative The derivative of a differential form along the flow of a vector field.
Lie derivative of a differential form The derivative {L}_X of a form along a vector field , characterized by Cartan’s formula.
Lie group A group that is also a smooth manifold, with smooth multiplication and inversion.
Lie-algebra-valued k-form A differential form whose values lie in a fixed Lie algebra.
Local connection 1-form A Lie algebra valued 1-form on an open set obtained by pulling back a principal connection along a local section
Local curvature 2-form The curvature 2-form expressed on the base via pullback along a local section.
Local curvature formula Local expression for the curvature of a principal connection in a chosen gauge.
Local frame of a vector bundle A choice of smooth local sections that form a basis of each fiber over an open set.
Local gauge transformation A smooth group-valued function on an open set that represents a gauge transformation in a chosen local trivialization.
Local gauge transformation law for a connection Under a change of local section, the local connection form transforms as A^g = g^{-1}Ag + g^{-1}dg.
Local trivialization A local trivialization identifies a bundle over an open set with a product of that open set and the fiber.
Maurer–Cartan equation for the left Maurer–Cartan form The left Maurer–Cartan form on a Lie group satisfies the structure equation dθ + 1/2[θ∧θ] = 0.
Moment map A map from a Hamiltonian Lie group action to the dual Lie algebra encoding infinitesimal symmetries of a symplectic form.
Naturality of Chern–Weil classes under pullback Chern–Weil forms and their de Rham classes commute with pullback of principal bundles.
Nontrivial principal bundle with no global section Illustration of the fact that a principal bundle is trivial exactly when it admits a global smooth section.
Orbit map The smooth map from a Lie group to a manifold sending a group element to its action on a fixed point.
Orbit of a group action The set of points reachable from a given point under a group action.
Orientation of a real vector bundle A choice of consistent orientation in each fiber of a real vector bundle, varying continuously across the base.
Oriented frame An ordered basis of a real vector space or fiber that is compatible with a chosen orientation.
Orthonormal frame bundle Principal O(n) subbundle of the frame bundle determined by a Riemannian metric.
Orthonormal frame bundle The principal O(n)-bundle of orthonormal frames determined by a bundle metric on a real rank-n bundle.
Paracompact manifold A smooth manifold whose underlying topological space is paracompact, enabling global constructions via partitions of unity.
Paracompact topological space A topological space in which every open cover has a locally finite open refinement.
Parallel section along a curve A section along a curve whose covariant derivative along the curve vanishes.
Parallel transport for an Ehresmann connection Transport along a base curve defined by taking the endpoint of its horizontal lift in the total space.
Parallel transport map along a curve Construction of the parallel transport map determined by a connection along a smooth curve.
Parallel transport respects concatenation of paths Parallel transport along a concatenated path equals the composition of parallel transports along the two pieces.
Partition of unity subordinate to an open cover A locally finite family of smooth functions that sum to one and have supports contained in prescribed open sets.
Pontryagin class via Chern–Weil theory Characteristic cohomology classes of a real vector bundle defined from curvature, using the complexification in Chern–Weil theory.
Principal action A smooth action that is both free and proper.
Principal bundle automorphism A principal bundle isomorphism from a principal bundle to itself, possibly covering a nontrivial base diffeomorphism.
Principal bundle isomorphism An invertible principal bundle morphism, equivalently an equivariant diffeomorphism of total spaces covering a base diffeomorphism.
Principal bundle morphism A smooth equivariant map between principal bundles covering a smooth map of the bases.
Principal bundle over S1 from a clutching function A principal G bundle over the circle can be constructed by gluing a cylinder using a group element or clutching data.
Principal bundle transition function The group-valued cocycle on overlaps that relates two equivariant trivializations of a principal bundle.
Principal connection A G-invariant choice of horizontal subspaces complementing the vertical tangent spaces in a principal bundle.
Principal G-bundle A smooth fiber bundle with a free and transitive right action of a Lie group on each fiber and local trivializations compatible with the action.
Principal H-subbundle An H-invariant submanifold of a principal G-bundle that is itself a principal H-bundle over the same base.
Product principal bundle (fiber product over the base) Construction of a principal G×H-bundle from principal G- and H-bundles over the same base.
Pullback bundle The fiber bundle over N obtained by pulling back a bundle over M along a smooth map f: N to M.
Pullback of covectors The contravariant map on cotangent spaces induced by a smooth map, defined by precomposing with the differential.
Pullback of differential forms Given a smooth map, pull back a k-form by applying the differential to each argument.
Pure gauge connection on a trivial bundle A flat connection obtained from the zero connection by a global gauge transformation.
Quotient manifold (for a free proper action) The smooth manifold structure on an orbit space arising from a free and proper Lie group action.
Quotient space of an action (orbit space) The topological space obtained by identifying points lying in the same orbit of a group action.
Rank of a vector bundle The (constant) dimension of the fibers of a vector bundle, viewed as real or complex vector spaces.
Reducing a GL(n)-structure to O(n) using a bundle metric A fiberwise inner product reduces the structure group of a frame bundle from GL(n) to O(n).
Reduction of structure group A way to replace the structure group G of a principal bundle by a subgroup H by choosing compatible H-frames in each fiber.
Reduction of structure group via H-valued transition functions Constructing a principal H-subbundle when transition functions take values in a subgroup H.
Regular value A point in the target such that the differential is surjective along its fiber.
Reproduction property The connection form evaluates to the generating Lie algebra element on each fundamental vector field.
Restricted holonomy group The identity-component holonomy generated by parallel transport around contractible loops.
Right principal action A smooth right action of a Lie group on a bundle total space that is free and transitive along each fiber.
Right Translation on a Lie Group For g G, the diffeomorphism R_g:G G, R_g(h)=hg, used to transport geometric data by right multiplication.
Section of Ad(P) A smooth choice of an element in each fiber of the adjoint bundle, equivalently a globally defined gauge function with conjugation gluing laws.
Smooth action of a Lie group on a manifold A smooth map defining a group action of a Lie group on a smooth manifold.
Smooth atlas A covering by coordinate charts whose overlap transition maps are smooth.
Smooth chart A local coordinate map from an open subset of a smooth manifold to an open subset of Euclidean space.
Smooth chart (coordinate chart) A homeomorphism from an open subset of a manifold to an open subset of Euclidean space, providing local coordinates.
Smooth embedding A smooth map that is an injective immersion and a homeomorphism onto its image.
Smooth fiber bundle A surjective submersion that is locally a product with a fixed model fiber.
Smooth immersion A smooth map whose differential is injective at every point.
Smooth manifold A topological manifold equipped with a maximal smooth atlas, enabling calculus in local coordinates.
Smooth map A map between smooth manifolds that becomes an ordinary smooth function in local coordinates.
Smooth submersion A smooth map whose differential is surjective at every point.
Solder form on the frame bundle The canonical R^n-valued 1-form on the frame bundle that identifies horizontal directions with tangent vectors on the base.
Special orthonormal frame bundle The principal SO(n)-bundle of oriented orthonormal frames for an oriented metric real rank-n bundle.
Special unitary frame bundle The principal SU(n)-bundle obtained by restricting to unitary frames with determinant one.
Splitting of the Atiyah sequence A right inverse TM to TP/G that is equivalent to choosing a principal connection.
Symmetric power bundle The vector bundle whose fiber at each point is the k-th symmetric power of the original fiber.
Tangent bundle The smooth vector bundle whose fiber at p is the tangent space T_pM.
Tangent space at a point The vector space of tangent vectors to a smooth manifold at a given point.
Tensor product vector bundle The bundle over a common base whose fiber is the tensor product of the fibers of two bundles.
Tensorial forms and ad(P)-valued forms Equivalence between horizontal equivariant Lie-algebra-valued forms on a principal bundle and differential forms on the base with values in the adjoint bundle.
TFAE: Flat principal bundles (principal G-bundle with connection) Equivalent conditions for a principal bundle connection to be flat, including vanishing curvature and homotopy-invariant parallel transport.
TFAE: Metric-compatible connections on a metric vector bundle Equivalent conditions for a connection to preserve a fiber metric, including skew connection forms and isometric parallel transport.
The tangent bundle of the 2-sphere is nontrivial The tangent bundle of the 2-sphere is a rank-2 real vector bundle that admits no global nowhere-zero vector field.
Theorem: A trivial principal bundle admits a global section Any principal bundle isomorphic to a product bundle has a canonical global section.
Theorem: Existence and uniqueness of horizontal lifts of curves Given a connection, any curve in the base has a unique horizontal lift through a chosen point in the fiber.
Theorem: Existence of principal connections on smooth manifolds Every principal bundle over a smooth manifold admits a principal connection, using partitions of unity.
Theorem: Global section implies a principal bundle is trivial A principal bundle admitting a smooth global section is isomorphic to the product bundle.
Theorem: Parallel transport defines a G-equivariant map between fibers Parallel transport along a curve yields a right G-equivariant diffeomorphism between principal bundle fibers.
Theorem: Principal connections are equivalent to splittings of the Atiyah sequence A principal connection is the same as a vector bundle splitting of the Atiyah sequence of a principal bundle.
Theorem: Pullback of a principal bundle is a principal bundle The pullback construction sends principal bundles to principal bundles functorially in the base map.
Theorem: Pullback of a principal connection is a principal connection A principal connection pulls back along a smooth map to a canonical connection on the pullback bundle.
Theorem: Reduction by cocycle (H-reduction iff H-valued transition functions exist) A principal G-bundle reduces to a subgroup H exactly when its transition functions can be chosen to land in H.
Torsion 2-form The R^n-valued 2-form on a frame bundle that measures failure of a connection to be torsion-free.
Transgression form A differential form whose exterior derivative is the difference of two characteristic forms coming from different connections
Transgression theorem (Chern–Simons) The difference of Chern–Weil forms for two connections is exact, with an explicit transgression form.
Transition function The change-of-trivialization data on overlaps, encoding how local bundle charts glue.
Transition functions from local sections How local sections determine transition functions on overlaps in a principal bundle.
Transition matrix of a local frame The matrix-valued function describing how two local frames are related on an overlap.
Trivial fiber bundle A fiber bundle globally isomorphic to a product M times F over the base.
Trivial principal bundle The product principal bundle M times G with its standard projection and right action.
Trivial vector bundle The product bundle M times V with constant fiber V and a global frame of constant sections.
Typical fiber A chosen model manifold F that locally represents every fiber of a smooth fiber bundle.
Unitary frame bundle The principal U(n)-bundle of unitary frames determined by a Hermitian metric on a complex rank-n bundle.
Universal principal bundle EG→BG A canonical principal G-bundle whose pullbacks classify principal G-bundles over paracompact bases.
Vector bundle A smooth fiber bundle whose fibers are vector spaces and whose local trivializations are fiberwise linear.
Vector bundle morphism A smooth map between total spaces of vector bundles that covers a base map and is linear on each fiber.
Vector field A smooth section of the tangent bundle; equivalently, an assignment of a tangent vector to each point varying smoothly.
Vertical subbundle The smooth subbundle of TE consisting of vectors tangent to the fibers of a surjective submersion.
Vertical tangent space The subspace of a tangent space consisting of vectors tangent to a fiber of a surjective submersion.
Vertical vector field A vector field on the total space of a fibered manifold that is tangent to every fiber.
Wedge product of differential forms An alternating product that combines a -form and an -form into a (k+)-form.
Yang–Mills connection A connection whose curvature is a critical point of the Yang–Mills functional, equivalently satisfying the Yang–Mills equation.
Yang–Mills equation The Euler–Lagrange equation for the Yang–Mills functional, expressed as a covariant divergence-free condition on curvature.
Yang–Mills functional The energy of a connection defined as the L2 norm of its curvature on a Riemannian manifold.

