Index - Knowlpedia

45 subjects and collections

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Subjects

Algebraic geometry130 knowls

$k$ -ideal of a blueprint A multiplicative ideal of a blueprint that is closed under subtraction encoded by additive relations.
Adic space A locally ringed space locally modeled on continuous valuations of a Huber pair.
Affine blue scheme A blueprinted space isomorphic to the spectrum of a blueprint.
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.
Algebraic curve A separated finite-type scheme of pure dimension one over a field.
Algebraic group A group scheme of finite type over a field.
Algebraic space An étale sheaf admitting a representable diagonal and a surjective étale atlas by a scheme.
Algebraic stack A stack in groupoids with a representable diagonal and a smooth surjective atlas by a scheme.
Algebraically closed field A field in which every nonconstant one-variable polynomial has a root.
Band as an ordered blueprint The fully faithful, coreflective embedding of bands into ordered blueprints.
Base change Pulling an object over a base back along a morphism to a new base.
Blue scheme A locally blueprinted space covered by spectra of blueprints.
Blueprint A commutative multiplicative monoid together with compatible formal additive relations.
Blueprint as an ordered blueprint The fully faithful embedding that regards additive equalities as an ordered blueprint with equality order.
Borel subgroup A maximal connected solvable subgroup of a connected linear algebraic group.
Chevalley lattice and integral model Integral Lie and representation lattices and the reductive Chevalley group scheme determined by a root datum.
Closed point A point whose singleton is closed in the underlying topological space of a scheme.
Coherent sheaf A sheaf of modules locally generated by finitely many sections with finitely generated relations.
Commutative monoid with zero as a blueprint The blueprint obtained from a commutative monoid with zero by imposing only the minimal additive relations.
Compactly supported étale cohomology Étale cohomology formed with the exceptional direct image, retaining only classes with proper support.
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.
D-module A quasi-coherent sheaf module over the sheaf of differential operators on a smooth algebraic variety.
Derived algebraic stack A derived stack with representable diagonal and a smooth surjective atlas by derived affine schemes.
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 locally free morphism A scheme morphism whose direct-image algebra is locally a finite-rank free module.
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.
Flag variety A projective homogeneous variety parametrizing flags, realized for a reductive group as a quotient by a parabolic subgroup.
Flat morphism A scheme morphism whose induced homomorphisms on local rings are flat.
Formal scheme A locally topologically ringed space locally modeled on formal spectra of complete adic rings.
Formal spectrum The affine formal scheme Spf(A) associated to a complete adic ring A.
Fundamental theorem of projective geometry A line-preserving bijection of projective spaces of dimension at least two is induced by a semilinear isomorphism.
G-torsor on a site A sheaf with a locally trivial simply transitive action of a group sheaf.
Galois descent, twisted forms, and inner forms Semilinear cocycle data that descend an algebraic group and distinguish general, inner, and pure inner forms.
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.
Geometric fiber The fiber of a scheme morphism after base change to an algebraic closure of a residue field.
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.
Harder–Narasimhan filtration The canonical filtration of a bundle by semistable pieces of strictly decreasing slope.
Hyperrings as ordered blueprints The fully faithful encoding of Krasner hyperrings by monomial inequalities in ordered blueprints.
Idyll as an ordered blue field The band definition of an idyll agrees with its purely positive ordered-blue-field realization.
Integrable algebraic connection An algebraic connection whose curvature vanishes.
Isocrystal A finite-dimensional vector space over a Witt-vector fraction field equipped with a bijective semilinear Frobenius.
Levi subgroup A reductive complement to the unipotent radical of a parabolic subgroup.
Lisse ell-adic sheaf An ell-adic sheaf that is locally a finite free constant sheaf in the étale topology.
Local diffeomorphism A smooth map that restricts near every point to a diffeomorphism onto an open neighborhood.
Localization of a blueprint The universal blueprint in which a chosen multiplicative subset becomes invertible.
Locally blueprinted space A topological space with a sheaf of blueprints whose stalks are local blueprints.
Locally free sheaf A sheaf of modules locally isomorphic to a finite direct sum of the structure sheaf.
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.
Locally topologically ringed space A topological space with a sheaf of topological rings whose underlying ringed space is locally ringed.
Maximal torus, character lattice, and weights A maximal torus with its geometric character and cocharacter lattices, carrying a Galois action over a nonsplit field.
Moduli stack of G-bundles on a curve The algebraic stack Bun_G whose families are principal G-bundles on a fixed curve.
Monoid blueprint to semiring blueprint The canonical morphism from the monoid blueprint of a semiring to its semiring blueprint.
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.
Ordered blue field A nontrivial ordered blueprint whose nonzero monomials are all units.
Ordered blue scheme A scheme locally modeled on spectra of ordered blueprints.