Knowlification3 knowls

Galois Extensions and Bundles — knowlified transcript A full conversation transcript with expandable definitions for its mathematical terminology.
New knowls for Galois Extensions and Bundles The new definitions and bridge results added while knowlifying the Galois Extensions and Bundles conversation.
New knowls for the conjectures collection The definitions added while knowlifying twelve selected formal conjectures and open problems.

Langlands letter36 knowls

$L$ -Group and Satake Parameter The semidirect product and the conjugacy class encoding unramified local data
$p$ -Adic Field A finite extension of with ring of integers and residue field
Adeles and Restricted Products The adele ring used for automorphic forms
Automorphic Form and Hecke Eigenvalues A function on whose unramified Hecke action yields Satake parameters
Borel–Mostow Normalizer Representative (Semisimple Class) Choosing representatives of semisimple classes in a torus normalizer, as used to parametrize Hecke characters
Characters Separate Semisimple Conjugacy Classes In a complex reductive group, semisimple classes are determined by values of irreducible characters
Chevalley Basis A root-adapted Lie algebra basis with integral structure constants
Chevalley Lattice and Integral Model A -lattice stable under a Chevalley -form, giving at good primes
Choosing Embeddings $\\overline{\\mathbb{Q}}\\hookrightarrow \\overline{\\mathbb{Q}}_p$ How a choice of -adic embedding fixes a decomposition group and conjugates Frobenius/Satake data
Contragredient (Dual) Representation The representation on given by
Coroots and the Weight–Coroot Pairing The integers that control dominance and duality
Dual Lattice The -dual and its role in dual root data
Eisenstein Series on a Reductive Group A series induced from a parabolic whose analytic continuation produces -functions
Euler Product and Determinant Local $L$ -Factor An -function defined as at unramified primes
Galois Descent, Twisted Forms, and Inner Forms Constructing -groups from -groups using a Galois action and a 1-cocycle
Galois Extension and Galois Group A finite extension that is normal and separable, with group
Global and Local Fields; Completions Number fields and their completions at places (e.g. , )
Group Algebra of a Lattice and Multiplicative Basis The algebra with basis elements and
Ideles, Hecke Characters, and Artin Reciprocity The idele class group and its link to abelian Galois groups; source of abelian -series
Langlands Dual Group The complex reductive group with dual root datum, denoted (called in the letter)
Langlands Functoriality and $L$ -Homomorphisms Maps that push forward Satake parameters
Langlands Letter An annotated reading of Langlands' letter to Weil
Langlands Letter Definitions Mathematical definitions for the Langlands letter
Langlands' Letter to Weil The 1967 letter that launched the Langlands program
Maximal Compact and Hyperspecial Subgroup Compact open subgroups ; hyperspecial at good places
Maximal Torus and Weight Lattice A maximal torus and its character lattice
Nonabelian $H^1(\\Gamma,G)$ and 1-Cocycles Cocycles with classify inner forms
Pinning and Pinned Automorphisms A choice of rigidifying and its outer automorphisms
Root Lattice, Weight Lattice, and Isogeny Forms How lattices between and parametrize central isogenies
Roots, Weyl Group, and Dominant Weights Roots , Weyl group , and the dominant chamber
Semisimple Element and Semisimple Conjugacy Class Elements diagonalizable in representations; conjugacy classes used for Satake parameters
Simply Connected Semisimple Algebraic Group A semisimple group with no nontrivial central isogeny covers (in the sense of algebraic groups)
Spherical Hecke Algebra and Satake Isomorphism The convolution algebra and its identification with functions on the dual torus
Split Reductive Algebraic Group A connected affine algebraic group with trivial unipotent radical and a split maximal torus
Unramified Extension of a $p$ -Adic Field A finite extension with ramification index , controlled by residue fields
Unramified Prime and Frobenius Element The conjugacy class in a Galois group controlling unramified local factors

Large deviations13 knowls

Contraction principle How a large deviation principle transfers through a continuous mapping.
Cramér transform The convex dual of a log moment generating function, giving a canonical large-deviation rate function.
Cramér's theorem Large deviations for empirical means of independent identically distributed real random variables.
Exponential tightness A compact-containment condition ensuring probabilities outside compacts decay exponentially fast.
Good rate function A rate function whose sublevel sets are compact.
Gärtner–Ellis theorem A large deviation principle obtained from limits of scaled log moment generating functions.
Laplace principle A variational limit for exponential integrals that encodes large-deviation behavior.
Large deviation principle Asymptotic exponential bounds for probabilities of rare events at a given speed.
Large Deviations Large deviation principles and rate functions
Log moment generating function The logarithm of the moment generating function, viewed as a convex functional of the parameter.
Rate function A lower semicontinuous function that governs exponential decay rates in large deviations.
Sanov's theorem Large deviations for empirical measures of an independent identically distributed sample.
Varadhan's lemma Asymptotic evaluation of exponential integrals under a large deviation principle.

Lie groups159 knowls

Abelian Lie algebra A Lie algebra whose bracket vanishes identically.
Abelian Lie group A Lie group with commutative multiplication.
Ad-invariance of the Killing form The Killing form satisfies B([x,y],z)=B(x,[y,z]).
Adjoint Action of a Lie Group The conjugation action of a Lie group on itself and the induced linear action on its Lie algebra.
Adjoint Representation of a Lie Algebra The representation sending an element to the linear map given by bracketing with it.
Adjoint representation: discrete kernel iff discrete center For connected Lie groups, ker(Ad)=Z(G), so Ad has discrete kernel exactly when the center is discrete.
Ado’s theorem Every finite-dimensional Lie algebra over characteristic 0 has a faithful finite-dimensional representation.
Baker–Campbell–Hausdorff formula A Lie series for the product exp(X)exp(Y) expressed as exp(BCH(X,Y)).
Bi-invariant differential form A differential form on a Lie group invariant under both left and right translations.
Bi-invariant metric A Riemannian metric on a Lie group invariant under left and right translations.
Bi-invariant metrics on compact Lie groups A compact Lie group always admits a bi-invariant Riemannian metric by averaging.
Cartan matrix The integer matrix encoding the simple-root geometry of a semisimple Lie algebra.
Cartan subalgebra A maximal nilpotent, self-normalizing subalgebra; in the semisimple case, a maximal toral subalgebra.
Cartan subalgebras are self-normalizing If h is a Cartan subalgebra, then its normalizer in g equals h.
Cartan’s criterion for semisimplicity A finite-dimensional Lie algebra over characteristic 0 is semisimple iff its Killing form is nondegenerate.
Cartan’s criterion for solvability A Lie algebra over characteristic 0 is solvable iff a certain trace pairing vanishes on g × [g,g].
Center of a Lie algebra Elements that bracket to zero with everything; equivalently, the kernel of ad.
Center of a Lie group Elements commuting with all group elements; a closed normal subgroup.
Center of a simple Lie algebra is trivial A simple Lie algebra has zero center, since the center is always an ideal.
Classification of complex simple Lie algebras Complex simple Lie algebras are classified by connected Dynkin diagrams of types A–G.
Closed subgroup of a Lie group A subgroup that is closed in the topology of the ambient Lie group.
Closed subgroup theorem A closed subgroup of a Lie group is an embedded Lie subgroup, and the quotient G/H is a smooth manifold.
Coadjoint representation of a Lie algebra The dual of the adjoint representation, acting on the dual space g*.
Commutator subgroup of a Lie group The subgroup generated by commutators, governing the abelianization of a Lie group.
Compact Lie algebra is reductive The Lie algebra of a compact Lie group splits as center ⊕ semisimple part.
Compact Lie group A Lie group that is compact as a manifold (equivalently, as a topological group).
Completely reducible Lie representation A representation that splits as a direct sum of irreducible subrepresentations.
Conjugation action of a Lie group The smooth action of a Lie group on itself by conjugation.
Connected Lie group A Lie group whose underlying smooth manifold is connected (equivalently, equal to its identity component).
Connected subgroup determined by its Lie algebra In a Lie group, a connected Lie subgroup is uniquely determined by its Lie algebra.
Coset space The quotient space of left cosets, a smooth manifold when is a closed Lie subgroup.
Covering Lie group A Lie group homomorphism that is a covering map; its kernel is discrete and central and it induces an isomorphism of Lie algebras.
Derivation of a Lie algebra A linear map with ; derivations form a Lie algebra containing the inner derivations.
Derived series of a Lie algebra The descending chain , used to define solvability.
Derived subalgebra The Lie subalgebra spanned by commutators; it measures how far is from abelian.
Derived subalgebra is an ideal For any Lie algebra g, its commutator subalgebra [g,g] is an ideal of g.
Differential of a Lie group homomorphism If is a Lie group homomorphism, then is a Lie algebra homomorphism.
Direct sum of Lie algebras The product vector space with componentwise bracket, modeling Lie algebras of product groups.
Discrete subgroup A subgroup that is discrete in the manifold topology; its Lie algebra is .
Dual (contragredient) representation Given a representation on , the induced representation on is ; infinitesimally, .
Dynkin diagram A graph encoding the angles and relative lengths among simple roots of a semisimple Lie algebra via the Cartan matrix.
Effective action A Lie group action with trivial kernel; equivalently, the only element acting as the identity on the space is .
Equivalent characterizations of nilpotency for Lie algebras Nilpotency can be tested via the lower central series, Engel’s condition on adjoints, or strict upper-triangular models.
Equivalent characterizations of semisimplicity for Lie algebras Semisimplicity is equivalent to nondegeneracy of the Killing form and to decomposition into simple ideals.
Equivalent characterizations of solvability for Lie algebras Solvability can be detected via the derived series, triangular representations, or Cartan’s trace criterion.
Example: The 3D simple Lie algebra of traceless complex matrices with standard relations.
Example: $SU(2)$ and its Lie algebra is simply connected; is 3D with Pauli-matrix commutators, and is a 2-fold cover.
Example: $U(1)$ (the circle group) has Lie algebra and exponential map with kernel .
Example: and rotations The Lie algebra of consists of real skew-symmetric matrices; exponentials are rotation matrices.
Example: strictly upper triangular matrices Strictly upper triangular matrices form a nilpotent Lie algebra under commutator; commutators move entries further above the diagonal.
Example: the Heisenberg Lie algebra A 3D nilpotent Lie algebra with basis and bracket .
Example: the sphere as a homogeneous space The -sphere is a homogeneous space via the standard transitive action.
Example: the torus $T^n$ The -torus is a connected abelian Lie group with Lie algebra and exponential map .
Example: upper triangular matrices (a solvable Lie algebra) Upper triangular matrices form a Lie algebra whose derived subalgebra is strictly upper triangular, giving an explicit derived series.
Existence of universal covering groups Every connected Lie group admits a unique (up to isomorphism) simply connected covering group compatible with multiplication.
Exponential map is a local diffeomorphism For any Lie group G, exp is a diffeomorphism from a neighborhood of 0 in its Lie algebra onto a neighborhood of the identity in G.
Exponential map of a Lie group The map sending a Lie-algebra element X to the time-one value of its one-parameter subgroup.
Exponentials and one-parameter subgroups The curve t ↦ exp(tX) is the unique one-parameter subgroup with initial velocity X.
Free smooth Lie group action A Lie group action is free if all stabilizers are trivial.
Fundamental representation An irreducible highest-weight representation whose highest weight is a fundamental weight.
General linear group The Lie group GL(V) of invertible linear maps on a finite-dimensional vector space.
General linear Lie algebra The Lie algebra gl(V) of all endomorphisms with commutator bracket.
Heisenberg group The basic nonabelian nilpotent Lie group, central extension of an abelian group.
Highest weight A dominant maximal weight that labels irreducible representations of semisimple Lie algebras.
Highest-weight representation A representation generated by a vector annihilated by the positive root spaces.
Highest-weight theorem Finite-dimensional irreducibles of a semisimple Lie algebra are classified by dominant integral highest weights.
Homogeneous space A manifold with a transitive Lie group action; equivalently a quotient G/H by a stabilizer.
Ideal in a Lie algebra A Lie subalgebra stable under bracketing with the whole algebra.
Inner derivation A derivation of the form ad_x(y) = [x,y].
Irreducible representation of a Lie algebra A representation with no nontrivial invariant subspaces.
Irreducible representation of a Lie group A group representation with no nontrivial invariant subspaces.
Kernel of the group adjoint representation For a connected Lie group, ker(Ad) equals the center.
Kernel of the Lie-algebra adjoint representation The kernel of the adjoint representation ad is the center of the Lie algebra.
Killing form The invariant bilinear form B(x,y)=tr(ad_x ad_y) on a Lie algebra.
Killing form nondegeneracy criterion A finite-dimensional Lie algebra is semisimple iff its Killing form is nondegenerate.
Left Maurer–Cartan form The canonical g-valued 1-form θ^L = (dL_{g^{-1}})_g on a Lie group.
Left Translation The diffeomorphism of a Lie group given by multiplying on the left by a fixed element.
Left-invariant differential form A differential form on a Lie group fixed by all left translations.
Left-Invariant Vector Field A vector field on a Lie group that is unchanged by all left translations.
Left-invariant vector fields form the Lie algebra Left-invariant vector fields are closed under bracket and identify with T_eG.
Levi decomposition Any finite-dimensional Lie algebra splits as a semidirect product of semisimple part and solvable radical.
Lie Algebra A vector space with a bilinear bracket operation that is antisymmetric and satisfies the Jacobi identity.
Lie algebra automorphism An invertible linear map preserving the Lie bracket.
Lie algebra homomorphism A linear map between Lie algebras that preserves the Lie bracket.
Lie algebra isomorphism A bijective Lie algebra homomorphism (equivalently, a bracket-preserving linear isomorphism).
Lie Algebra of a Lie Group The tangent space at the identity of a Lie group, equipped with a canonical bracket from invariant vector fields.
Lie algebra of a product The Lie algebra of a product Lie group is the direct sum of the Lie algebras.
Lie algebra of a subgroup lemma A Lie subgroup has Lie algebra equal to its tangent space at the identity, viewed as a Lie subalgebra.
Lie correspondence Connected Lie subgroups correspond to Lie subalgebras via the tangent space at the identity.
Lie Group Homomorphism A smooth map between Lie groups that is also a group homomorphism.
Lie Groups and Lie Algebras Core theory of Lie groups, Lie algebras, representations, and structure theory at graduate level.
Lie subalgebra A linear subspace closed under the Lie bracket.
Lie Subgroup A subgroup of a Lie group that carries a compatible immersed submanifold structure.
Lie’s third theorem Every finite-dimensional Lie algebra is the Lie algebra of a connected, simply connected Lie group.
Logarithm map A local inverse to the exponential map near the identity of a Lie group.
Lorentz group The group of linear transformations preserving the Minkowski bilinear form.
Lower central series of a Lie algebra A descending sequence defined by iterated commutators, used to define nilpotent Lie algebras.
Maurer–Cartan equation The structure equation satisfied by the Maurer–Cartan form on a Lie group.
Maximal torus theorem In a compact connected Lie group, maximal tori exist and are all conjugate.
Nilpotent implies solvable Every nilpotent Lie algebra is solvable (derived series terminates).
Nilpotent Lie algebra A Lie algebra whose lower central series reaches zero after finitely many steps.
Normal Lie subgroup A Lie subgroup invariant under conjugation; infinitesimally, it corresponds to an ideal.
One-parameter subgroup A smooth homomorphism from (R,+) into a Lie group, generated by a Lie algebra element.
One-parameter subgroups as integral curves Exponentials give flows of invariant vector fields; invariant flows recover one-parameter subgroups.
Orbit of a Lie group action The set of points reachable from x under the action; it is an immersed homogeneous space.
Orbit space The quotient of a G-manifold by the equivalence relation of lying in the same orbit.
Orthogonal group The Lie group of linear transformations preserving a nondegenerate symmetric bilinear form.
Orthogonal Lie algebra The Lie algebra of the orthogonal group: skew-symmetric endomorphisms (or their indefinite analogues).
Outer derivation A derivation not arising as an inner derivation; measured by Der(g)/ad(g).
Peter–Weyl theorem Finite-dimensional unitary representations of a compact Lie group span the regular representation.
Poincaré group The isometry group of Minkowski space: translations semidirect the Lorentz group.
Positive root A choice of “half” of a root set, compatible with addition, used to organize roots into positive and negative.
Principal Homogeneous Space A space with a free and transitive action of a Lie group, also called a torsor.
Product Lie group The Cartesian product of Lie groups, with componentwise multiplication, is again a Lie group.
Proper action A smooth Lie group action is proper if the action graph map is proper; this guarantees good quotient behavior.
Quotient Lie algebra If i is an ideal in g, then g/i inherits a canonical Lie bracket.
Quotient Lie group If N is a closed normal Lie subgroup of G, then G/N carries a natural Lie group structure.
Reductive Lie algebra A Lie algebra that decomposes as a direct sum of its center and a semisimple ideal.
Representation of a Lie Algebra A Lie algebra homomorphism from a Lie algebra to endomorphisms of a vector space.
Representation of a Lie Group A smooth homomorphism from a Lie group to the group of invertible linear maps on a vector space.
Right Maurer–Cartan form The canonical g-valued 1-form on a Lie group obtained by translating tangent vectors to the identity on the right.
Right Translation The diffeomorphism of a Lie group given by multiplying on the right by a fixed element.
Right-invariant differential form A differential form on a Lie group fixed by all right translations, determined by its value at the identity.
Right-Invariant Vector Field A vector field on a Lie group that is unchanged by all right translations.
Root of a Lie algebra A nonzero weight for the adjoint action of a Cartan subalgebra on a semisimple Lie algebra.
Root space The eigenspace g_α for the adjoint action of a Cartan subalgebra corresponding to a root α.
Root space decomposition Decomposition of a semisimple Lie algebra into a Cartan subalgebra plus root spaces for the adjoint action.
Root system A finite set of vectors closed under reflections and satisfying integrality; the combinatorial data behind semisimple Lie theory.
Schur orthogonality for compact Lie groups Matrix coefficients of distinct irreducible unitary representations are orthogonal in L²(G), with a sharp normalization.
Semisimple Lie algebra A Lie algebra with no nonzero solvable ideals; equivalently, one with nondegenerate Killing form (char 0).
Semisimple Lie algebra as a direct sum of simple ideals A finite-dimensional semisimple Lie algebra splits uniquely as a direct sum of simple Lie algebras.
Simple Lie algebra A non-abelian Lie algebra with no ideals other than 0 and itself.
Simple root A minimal positive root; simple roots form a basis for the root system and generate all positive roots.
Simply connected Lie group A Lie group whose underlying manifold is simply connected (connected with trivial fundamental group).
Simply connected Lie groups are determined by their Lie algebras Connected simply connected Lie groups with isomorphic Lie algebras are isomorphic as Lie groups.
Smooth action of a Lie group A Lie group action on a manifold given by a smooth map G×M→M satisfying the action axioms.
Solvable Lie algebra A Lie algebra whose derived series eventually becomes zero; the Lie-algebra analogue of solvable groups.
Special linear group The matrix Lie group SL(n,F) of determinant-1 invertible matrices.
Special linear Lie algebra The Lie algebra sl(n,F) of trace-zero matrices with bracket [X,Y]=XY−YX.
Special orthogonal group The determinant-1 subgroup of the orthogonal group preserving a quadratic form.
Special unitary group The compact matrix Lie group SU(n) preserving a Hermitian form with determinant 1.
Special unitary Lie algebra The Lie algebra of : traceless skew-Hermitian matrices with the commutator bracket.
Spin group For n at least 2, the connected double cover Spin(n) of SO(n), constructed inside the even Clifford algebra.
Stabilizer (isotropy subgroup) in a Lie group action For a smooth action , the stabilizer fixes a point and is a closed Lie subgroup.
Structure of compact connected Lie groups A compact connected Lie group is a torus times a compact semisimple group modulo a finite central subgroup.
Structure of connected abelian Lie groups Every connected abelian Lie group is isomorphic to R^n × T^m.
Subrepresentation of a Lie algebra An invariant subspace for a Lie algebra representation, i.e. a -submodule.
Symplectic group The Lie group of linear transformations preserving a nondegenerate alternating form on .
Symplectic Lie algebra The Lie algebra of the symplectic group: matrices satisfying with commutator bracket.
Tensor product of representations The diagonal action on : for Lie algebras acts by Leibniz, for Lie groups by tensoring operators.
Transitive Lie group action A smooth action is transitive if it has a single orbit; equivalently for a stabilizer .
Unitary group $U(n)$ The compact Lie group of complex matrices preserving the standard Hermitian inner product.
Unitary Lie algebra The Lie algebra of : skew-Hermitian matrices with the commutator bracket.
Universal covering group A simply connected covering Lie group of a connected Lie group , unique up to isomorphism.
Weight of a representation A functional occurring as a simultaneous eigenvalue for the action of a Cartan subalgebra.
Weight space For a representation relative to a Cartan subalgebra, is the simultaneous eigenspace with weight .
Weights in the dual Cartan Weights are elements of ; integrality conditions define weight lattices tied to maximal tori and characters.
Weyl group A finite reflection group defined as (or via root reflections) acting on the Cartan and its dual.
Weyl’s theorem on complete reducibility Finite-dimensional representations of semisimple Lie algebras (and compact Lie groups) split as direct sums of irreducibles.