Ordered blueprint An ordered semiring together with a multiplicative monoid of generators.
Ordered blueprint with unique weak inverses An ordered blueprint in which every element has exactly one additive weak inverse.
Ordered blueprinted space A topological space equipped with a sheaf of ordered blueprints.
Parabolic subgroup A subgroup of a reductive algebraic group whose homogeneous quotient is proper.
Perfect complex A complex locally quasi-isomorphic to a bounded complex of finite-rank projective modules.
Perfectoid field A complete nondiscretely valued nonarchimedean field whose residue-level Frobenius is surjective.
Perfectoid space An adic space locally modeled on perfectoid affinoid algebras.
Pinning and pinned automorphisms A Borel, maximal torus, and simple-root vectors that canonically lift automorphisms of a based root datum.
Pointed algebraic curve An algebraic curve equipped with an ordered or labeled collection of marked points.
Pre-addition on a monoid An additive and multiplicative equivalence relation on formal sums of elements of a monoid with zero.
Principal G-bundle on a scheme A G-torsor over a scheme in a specified Grothendieck topology.
Proj of a graded ring The scheme built from homogeneous prime ideals of a graded ring.
Projective geometry The incidence geometry of lines and subspaces in a vector space, viewed projectively.
Projective line The one-dimensional projective space formed by lines in a two-dimensional vector space.
Projective morphism A scheme morphism that factors as a closed immersion into relative projective space.
Projective semilinear group The collineation group obtained from invertible semilinear maps modulo scalar maps.
Projective space The scheme of one-dimensional linear subspaces of a vector space, covered by affine coordinate charts.
Projective transformation A transformation of projective space induced by an invertible linear map.
Punctured algebraic curve The complement of a finite reduced divisor in an algebraic curve.
Quasi-coherent sheaf A sheaf of modules on a scheme that locally comes from a module on each affine chart.
Quasi-split reductive group A connected reductive group having a Borel subgroup defined over the base field.
Reductive algebraic group A smooth connected affine algebraic group with trivial geometric unipotent radical.
Regular local ring A Noetherian local ring whose maximal ideal requires exactly its Krull dimension many generators.
Regular scheme A locally Noetherian scheme whose local rings are regular.
Relative Kähler differentials The module or sheaf that universally records first-order variation relative to a base.
Ringed space A topological space equipped with a sheaf of rings.
Root, weight, and isogeny lattices Intermediate character lattices between the root and weight lattices encode central isogeny forms.
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.
Semiring as a blueprint The blueprint whose additive relations record every equality in a commutative semiring.
Semiring completion of a blueprint The semiring obtained by quotienting formal sums by all additive relations of a blueprint.
Semisimple element and conjugacy class An algebraic-group element whose image in a faithful representation is diagonalizable.
Semisimple representatives in a torus normalizer The normalizer theorem used in the letter to place semisimple elements of a disconnected reductive extension in standard torus data.
Sheaf A system of local data on open sets that can be uniquely glued when compatible.
Sheaf of differential operators The filtered sheaf D_X generated locally by functions and vector fields acting as differential operators.
Sheaf of groups A sheaf whose sections form groups compatibly with restriction.
Sheaf of modules A sheaf carrying module structures over a fixed sheaf of rings, compatibly with restriction.
Sieve on an object A collection of morphisms into one object that is closed under precomposition.
Simply connected semisimple algebraic group A semisimple algebraic group with no nontrivial central isogeny cover.
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.
Smooth morphism A locally finitely presented flat morphism with geometrically regular fibers.
Smooth projective curve A smooth one-dimensional projective scheme over a field, the standard global base for geometric Langlands.
Spectrum of a blueprint The prime spectrum of a blueprint with its Zariski topology and structure sheaf.
Split reductive algebraic group A connected reductive group possessing a maximal torus split over the base field.
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.
Tangent sheaf The sheaf of derivations, equivalently the dual of the sheaf of Kähler differentials.
The projective line as a rank-one flag variety The projective line is the complete flag variety of a two-dimensional vector space.
Tilt and untilt of a perfectoid field The passage between perfectoid fields of mixed and positive characteristic via inverse-limit Frobenius.
Torsor condition The condition that two points in the same fiber differ by a unique group element.
Unipotent radical The largest smooth connected normal unipotent subgroup of a linear algebraic group.
Unique-weak-inverse reflection of an ordered blueprint The universal map from an ordered blueprint to one with unique weak inverses.
Unramified morphism A locally finite type scheme morphism whose relative differentials vanish.
Unramified reductive group A reductive group over a nonarchimedean local field that is quasi-split and splits over an unramified extension.
v-stack A stack on perfectoid spaces for the very fine v-topology.
Weil restriction A representable restriction-of-scalars functor for schemes over a finite locally free base.
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.