Linear algebra36 knowls

Banach space A complete normed vector space.
Basis existence theorem Every vector space has a basis.
Bilinear form A function of two vector arguments that is linear in each argument.
Cauchy–Schwarz inequality In an inner product space, the absolute value of an inner product is at most the product of norms.
Cayley–Hamilton theorem A square matrix satisfies its own characteristic polynomial.
Characteristic polynomial Polynomial det(tI - A) attached to a square matrix or linear operator.
Closed linear subspace A linear subspace that is closed in the topology induced by the ambient norm.
Compact operator A linear operator whose unit ball image has compact closure.
Determinant A scalar invariant of a square matrix measuring volume scaling and invertibility.
Eigenspace Set of vectors sent to scalar multiples of themselves for a fixed eigenvalue.
Eigenvalue A scalar for which a linear operator has a nonzero vector it only scales.
Eigenvector A nonzero vector that is scaled by a linear operator.
Euclidean norm The norm induced by an inner product on a Euclidean space.
Euclidean space A finite-dimensional real inner product space.
Hilbert space A complete inner product space.
Inner product A positive-definite product on a vector space that defines lengths and angles.
Inner product space A vector space equipped with an inner product.
Invariant subspace of an operator A linear subspace mapped into itself by a linear operator.
Linear Algebra Vector spaces, linear maps, inner products, and spectral theory
Linear map A function between vector spaces that respects addition and scalar multiplication.
Linear operator A linear map from a vector space to itself.
Matrix A rectangular array of numbers, symbols, or expressions arranged in rows and columns.
Minimal polynomial Smallest-degree monic polynomial that annihilates a linear operator.
Norm A function assigning a nonnegative length to vectors.
Normed vector space A vector space together with a norm, giving a notion of distance and convergence.
Operator norm Norm of a linear map defined by its maximal expansion of unit vectors.
Orthogonal complement The subspace of vectors orthogonal to every vector in a given subset.
Orthogonal projection The bounded linear operator projecting a Hilbert space onto a closed subspace along its orthogonal complement.
Orthogonality Condition that two vectors have inner product equal to zero.
Orthonormal basis A basis whose vectors have unit length and are pairwise orthogonal.
Rank The rank of a linear map or matrix is the dimension of its image.
Rank–nullity theorem For a linear map on a finite-dimensional space, dimension equals rank plus nullity.
Surjective linear isometry A bijective linear map that preserves the norm exactly.
Trace Sum of diagonal entries of a square matrix, invariant under change of basis.
Unit sphere of a normed space The set of vectors having norm exactly one.
Vector space A set with addition and scalar multiplication satisfying the vector space axioms.

Measure theory49 knowls

Almost everywhere Holding except on a set of measure zero.
Almost everywhere convergence Convergence of functions pointwise outside a null set.
Almost-everywhere equality Two functions are a.e. equal if they differ only on a null set.
Borel sigma-algebra The sigma-algebra generated by the open sets of a topological space.
Carathéodory construction A method that turns an outer measure into a measure by selecting Carathéodory measurable sets.
Carathéodory measurable set A set that satisfies Carathéodory’s splitting condition for an outer measure.
Change of variables for pushforward measures Identity relating integrals with respect to a pushforward measure to composition with the underlying map.
Characteristic function (indicator function) The function that records membership in a set by 0/1 values.
Continuity from above For decreasing measurable sets, the measure of the intersection is the limit of the measures under a finiteness hypothesis.
Continuity from below For increasing measurable sets, the measure of the union is the limit of the measures.
Convergence in $L^p$ Norm convergence in an Lp space.
Convergence in measure A mode of convergence where the set of large errors has measure tending to zero.
Dominated convergence theorem If measurable functions converge almost everywhere and are dominated by an integrable function, then integrals and L1 norms converge.
Essential supremum Least upper bound of a measurable function after ignoring a null set.
Fatou's lemma For nonnegative measurable functions, the integral of the liminf is bounded by the liminf of the integrals.
Fubini's theorem Interchange of iterated integrals for absolutely integrable functions on a product measure space.
Indicator function A function that equals 1 on a set and 0 outside it.
Jensen's inequality for integrals A convexity inequality comparing a convex function of an average with the average of a convex function.
Jordan content A finite-additivity notion of volume for certain bounded subsets of Euclidean space.
L-infinity function A measurable function that is essentially bounded on a measure space.
L^1 function A measurable function with finite integral of absolute value, modulo a.e. equality.
L^p norm Norm from integrating the pth power of absolute value, or essential supremum when p is infinity.
L^p space Measurable functions with finite Lp norm, identified up to equality almost everywhere.
Lebesgue criterion for Riemann integrability A bounded function on a closed interval is Riemann integrable exactly when its discontinuities form a Lebesgue null set.
Lebesgue integrable function A measurable function whose absolute value has finite Lebesgue integral.
Lebesgue integral Integral of a measurable function defined from its positive and negative parts.
Lebesgue integral of a nonnegative function Definition of the Lebesgue integral for nonnegative measurable functions.
Lebesgue measure The standard complete translation-invariant measure on Euclidean space built from covering by rectangles.
Lebesgue number lemma refinement lemma On a compact set, an open cover can be refined by finitely many small balls subordinate to it
Measurable function A function whose preimages of measurable sets are measurable.
Measurable rectangle A product set whose factors are measurable in their respective spaces.
Measurable set A subset that belongs to the sigma-algebra of a measurable space.
Measurable space A set equipped with a sigma-algebra of measurable subsets.
Measure A countably additive function on a sigma-algebra assigning sizes to sets.
Measure space A measurable space equipped with a measure.
Measure Theory Sigma-algebras, measures, and Lebesgue integration foundations
Minkowski inequality in Lp Triangle inequality for the Lp norm.
Monotone convergence theorem For an increasing sequence of nonnegative measurable functions, the integral of the limit equals the limit of the integrals.
Null set A measurable set of measure zero.
Outer measure A monotone, countably subadditive set function defined on all subsets.
Premeasure A countably additive set function defined on a set algebra.
Product measure A measure on a product space determined by its values on measurable rectangles.
Pushforward measure The measure obtained by transporting a measure through a measurable map.
Set algebra A collection of subsets closed under complements and finite unions.
Set of measure zero in ℝ^k A set that can be covered by countably many rectangles (or balls) with arbitrarily small total volume.
Sigma-algebra A collection of subsets closed under complements and countable unions.
Simple function A measurable function that takes only finitely many values.
Tonelli's theorem Interchange of integrals for nonnegative measurable functions on a product measure space.
Uniform integrability A condition preventing L1 functions from concentrating too much mass on large values or small sets.

Posts2 knowls

Research Advice Analysis Research Advice Analysis
Semigroup–Quasigroup Structure Semigroup–Quasigroup Structure

Probability38 knowls

Central limit theorem The classical limit theorem stating that normalized sums of i.i.d. variables converge in distribution to a normal law.
Characteristic function The complex-valued function t ↦ E[exp(i t X)] associated with a real-valued random variable.
Chebyshev's inequality Upper bound on deviation probability using variance.
Chernoff bound Exponential tail bound using moment generating functions.
Conditional expectation Expectation of a random variable given partial information represented by a sigma-algebra
Conditional probability Probability of an event given another event or a sigma-algebra representing available information
Correlation coefficient Normalized covariance giving a scale-free measure of linear association between two random variables
Covariance Expectation of a centered product measuring joint linear variability of two random variables
Cumulant A numerical summary of a distribution given by derivatives of the cumulant generating function at zero.
Cumulant generating function The logarithm of the moment generating function, when the latter is finite near zero.
Differential entropy The entropy of a continuous distribution defined via an integral of the log-density.
Distribution (law) The probability measure induced by a random variable on its state space.
Expectation The integral of a random variable with respect to the underlying probability measure.
Expectation of a function of a random variable Compute the expectation of a transformed random variable using the distribution of the original.
Gibbs' inequality (nonnegativity of KL divergence) The Kullback–Leibler divergence is always nonnegative, and it is zero only when the two distributions are identical.
i.i.d. sequence A sequence of random variables that are independent and identically distributed.
Identically distributed random variables Two random variables with the same probability law.
Independence of events A condition ensuring knowledge of one event does not change the probability of another
Independence of random variables Definition of when random variables have factorizing joint probabilities.
Independence of sigma-algebras A condition ensuring events measurable with respect to different sigma-algebras are independent
Markov inequality An upper bound on the tail probability of a nonnegative random variable using its expectation.
Maximum entropy principle A rule for selecting a probability distribution by maximizing entropy subject to known constraints.
Moment Expected power of a random variable, used to summarize features of its distribution
Moment generating function Function of a real parameter defined by the expected exponential of t times a random variable
Pinsker's inequality An inequality bounding total variation distance by the square root of Kullback–Leibler divergence.
Probability measure A measure on a sigma-algebra with total mass 1.
Probability of an event The number assigned by a probability measure to an event.
Probability space A sample space with a sigma-algebra of events and a probability measure.
Probability Theory Foundations of probability theory and information theory
Radon–Nikodym theorem Existence and uniqueness of a density for one measure that is absolutely continuous with respect to another.
Random variable A measurable real-valued function on a probability space.
Random vector A measurable map from a probability space into a finite-dimensional real vector space.
Relative entropy (KL divergence) A directed measure of discrepancy between two probability distributions, defined by an expectation of a log-likelihood ratio.
Shannon entropy A measure of uncertainty of a discrete random variable, defined from its probability mass function.
Strong law of large numbers Sample averages of iid variables converge almost surely to the mean.
Total variation distance A distance between two probability distributions defined by the largest possible difference they assign to the same event.
Variance A measure of how spread out a random variable is around its mean.
Weak law of large numbers Sample averages of iid variables converge in probability to the mean.

Quantum foundations18 knowls

Bounded Operator on a Hilbert Space A linear operator whose action does not increase vector norms by more than a fixed constant; equivalently, a continuous linear map.
Density Operator A positive semidefinite trace-one operator representing the state of a quantum system, allowing both pure and statistical mixtures.
Finite-Dimensional Complex Hilbert Space A finite-dimensional complex inner product space, automatically complete, used as the state space in finite-dimensional quantum theory.
Golden-Thompson inequality Trace inequality bounding Tr exp(A+B) by Tr(exp(A)exp(B)) for Hermitian matrices.
Mixed quantum state A quantum state described by a density operator that is not a rank-one projector.
Normal operator A bounded operator that commutes with its adjoint.
Normalized state vector A unit vector representing a pure quantum state up to multiplication by a phase.
Partial trace Linear map that traces out one tensor factor to produce a reduced operator.
Positive operator-valued measure A family of positive operators summing to the identity and representing a generalized quantum measurement.
Positive semidefinite operator A self-adjoint operator whose quadratic form is nonnegative on every vector.
Pure quantum state A quantum state represented by a rank-one projector onto a unit vector.
Quantum Foundations Quantum mechanical foundations for statistical mechanics
Quantum relative entropy Noncommutative generalization of Kullback-Leibler divergence for density operators.
Rank-one projector The orthogonal projection onto the line spanned by a normalized vector.
Self-Adjoint Operator (Observable) A linear operator equal to its adjoint; in quantum theory it represents an observable with real measurement outcomes.
Spectrum of a Self-Adjoint Operator in Finite Dimension For a finite-dimensional self-adjoint operator, the spectrum is exactly the set of its real eigenvalues and yields a spectral decomposition.
Trace of an Operator A basis-independent scalar associated to a linear operator, equal to the sum of diagonal entries or eigenvalues in finite dimension.
Von Neumann entropy Entropy of a quantum state defined as minus the trace of rho log rho.