Analysis: geometric and quantitative tools17 knowls

Absolute continuity A strong continuity condition on an interval controlling total change over collections of small subintervals
Ahlfors–David regular set A set carrying a measure whose mass in centered balls is uniformly comparable to a fixed power of the radius.
Analysis Real analysis, metric spaces, and function spaces.
Box porosity A discrete multiscale porosity condition expressed by an empty child in every occupied cube.
Dyadic annulus A radial shell whose outer radius is twice its inner radius, used to organize estimates by scale.
Finitely overlapping family of sets A family of sets whose multiplicity function is uniformly bounded.
Line-section decay for line-porous sets A line-porous set occupies only a power-decaying fraction of every line segment in the controlled scale range.
Locally finite sum of smooth functions A family with locally finite supports may be summed and differentiated as a finite sum near each point.
Lp estimate for sums with bounded overlap A multiplicity bound controls the Lp norm of a countable sum in terms of the individual Lp norms.
Measure decay for porous sets Truncated ball porosity forces a quantitative power saving in local Lebesgue measure.
Minkowski thickening The closed or open metric neighborhood obtained by adding a radius-r ball to a set.
Porosity on balls A quantitative hole condition requiring every ball in a range of scales to contain a comparably sized ball disjoint from the set.
Porosity on lines A quantitative hole condition imposed on every line segment in a prescribed range of scales.
Quantitative unique continuation An estimate that gives an explicit lower bound for the mass of a function on an observation set from analytic or spectral constraints.
Smooth convergence of a series from derivative bounds Summability of every compact derivative seminorm gives a smooth sum with termwise differentiation.
Unique ergodicity The property that a continuous dynamical system has exactly one invariant probability measure.
Vanishing products from separated auxiliary supports Closed support separation makes all differentiated cross products vanish, including after evaluation along a smooth auxiliary map.

Asymptotics29 knowls

Absorbing logarithmic losses into a power Every fixed power of a logarithm grows more slowly than any negative power of a small parameter.
Asymptotic equivalence A ratio tending to one in a stated limiting regime.
Asymptotic expansion Approximation to every finite order in an asymptotic scale, without a convergence requirement.
Asymptotic scale An ordered sequence whose successive functions are negligible relative to their predecessors.
Asymptotic summation by shrinking cutoffs Increasing decay orders with stage-independent derivative losses admit a smooth locally finite realization and quantitative tails.
Asymptotics Asymptotic methods and approximations
Big-O notation An eventual bound by a fixed multiple of a comparison function.
Comparable positive functions Two positive quantities bounded above and below by fixed multiples of one another.
Concentration scale A shrinking spatial length used to locate the region occupied by a family of profiles.
Diagonal choice of shrinking scales Finitely many vanishing requirements at each stage can be met by one geometrically shrinking sequence.
Entropy and multinomial coefficients Approximations and bounds relating multinomial coefficients to Shannon entropy.
Exponential decay in the square of a logarithm A Gaussian in log one over the scale dominates every fixed inverse power and logarithmic loss.
Finite truncation of an expansion The partial sum through a chosen order and its associated remainder.
Formal expansion A sequence of coefficients manipulated order by order without asserting analytic convergence.
Hierarchy of parameter choices An ordered choice of small or large parameters that makes the permitted dependencies explicit.
Infinite-order decay with derivatives Every fixed derivative is bounded by every positive power of a chosen vanishing scale.
Laplace's method Asymptotic evaluation of integrals dominated by a single interior maximizer of the exponent.
Leading order The first nonvanishing contribution in an ordered asymptotic expansion.
Leading term of an asymptotic expansion The first nonzero coefficient multiplied by its comparison function in an ordered expansion.
Little-o notation A quantity negligible relative to a comparison function in a specified limit.
Logarithmic selection of an iteration level A floor of a logarithm chooses an integer iterate whose size is comparable to a target scale.
Method of types Counting and probability estimates for sequences grouped by their empirical distribution.
Perturbation A change from a reference object, with smallness measured in a stated space or parameter.
Power law Dependence on a positive scale through a fixed real exponent.
Quadratic self-interaction The quadratic term involving a perturbation with itself in an exact expansion.
Remainder estimate A quantitative error bound after subtracting a stated approximation.
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.
Uniform parameter estimate An estimate whose constants and validity domain are independent of the declared parameters.