Real analysis242 knowls

Abel test A convergence test for sums of products when one series converges and the other factor is monotone and bounded.
Abel's theorem A boundary limit theorem relating a convergent series to its associated power series near the radius 1.
Absolute convergence implies Cauchy An absolutely convergent series has Cauchy partial sums.
Absolute convergence implies convergence If the series of absolute values converges, then the original series converges.
Absolute value The standard absolute value function on the real numbers.
Absolute value preserves integrability If a function is Riemann integrable then so is its absolute value, with a triangle inequality.
Absolutely convergent series A series that converges after taking absolute values term-by-term.
Additivity and linearity lemmas for Riemann and Riemann–Stieltjes integrals Linearity in the integrand and additivity over subintervals for Riemann and Riemann–Stieltjes integrals
Algebraic properties of sup and inf Supremum and infimum behave predictably under inclusion, translation, scaling, and unions
Alternating series test A convergence test for alternating series with decreasing term magnitudes tending to zero.
Antiderivative A function whose derivative equals a given function.
Archimedean Property Natural numbers are unbounded in the real numbers.
Arzelà–Ascoli theorem On a compact metric space, a uniformly bounded equicontinuous sequence of continuous functions has a uniformly convergent subsequence.
Banach Fixed Point Theorem A contraction on a complete metric space has a unique fixed point, found by iteration
Basic properties of lim sup and lim inf Key identities and inequalities for limsup and liminf of a sequence
Basic Properties of limsup and liminf Standard inequalities and identities involving limit superior and limit inferior.
Bounded above A set of real numbers that has an upper bound.
Bounded below A set of real numbers that has a lower bound.
Bounded derivative implies uniform continuity A differentiable function with bounded derivative is Lipschitz, hence uniformly continuous.
Bounded sequence A sequence whose terms all lie within some fixed distance from the origin.
C^2 implies equal mixed partials If f has continuous second partial derivatives, then mixed partials commute
Cauchy condensation test A convergence test for nonincreasing nonnegative series using dyadic subsequences.
Cauchy Criterion in Rk In Euclidean space, a sequence converges exactly when it is Cauchy.
Cauchy mean value theorem A two-function mean value theorem relating ratios of increments to ratios of derivatives.
Cauchy product A convolution-style product of two series.
Cauchy–Hadamard theorem A formula for the radius of convergence of a power series using a limsup of nth roots of coefficients.
Chain rule Derivative of a composition equals the composition of derivatives.
Chain rule (multivariable) The derivative of a composition is the composition (product) of derivatives
Change of variables formula A multivariable substitution rule involving the Jacobian determinant.
Class C^k function A function with continuous derivatives up to order k.
Class C^k map A map with continuous partial derivatives up to order k.
Comparison Test A nonnegative series is controlled by a larger or smaller nonnegative series.
Completeness Axiom Every nonempty set of real numbers that is bounded above has a least upper bound.
Completeness Equivalences Several standard statements that are equivalent forms of completeness of the real numbers.
Composition preserves Riemann integrability Composing a Riemann integrable function with a continuous function preserves integrability.
Conditionally convergent series A convergent series that is not absolutely convergent.
Constraint set A subset defined by one or more equations or inequalities that restrict admissible points
Continuity at a point The epsilon-delta condition that a function preserves closeness near a given point.
Continuity on a set A function is continuous on a set if it is continuous at every point of that set.
Continuity via sequences In metric spaces, f is continuous at x iff it preserves limits of sequences converging to x
Continuous functions are Riemann integrable Every continuous function on a closed interval has a Riemann integral
Continuous functions are Riemann integrable A function continuous on a closed interval is Riemann integrable.
Convergent series A series whose partial sums approach a finite limit.
Convergent series terms go to zero If a series converges, its terms must converge to 0
Corollary of the M-test If the sum of supremum norms is finite, then the corresponding series of functions converges uniformly.
Critical point A point where the first derivative of a scalar function vanishes
Critical value A value attained at some point where the derivative is not surjective
Darboux's theorem Derivatives satisfy the intermediate value property even when they are not continuous.
Density of the Irrationals Between any two real numbers there is an irrational number.
Density of the Rationals Between any two real numbers there is a rational number.
Density of ℝ \\ ℚ in ℝ Between any two real numbers there is an irrational number
Derivative The limit of the difference quotient, measuring instantaneous rate of change.
Derivative sign implies monotonicity A nonnegative derivative forces a function to be nondecreasing, and a nonpositive derivative forces it to be nonincreasing.
Derivative zero implies constant If the derivative of a differentiable function is zero everywhere on an interval, the function is constant.
Determinant nonvanishing implies local invertibility lemma Invertibility is stable under small perturbations, with a quantitative bound on the inverse
Difference quotient The ratio (f(x)-f(a))/(x-a) measuring average rate of change from a to x.
Differentiability at a point (one variable) A function is differentiable at a point if the limit defining its derivative exists there.
Differentiability criterion Characterization of differentiability via a best linear approximation.
Differentiability implies continuity If a function is differentiable at a point, then it is continuous at that point.
Differentiability in one variable The property of having a finite derivative at a point or on an interval.
Differentiable map Differentiability for maps between Euclidean spaces via a best linear approximation
Differentiation rules Formulas for derivatives of sums, products, quotients, and compositions.
Dini's theorem On a compact space, monotone pointwise convergence of continuous functions to a continuous limit is uniform.
Directional derivative Rate of change of a function in a specified direction
Dirichlet test A convergence test for sums of products using bounded partial sums and monotone factors.
Discontinuity point A point where a function fails to be continuous
Divergent series A series whose partial sums do not converge to a finite limit.
Equicontinuity A uniform form of continuity shared by all functions in a family.
Equicontinuity + pointwise boundedness implies uniform boundedness on compact sets On a compact domain, equicontinuity upgrades pointwise bounds to a global bound
Equicontinuity and dense sets lemma On a compact metric space, equicontinuity allows pointwise Cauchy behavior on a dense set to upgrade to uniform Cauchy behavior.
Equicontinuity–boundedness criterion On a compact metric space, equicontinuity plus pointwise boundedness implies uniform boundedness.
Equicontinuous family A family of functions that satisfies the equicontinuity condition at every point.
Equivalent definitions of continuity (metric spaces) Epsilon–delta, sequential continuity, and open-set preimages are equivalent in metric spaces
Every bounded sequence in R^k has a convergent subsequence A direct corollary form of the Bolzano–Weierstrass theorem
Field axioms Axioms for addition and multiplication in a field, as used for the real numbers.
Finite subcover lemma A compact set has a finite subcover for every open cover
Fixed point A point x satisfying T(x)=x for a self-map T
Fréchet derivative The derivative of a multivariable function as a best linear approximation at a point
Fubini theorem for Riemann integrals For continuous functions on a rectangle, iterated integrals exist and agree with the double Riemann integral.
Function of bounded variation A function whose total variation on an interval is finite.
Fundamental theorem of calculus I The integral defines an antiderivative at points where the integrand is continuous.
Fundamental theorem of calculus II A Riemann integral can be computed from any antiderivative.
Global extrema A continuous real function on a compact set attains its maximum and minimum.
Global maximum and global minimum A point where a function attains the largest/smallest value on its entire domain.
Gradient Vector of first partial derivatives of a scalar function
Greatest Lower Bound Theorem Nonempty subsets of R that are bounded below have an infimum in R
Hessian matrix Matrix of second partial derivatives of a scalar function
Higher derivatives Derivatives of order two and higher, defined iteratively.
Image (range) The set of values a function actually attains.
Implicit function theorem Solves an equation F(x,y)=0 locally for y as a function of x under a nondegeneracy condition.
Implicitly defined function A function specified indirectly by an equation involving its inputs and outputs
Infimum The greatest lower bound of a nonempty set of real numbers.
Integral test A convergence test that compares a nonnegative decreasing series to an improper integral.
Integration by parts An identity relating the integral of a product to boundary terms and another integral.
Integration by parts for Riemann–Stieltjes integrals A boundary-term identity relating two Riemann–Stieltjes integrals.
Integrator function The function whose increments weight the sums in a Riemann–Stieltjes integral.
Interchanging limit and integral Under uniform convergence, the limit of Riemann integrals equals the Riemann integral of the limit.
Intermediate value theorem A continuous function on an interval takes all values between its endpoint values.
Interval A subset of the real line that contains every point between any two of its points.
Inverse function theorem in one dimension A differentiable function with nonzero derivative has a differentiable local inverse.
Inverse function theorem in R^k A map with invertible derivative at a point has a differentiable local inverse.
Isolated point A point of a set that has a neighborhood containing no other points of the set.
Iterated integral A repeated one-variable integration over a rectangle or product of intervals.
Jacobian determinant Determinant of the Jacobian matrix for a map from Rn to Rn
Jacobian matrix Matrix of first partial derivatives of a multivariable map
Jordan decomposition lemma A bounded variation function can be written as a difference of two increasing functions.
L'Hôpital's rule A method for evaluating certain indeterminate limits by comparing derivatives.
Lagrange multiplier condition Necessary first-order condition for constrained extrema in terms of gradients.
Lagrange multipliers theorem Constrained extrema give critical points of a Lagrangian under a regularity hypothesis.
Least Upper Bound Theorem Nonempty subsets of R that are bounded above have a supremum in R
Limit Algebra for Sequences Rules for limits of sums, products, and quotients of convergent sequences.
Limit at a point The epsilon-delta definition of the limit of a function as x approaches a.
Limit at infinity The epsilon-M definition of the limit of a function as x goes to plus or minus infinity.
Limit Comparison Test Two positive series with asymptotically proportional terms converge or diverge together.
Limit inferior The eventual lower limiting value of a real sequence.
Limit inferior (lim inf) The smallest limit point of a bounded sequence, or equivalently the supremum of infima of tails.
Limit of a function at a point The value L that f(x) approaches as x approaches x0, defined by an ε–δ condition.
Limit of a sequence A point x such that x_n becomes arbitrarily close to x as n→∞.
Limit point (accumulation point, cluster point) A point x such that every neighborhood of x contains a point of the set different from x.
Limit superior The eventual upper limiting value of a real sequence.
Limit superior (lim sup) The largest limit point of a bounded sequence, or equivalently the infimum of suprema of tails.
Linearity of the Riemann–Stieltjes integral The Riemann–Stieltjes integral is linear in both the integrand and the integrator when the relevant integrals exist.
Local diffeomorphism corollary Nonvanishing Jacobian determinant implies a map is a diffeomorphism in a neighborhood of each point.
Local extremum A point where a function attains a local maximum or local minimum.
Local implicit-function parameterization Near a regular point, a level set is locally the graph of a differentiable map.
Local maximum and local minimum A point where a function attains a maximum/minimum relative to nearby points.
Lower sum A Riemann lower sum built from infima on each subinterval.
Lower sum (Riemann) A weighted sum of infima of f over subintervals of a partition.
M-test continuity and integration corollary Under the M-test, a function series converges uniformly, giving continuity and term-by-term integration
Maximum The largest element of a set of real numbers, when it exists.
Mean value estimate A bound on the change in a function in terms of a bound on its derivative.
Mean value inequality A bound on the change of a differentiable map using a bound on its derivative.
Mean value theorem A differentiable function attains its average slope at some interior point.
Mean value theorem for integrals A continuous function attains its average value somewhere on the interval.
Mertens theorem on Cauchy products Convergence of the Cauchy product under absolute convergence of one factor
Mertens' theorem A condition ensuring the Cauchy product of two series converges to the product of their sums.
Mesh of a partition The length of the longest subinterval in a partition.
Minimum The smallest element of a set of real numbers, when it exists.
Mixed partial derivative A second partial derivative taken with respect to two different coordinates
Modulus (absolute value) on ℂ The nonnegative magnitude |z| of a complex number z, equal to its distance from 0.
Monotone function A function that preserves or reverses order on an interval.
Monotone sequence A real sequence that is nondecreasing or nonincreasing.
Monotone Sequence Convergence Theorem Every bounded monotone real sequence converges, with limit given by a supremum or infimum.
Monotone sequence of functions A sequence of functions that is monotone at each point of the domain.
Monotone Subsequence Lemma Every real sequence has a monotone subsequence.
Multiple Riemann integral Riemann integration of a bounded function over a rectangular region in Euclidean space.
Newton–Leibniz formula If F is an antiderivative of f, then the integral of f equals F(b)-F(a)
One-sided limit A limit taken from the left or from the right of a point.
Order axioms Axioms for a total order compatible with addition and multiplication on the real numbers.
Oscillation The amount a function varies on a set or interval.
Oscillation criterion for Riemann integrability A bounded function is Riemann integrable exactly when its total oscillation can be made small by a partition.
Partial derivative Derivative of a multivariable function with respect to one coordinate
Partial sums The finite sums obtained by truncating a series.
Partition of an interval A finite increasing sequence of points that subdivides a closed interval.
Pointwise bounded family A family of functions that is bounded at each fixed point of the domain.
Pointwise convergence Convergence of a sequence of functions at each fixed point of the domain.
Polynomial A finite linear combination of powers of a variable with real coefficients.
Positive derivative implies increasing If a differentiable function has positive derivative everywhere on an interval, then it is strictly increasing.
Power series A series in powers of (x minus a center), defining a function on an interval of convergence.
Power series is analytic on its disk of convergence Within its radius of convergence, a power series defines a function with derivatives given by termwise differentiation.
Preimage (inverse image) The set of inputs that a function maps into a given subset of the codomain.
Ratio Test A series converges absolutely if successive terms shrink by a uniform factor less than one.
Real Analysis Sequences, series, continuity, differentiation, and integration on the real line
Rearrangement of a series A series obtained by permuting the terms of another series.
Rearrangement theorem for absolutely convergent series Any rearrangement of an absolutely convergent series converges to the same sum.
Refinement lemma for upper and lower sums Refining a partition decreases upper sums and increases lower sums.
Refinement of a partition A partition that contains all points of another partition.
Regular point A point where a differentiable map has maximal rank or a surjective derivative
Regular point and critical point Points where the derivative of a map has maximal rank, versus points where it fails to
Regular value and critical value Values whose preimages contain only regular points, versus values hit at some critical point
Remainder term in Taylor's theorem The difference f(x)−T_k f(x;a), measuring Taylor approximation error.
Reverse triangle inequality The difference of norms is bounded by the norm of the difference
Riemann algebra Riemann integrable functions are closed under products, forming an algebra.
Riemann integrability implies boundedness A Riemann integrable function on a closed interval must be bounded.
Riemann integrability of monotone functions Every monotone function on a closed interval is Riemann integrable.
Riemann integrability with finitely many discontinuities A bounded function with only finitely many discontinuities is Riemann integrable.
Riemann integrable function A bounded function whose upper and lower sums can be made arbitrarily close.
Riemann integral The common value determined by Riemann sums when a function is integrable.
Riemann linearity Linearity of the Riemann integral with respect to addition and scalar multiplication.
Riemann rearrangement theorem A conditionally convergent real series can be rearranged to converge to any real value or diverge.
Riemann sum A finite weighted sum approximating an integral using a tagged partition.
Riemann–Stieltjes integrability theorem Continuity of the integrand and bounded variation of the integrator guarantee Riemann–Stieltjes integrability.
Riemann–Stieltjes integral An integral defined using increments of an integrator function.
Right derivative and left derivative One-sided derivatives defined by one-sided limits of the difference quotient.
Rolle's theorem A differentiable function equal at two endpoints has a critical point in between.
Root test A convergence test using the limsup of the nth roots of the term magnitudes.
Schwarz–Clairaut theorem Under continuity, mixed second partial derivatives agree.
Second derivative tests Using the second derivative to classify local maxima and minima at critical points.
Separates points A property of a family of functions distinguishing any two different points.
Series An infinite sum understood through its sequence of partial sums.
Series of functions An infinite sum of functions defined through its partial sums.
Set of discontinuities The set of points where a function is discontinuous.
Space of continuous functions The set of all real-valued continuous functions on a given topological space.
Squeeze Theorem A sequence trapped between two sequences with the same limit has that limit as well.
Step function A function that is constant on each subinterval of some partition.
Stone–Weierstrass theorem A subalgebra of continuous functions on a compact space that separates points and contains constants is dense in the full algebra.
Subalgebra of continuous functions A subset of continuous functions closed under linear combinations and pointwise products.
Subsequence A sequence obtained by restricting to a strictly increasing sequence of indices.
Substitution rule A change of variables formula for one-dimensional Riemann integrals.
Substitution rule (change of variables) for Riemann integrals A one-dimensional change of variables formula for definite integrals
Sufficient condition for differentiability Continuity of partial derivatives at a point implies differentiability of a multivariable function there.
Supremum The least upper bound of a nonempty set of real numbers.
Supremum and Infimum Algebra How supremum and infimum interact with basic set operations such as translation and scaling.
Supremum Approximation Lemma A supremum can be approximated from below by elements of the set.
Supremum norm A norm on bounded functions given by the supremum of absolute values.
Tagged partition A partition together with a chosen sample point in each subinterval.
Taylor polynomial The polynomial built from derivatives of a function at a point.
Taylor's Theorem in several variables Approximates a smooth multivariable function by a polynomial in a neighborhood of a point
Taylor's theorem with remainder Taylor expansion with an explicit remainder term for a smooth real function.
Term-by-term differentiation for power series Inside the radius of convergence, a power series can be differentiated by differentiating each term.
Term-by-term integration of a power series Inside its radius of convergence, a power series can be integrated by integrating each term.
Term-by-term operations for power series Within the common disk of convergence, power series can be added, scaled, and multiplied by operating on coefficients.
Terms go to zero A necessary condition for a series to converge is that its terms tend to zero.
Total boundedness characterization via ε-nets A set is totally bounded iff it has a finite ε-net for every ε>0
Total derivative (Fréchet derivative in ℝ^k) The linear map Df(a) giving the best first-order approximation f(a+h)=f(a)+Df(a)h+o(‖h‖).
Total variation The supremum of sums of absolute increments over all partitions.
Triangle inequality The fundamental inequality relating the distance between three points in a metric space.
Uniform Cauchy criterion For functions into a complete metric space, uniform convergence is equivalent to being uniformly Cauchy.
Uniform Cauchy sequence A Cauchy condition for function sequences with a uniform bound over the domain.
Uniform continuity Continuity where a single delta works for the whole set, not point by point.
Uniform continuity preserves Cauchy sequences Uniformly continuous maps send Cauchy sequences to Cauchy sequences
Uniform convergence Convergence of functions with an error bound that is uniform in the domain variable.
Uniform convergence (sequence of functions) Convergence f_n→f with a single N(ε) working for all x in the domain.
Uniform convergence and differentiation If derivatives converge uniformly and one point converges, then the functions converge uniformly and the limit may be differentiated term by term.
Uniform convergence and integration A uniformly convergent series of Riemann integrable functions may be integrated term by term.
Uniform convergence implies pointwise convergence Uniform convergence guarantees pointwise convergence at every point.
Uniform convergence in supremum norm For bounded real-valued functions, uniform convergence is equivalent to convergence in the supremum norm.
Uniform convergence of power series on compact sets A power series converges uniformly (and absolutely) on every compact subset inside its interval of convergence.
Uniform convergence on compact sets Uniform convergence on every compact subset of the domain.
Uniform convergence preserves boundedness A uniform limit of bounded functions is bounded, and a uniformly convergent sequence of bounded functions is uniformly bounded.
Uniform limit of continuous functions is continuous A uniform limit of continuous functions is continuous.
Uniform limit of integrable functions A uniform limit of Riemann integrable functions on a closed interval is Riemann integrable.
Uniform limit theorem The uniform limit of continuous functions is continuous.
Uniform limit theorem for continuity A uniform limit of continuous functions is continuous
Uniform metric A metric on bounded functions defined by the supremum of pointwise distances.
Uniformly bounded family A family of functions bounded by a single constant on the whole domain.
Uniqueness of Supremum and Infimum A set has at most one least upper bound and at most one greatest lower bound.
Upper sum A Riemann upper sum built from suprema on each subinterval.
Upper sum (Riemann) A weighted sum of suprema of f over subintervals of a partition.
Weierstrass approximation theorem Every continuous function on a closed interval can be uniformly approximated by polynomials.
Weierstrass M-test A comparison test giving uniform convergence of a series of functions from an absolutely convergent numerical majorant.
Zero derivative implies constant If f' vanishes everywhere on an interval, the function is constant