Category theory62 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).
Algebra object An internal associative unital algebra in a monoidal category.
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.
Core of a category The maximal subgroupoid of a category, retaining all objects but only the isomorphisms.
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).
Faithful functor A functor that is injective on every hom-set.
Full Subcategory A subcategory that contains every morphism of the ambient category between its objects.
Fully faithful functor A functor that induces bijections on all hom-sets.
Functor A map between categories that preserves identities and composition.
Group object An object carrying multiplication, identity, and inverse morphisms that satisfy the group axioms internally.
Groupoid A category in which every morphism is invertible.
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.
Monoidal category A category with an associative tensor product and a tensor unit, coherently up to specified isomorphisms.
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.
Symmetric monoidal category A monoidal category whose tensor factors can be exchanged by a coherent involutive braiding.
Tannakian category A rigid abelian tensor category reconstructed as representations of an affine group scheme by a fiber functor.
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.

Coalgebras3 knowls

Bialgebra An algebra and coalgebra whose multiplication and comultiplication are compatible.
Coalgebra A module with coassociative comultiplication and a counit.
Hopf algebra A bialgebra with an antipode implementing algebraic inversion.

Commutative algebra50 knowls

$I$ -adic completion The inverse limit A-hat = lim A/I^n that completes a ring along the powers of an ideal.
$I$ -adic topology The linear topology on a ring or module whose neighborhoods of zero are the powers of an ideal.
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.
Fractional ideal A nonzero submodule of a fraction field whose denominators can be cleared.
Going-down theorem For an integral extension of domains with integrally closed base, prime chains descend inside a prescribed 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 homomorphism of commutative rings A→B is integral when every element of B satisfies a monic polynomial 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 For every simple right R-module S, the Jacobson radical satisfies S J(R) = 0.
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 For a multiplicative subset S of a commutative ring R, the localization S^{-1}M is the module obtained by making every element of S act invertibly.
Localization of a ring The universal ring in which every element of a multiplicative subset becomes invertible.
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 The field obtained from a local ring by reducing modulo its 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.