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Shale paper34 knowls

C*-Algebra of Field Observables 𝔄 The norm-closed algebra generated by Weyl unitaries for the CCR
Complex Structure Λ on K An operator Λ with Λ² = −I giving K the structure of a complex Hilbert space
Creation and Annihilation Operators Operators adding/removing one symmetric tensor factor in bosonic Fock space
Decomposition K = M ⊕ M A choice of real subspace M giving coordinates (x,y) for phase space K
Determinant on I + Trace-Class Extension of det via det(I+A)=exp(tr log(I+A)) for trace-class A
Duality Transform D (Segal) Unitary map identifying bosonic Fock space with Gaussian L₂(M,n)
Field Automorphism θ(T) from a Symplectic Map The *-automorphism induced by sending Weyl operators V(z) to V(Tz)
Fock–Cook Quantization The standard bosonic CCR representation built from creation/annihilation operators
Gaussian Measure on a Hilbert Space (Segal) An infinite-dimensional normal distribution built from finite-dimensional projections
Hilbert–Schmidt Operator An operator with finite ℓ²-norm of matrix coefficients (Schatten class 2)
Jacobian X(T) in Shale's Gaussian Setup The Radon–Nikodym derivative of the transformed Gaussian measure n(T) with respect to n
Kadison Transitivity (Used in §6) In an irreducible representation, algebra elements can move one vector to another
Polar Decomposition Writing T as a unitary/orthogonal part times a positive part
Projective Unitary Representation A group action by unitaries defined only up to phase (unitary rays)
Radon–Nikodym Derivative The density dν/dμ of one measure with respect to another
Restricted General Linear Group rGL(H) Invertible operators whose positive part differs from I by a Hilbert–Schmidt operator
Restricted Symplectic Group rSp(K) The implementable symplectic transformations in Shale's Fock representation
Segal/Shale Representation 𝔘(T) on L₂(M,n) Unitary action of rGL(M) on Gaussian L₂ by change of variables and a Jacobian
Shale's Paper Definitions from Shale's 1962 paper on linear symmetries of the free boson field
Shale's Subgroups GL(H)₀, GL(H)₁, GL(H)₂ 'Tame', trace-class, and Hilbert–Schmidt perturbations of the identity
Single Particle Structure Σ(H) Segal's package (K,B) extracted from a complex Hilbert space H for CCR quantization
Spectral Theorem for Compact Selfadjoint Operators Diagonalization by an orthonormal eigenbasis with eigenvalues → 0
State, Pure State, Regular State (CCR context) Positive normalized functionals, with purity and CCR-regularity conditions
Strong vs Weak Operator Topology Two common convergence notions for bounded operators on a Hilbert space
Symmetric Fock Space S(H) The bosonic Fock space ⊕_{n≥0} Sym^n(H) with vacuum vector
Symmetric Tensor Product (·)_s Averaging over permutations to land in the symmetric subspace
Symplectic Form A nondegenerate skew-symmetric bilinear form B on a real vector space
Symplectic Group Sp(K) Bounded invertible real-linear maps preserving the symplectic form B
Symplectic Hilbert Space (K,B) A real Hilbert space K equipped with a continuous nondegenerate skew form B
Tame Function (Segal) A function on an infinite-dimensional Hilbert space depending on finitely many coordinates
Trace-Class Operator An operator with absolutely summable singular values (Schatten class 1)
Weak Continuity of a Representation Continuity of matrix coefficients (π(g)x,y) in the group parameter g
Weyl CCR Quantization Encoding canonical commutation relations via Weyl unitaries V(z)=exp(iR(z))
Wiener Transform W A unitary transform on Gaussian L₂ intertwining T with T^{*-1}

Shared foundations61 knowls

Axiom of Choice Every family of nonempty sets has a choice function.
Bijective function A function that is both one-to-one and onto
Binary operation A function that combines two elements of a set to produce another element of the same set
Cardinality The size of a set, understood up to bijection
Cartesian product The set of all ordered pairs from two sets.
Codomain The target set in the definition of a function
Complement The elements of an ambient universe that are not in a given set.
Complex conjugate The map a+bi ↦ a-bi on complex numbers.
Complex numbers Numbers of the form a+bi with i^2=-1, forming a field extending the reals.
Composition of functions Forming a new function by applying one function after another
Composition of functions The function obtained by applying one function after another.
Contraction mapping A self-map that strictly shrinks distances by a uniform factor <1
Countable set A set that can be listed in a sequence, possibly with finitely many elements
Divisibility The relation that one integer is an exact multiplicative factor of another.
Domain The set of allowed inputs of a function
Empty set The unique set that contains no elements.
Equivalence class The set of all elements equivalent to a given element under an equivalence relation.
Equivalence relation A relation that formalizes when two elements should be regarded as the same type.
Every bounded infinite subset of R^k has a limit point A bounded infinite set in Euclidean space has an accumulation point
Factorial The product of all positive integers up to a given nonnegative integer.
Function A relation that assigns each input exactly one output
Graph of a function The set of ordered pairs consisting of each input and its output
Identity function The function that maps every element of a set to itself
Image The set of outputs a function attains on a given subset of inputs
Indexed family of sets A collection of sets labeled by elements of an index set.
Injective function A function that never takes the same value on two different inputs
Integers The set of whole numbers, positive, negative, and zero.
Intersection The set of elements that belong to all of the given sets.
Inverse function A function that undoes a bijective function
Lower bound An element that is less than or equal to every element of a subset in an ordered set.
Mathematical induction A principle for proving statements for all natural numbers.
Natural numbers The set of nonnegative integers used for counting and indexing.
Ordered pair A two-component object where order matters.
p-adic integers The compact ring obtained as the inverse limit of the residue rings modulo powers of a prime.
p-adic valuation The exponent of a prime in an integer, extended to a valuation on rational numbers.
Partial order A binary relation that is reflexive, antisymmetric, and transitive.
Partition A way to break a set into disjoint nonempty blocks that cover it.
Power set The set of all subsets of a given set.
Preimage The set of inputs that a function sends into a specified subset of its codomain
Proper subset A subset that is strictly smaller than the set it sits inside.
Quotient set The set of equivalence classes of a set under an equivalence relation
Rational numbers Numbers expressible as a ratio of two integers with nonzero denominator.
Real numbers The complete ordered number system containing the rationals.
Relation A set of ordered pairs encoding which elements are related.
Restriction of a function A function obtained by limiting the domain to a subset
Sequence A function from the natural numbers to a set.
Set A fundamental object determined entirely by which elements it contains.
Set difference The elements of one set that are not in another set.
Shared Foundations Basic set theory, logic, and function concepts used across all mathematics
Subset A set contained in another set, element by element.
Surjective function A function whose outputs cover the entire codomain
Symmetric difference The elements that lie in exactly one of two sets.
Total order A partial order in which any two elements are comparable.
Total order (linear order) A partial order in which every pair of elements is comparable.
Union The set of elements that belong to at least one of the given sets.
Upper bound An element that is greater than or equal to every element of a subset in an ordered set.
Well-ordered set A totally ordered set in which every nonempty subset has a least element.
Well-ordering principle Every nonempty subset of the natural numbers has a least element.
Well-ordering theorem Every set can be given a well-order.
ZFC axioms Standard axioms of set theory: Zermelo-Fraenkel axioms plus the Axiom of Choice.
Zorn's lemma A maximal-element principle for partially ordered sets.

Stat mech quantum8 knowls

Finite quantum statistical system A finite-dimensional Hilbert space with an observable algebra and a self-adjoint Hamiltonian.
KMS condition in finite quantum systems An analytic boundary condition on equilibrium correlation functions at inverse temperature beta.
Observable algebra A unital star-algebra of operators whose self-adjoint elements represent observables.
Quantum expectation value For a density operator rho and observable A, the expectation is Tr(rho A).
Quantum Gibbs state For a finite quantum system, the Gibbs state is exp(-beta H) normalized by its trace.
Quantum Hamiltonian A self-adjoint energy operator that generates time evolution and determines thermal weights.
Quantum partition function For a finite quantum system with Hamiltonian H, the canonical partition function is Tr(exp(-beta H)).
Quantum Statistical Mechanics Quantum ensembles and KMS states

Topology109 knowls

Baire category theorem In a complete metric space, countable intersections of dense open sets are dense.
Baire space A topological space in which countable intersections of dense open sets are dense
Basis of a topology A collection of sets whose unions give all open sets.
Bolzano–Weierstrass theorem Every bounded sequence in Euclidean space has a convergent subsequence
Boundary The set of points where every neighborhood meets both the set and its complement.
Bounded set A subset of a metric space that lies inside some ball of finite radius.
Cantor intersection theorem Nested closed sets with diameters going to zero intersect in a single point in a complete metric space
Category Argument Template A standard Baire category method for producing a dense or residual set by intersecting dense open sets.
Cauchy sequence A sequence in a metric space whose terms become arbitrarily close to each other.
Cauchy sequence is bounded In a metric space, every Cauchy sequence is bounded
Closed ball The set of points within a given radius of a center point in a metric space, using non-strict inequality.
Closed manifold A compact manifold without boundary.
Closed set A subset whose complement is open in the ambient space.
Closed sets are complements of open sets A set is closed iff its complement is open; closed sets are stable under intersections
Closed subset of compact set is compact A closed subset of a compact set is compact
Closure The smallest closed set containing a given subset.
Compact iff complete and totally bounded A metric space is compact exactly when it is complete and totally bounded.
Compact set A set in which every open cover has a finite subcover.
Compact subset of a Hausdorff space is closed In a Hausdorff space, every compact subset is closed.
Compact-to-Hausdorff homeomorphism criterion A continuous bijection from a compact space to a Hausdorff space is a homeomorphism.
Compactness implies boundedness In a metric space, every compact set is bounded
Compactness implies closedness In a Hausdorff space, every compact set is closed
Compactness implies completeness A compact metric space is complete: every Cauchy sequence converges.
Compactness implies total boundedness In a metric space, every compact set can be covered by finitely many small balls.
Compactness of graphs lemma The graph of a continuous map from a compact space is compact in the product.
Complete metric space A metric space in which every Cauchy sequence converges to a point of the space.
Complete metric space is Baire Every complete metric space is a Baire space.
Connected component A maximal connected subset of a topological space.
Connected set A set that cannot be split into two disjoint nonempty open pieces in the subspace topology.
Connected subsets of R are intervals A subset of the real line is connected exactly when it is an interval.
Connectedness criteria in R Equivalent ways to recognize when a subset of the real line is connected.
Continuity via open sets A function is continuous iff the preimage of every open set is open
Continuous attains max/min on compact set A continuous real-valued function on a compact set achieves a maximum and a minimum
Continuous bijection from compact is a homeomorphism criterion A continuous bijection from a compact space to a Hausdorff space has continuous inverse
Continuous function on a compact set is bounded Continuous functions on compact domains have finite upper and lower bounds
Continuous function on a compact set is uniformly continuous On compact domains, continuity automatically upgrades to uniform continuity
Continuous functions on compact sets are bounded A continuous real-valued function on a compact set has finite sup norm
Continuous group action A group action whose action map is continuous in the group element and the point.
Continuous image of a connected set is connected Continuous maps preserve connectedness.
Continuous image of compact set is compact A continuous map sends compact sets to compact sets
Continuous map A function whose preimage of every open set is open.
Convergence in product metric spaces A sequence in X×Y converges iff each coordinate sequence converges
Convergent sequence A sequence whose terms eventually remain in every neighborhood of a limit point.
Convergent sequence is Cauchy In a metric space, every convergent sequence is Cauchy
Cover A family of subsets whose union contains a given set.
Curve A continuous map from an interval of real numbers into a space.
Dense set A subset whose closure is the whole space.
Derived set The set of all limit points of a subset.
Diameter The supremum of all distances between pairs of points in a set within a metric space.
Epsilon-net A set of points that approximates a set in a metric space within a fixed tolerance.
Equivalent metrics Two metrics on the same set that generate the same open sets, hence the same topology.
Extreme value theorem A continuous function on a compact set attains its maximum and minimum.
Finite intersection property A property of a family of sets where every finite subfamily has nonempty intersection.
Finite intersection property theorem Compactness is equivalent to nonempty intersection for families of closed sets with the finite intersection property
Hausdorff space A space where any two distinct points have disjoint neighborhoods.
Heine-Cantor theorem A continuous function on a compact metric space is uniformly continuous.
Heine–Borel theorem In Euclidean space, compactness is equivalent to being closed and bounded.
Homeomorphism A bijective continuous map with a continuous inverse.
Homotopy equivalence A pair of maps inverse to one another up to continuous deformation.
Homotopy sphere A closed manifold homotopy equivalent to a standard sphere.
Hölder continuity A condition controlling how fast a function can change, generalizing Lipschitz continuity by allowing a power less than one.
Image of a compact connected set is an interval A continuous real-valued function on a compact connected space has an interval as its image.
Interior The largest open set contained in a given subset.
Intersection of Dense Open Sets is Dense In a topological space, the intersection of two dense open sets is again dense (and open).
Irreducible space A nonempty topological space that is not the union of two proper closed subsets.
Isometry A distance-preserving map between metric spaces.
Lebesgue number lemma Every open cover of a compact metric space has a uniform scale that fits inside the cover.
Limit point A point whose every neighborhood meets a set away from that point.
Lipschitz continuity A strong form of continuity where distances in the image are bounded by a constant times distances in the domain.
Locally compact group A topological group whose underlying topological space is locally compact.
Locally compact space A topological space in which every point has a neighborhood with compact closure.
Meager set A set that is a countable union of nowhere dense sets
Metric A distance function on a set satisfying positivity, symmetry, and the triangle inequality.
Metric space A set equipped with a metric that measures distances between its points.
Metric sphere The set of points at exactly a fixed distance from a given center point in a metric space.
Metric-induced topology The topology on a metric space in which a set is open if it contains an open ball around each of its points.
Neighborhood A set that contains an open set around a given point.
Nested interval theorem A nested sequence of nonempty closed intervals in the real line has nonempty intersection
Nowhere dense set A set whose closure has empty interior
Open ball The set of points within a given radius of a center point in a metric space, using strict inequality.
Open cover A cover consisting entirely of open sets in a topological space.
Open set A subset that belongs to the chosen topology on a space.
Open sets form a topology In a metric space, unions of open sets are open and finite intersections of open sets are open
Path A continuous map from the unit interval into a space.
Path-connected set A set in which any two points can be joined by a continuous path lying in the set.
Product topology The standard topology on a product of spaces, generated by cylinder sets.
Quotient topology The finest topology on a codomain that makes a given surjection continuous.
Refinement of an open cover A cover that is finer than another, with each set contained in a member of the original cover.
Relatively compact set A subset whose closure is compact in the ambient space.
Residual set A set whose complement is meager
Separable space A topological space containing a countable dense subset.
Separated sets Two sets in a topological space that do not meet each other's closure.
Sequential characterization of closed sets In a metric space, a set is closed iff it contains limits of all convergent sequences from it.
Sequential characterization of closure In a metric space, a point lies in the closure of a set iff it is the limit of a sequence from the set.
Sequential compactness equals compactness In metric spaces, compactness is equivalent to sequential compactness
Sequentially compact set A set where every sequence has a convergent subsequence with limit in the set.
Subbasis of a topology A collection of sets whose finite intersections form a basis.
Subspace topology The topology on a subset obtained by intersecting with open sets of the ambient space.
T0 space A space where distinct points can be distinguished by membership in an open set.
T1 space A space in which every singleton set is closed.
Topological group A group with a topology making multiplication and inversion continuous.
Topological manifold A Hausdorff, second-countable space locally homeomorphic to a fixed Euclidean space.
Topological space A set equipped with a topology, specifying which subsets are open.
Topology Topological spaces, metric spaces, compactness, connectedness, and Baire category
Topology A collection of subsets declared open, closed under unions and finite intersections.
Totally bounded implies Cauchy subsequence Every sequence in a totally bounded metric space has a Cauchy subsequence
Totally bounded set A set in a metric space that can be covered by finitely many small balls for every radius.
Uniformly continuous map A map between metric spaces where one delta works uniformly for all points for a given epsilon.
Uniqueness of limits in Hausdorff spaces In a Hausdorff space, a convergent sequence has at most one limit.