Complex analysis90 knowls

Affine octonionic line A translate in the octonionic plane of an eight-dimensional direction parametrized by the octonionic projective line.
Analytic continuation Extension of a holomorphic function through overlapping domains, uniquely controlled by the identity theorem.
Anti-Möbius transformation An orientation-reversing conformal automorphism of the Riemann sphere.
Argument principle The logarithmic derivative counts zeros minus poles inside a contour.
Automorphisms of the Riemann sphere Every holomorphic automorphism of the Riemann sphere is a Möbius transformation.
Casorati–Weierstrass theorem Near an essential singularity, the image of every punctured neighborhood is dense in the complex plane.
Cauchy estimate for holomorphic derivatives A bound on a complex neighborhood controls all derivatives on a smaller set with factorial constants.
Cauchy integral formula A holomorphic function and all its derivatives are recovered from boundary values.
Cauchy integral theorem Holomorphic functions have zero integral around null-homologous closed contours.
Cauchy–Fueter operators The first-order quaternionic differential operators analogous to the complex d-bar and d operators.
Cauchy–Riemann criterion A regularity-sensitive criterion for a function to be holomorphic.
Cauchy–Riemann equations The coordinate equations characterizing complex-linear real derivatives.
Classification of isolated singularities An isolated singularity is removable, a pole, or essential according to its Laurent principal part.
Common holomorphic neighborhood for a parameter family A single complex domain and bound that work for every member of a family near a real parameter set.
Complex Analysis Holomorphic functions, contour integration, singularities, conformal maps, and the projective geometry of the Riemann sphere.
Complex contour integral The integral of a complex-valued function along an oriented parametrized curve.
Complex derivative The derivative defined by a complex difference quotient independent of direction.
Complex domain A connected open subset of the complex plane.
Complex Monge–Ampère operator The determinant of the Levi matrix, extended as a measure for suitable plurisubharmonic functions.
Complex projective structure An atlas of local coordinates in the Riemann sphere whose transition maps are Möbius.
Continuity of quaternionic Monge–Ampère measures Locally uniform convergence of continuous quaternionic PSH functions implies weak convergence of their Hessian measures.
Cross-ratio A Möbius-invariant coordinate of an ordered quadruple on the projective line.
Cross-ratio invariance under Möbius transformations Every Möbius transformation preserves the cross-ratio of an ordered quadruple.
Cross-ratio-preserving bijections are Möbius A sphere bijection preserving every complex cross-ratio is a Möbius transformation.
Dirichlet theorem for the quaternionic Monge–Ampère equation Existence and uniqueness of a continuous quaternionic PSH solution on a bounded strictly pseudoconvex domain.
Entire function A complex-valued function holomorphic on the whole complex plane.
Entire function of several complex variables A complex-valued function holomorphic on all of complex Euclidean space.
Equal Schwarzians differ by Möbius postcomposition Two locally univalent maps on a connected domain have the same Schwarzian exactly when one is a Möbius postcomposition of the other.
Fundamental theorem of algebra (complex analysis) Every nonconstant complex polynomial has a complex root, with a proof via Liouville's theorem.
Generalized circle A circle in the complex plane or a straight line completed by the point at infinity.
Great Picard theorem Near an essential singularity, every complex value with at most one exception occurs infinitely often.
Harmonic function A twice differentiable function annihilated by the Laplacian, equivalently a continuous function with the local mean-value property.
Holomorphic dependence of a parameter integral Local integrable domination allows integration of a holomorphic family and passage of complex derivatives through the integral.
Holomorphic functions are analytic Complex differentiability on an open set forces a local convergent power-series expansion.
Holomorphic germ The local behavior of a holomorphic function near one point, modulo restriction to smaller neighborhoods.
Holomorphic L2 mean-value estimate The value of a holomorphic function is controlled by its L2 norm on any contained Euclidean ball.
Holomorphic logarithm of a nonvanishing function A branch L with exp(L)=f, available on a simply connected domain for a zero-free holomorphic function.
Holomorphic projective connection Local holomorphic coefficients transforming by the Schwarzian cocycle on a Riemann surface.
Hörmander L2 theorem for the d-bar equation A strictly plurisubharmonic weight gives a weighted L2 solution of the d-bar equation for closed (0,1)-forms.
Identity theorem Holomorphic functions agreeing on a set with an interior accumulation point agree everywhere on a domain.
Laurent series The unique expansion of a holomorphic function in positive and negative powers on an annulus.
Levi form of a function The Hermitian form defined by the mixed complex second derivatives of a twice differentiable real-valued function.
Liouville's theorem in complex analysis Every bounded entire function is constant.
Little Picard theorem A nonconstant entire function assumes every complex value with at most one exception.
Locally uniform limit theorem for holomorphic functions Uniform convergence on compact subsets preserves holomorphicity and all fixed-order derivatives.
Logarithmic derivative The meromorphic function f prime over f, whose residues record zeros and poles.
Maximum modulus principle A nonconstant holomorphic function cannot attain a local maximum of its modulus.
Maximum principle for harmonic functions A nonconstant harmonic function cannot attain an interior maximum or minimum.
Meromorphic function A function holomorphic away from isolated poles, equivalently locally a quotient of holomorphic functions.
Mixed quaternionic Monge–Ampère measure The polarized measure obtained from quaternionic Hessians of several plurisubharmonic functions.
Monodromy theorem Analytic continuation along all paths in a simply connected domain produces a single-valued holomorphic function.
Montel theorem Every locally uniformly bounded family of holomorphic functions is normal.
Möbius characterization by the Schwarzian A locally univalent holomorphic function has zero Schwarzian exactly when it is Möbius.
Möbius transformation A fractional linear automorphism of the Riemann sphere.
Möbius transformation group The group of fractional linear automorphisms of the Riemann sphere.
Nonvanishing on a neighborhood of a compact set A positive lower bound on a compact set persists on a sufficiently small neighborhood.
Normal family A family of holomorphic functions for which every sequence has a locally uniformly convergent subsequence.
Octonionic Hessian The octonionic Hermitian matrix of mixed Dirac derivatives of a real-valued function on the octonionic plane.
Octonionic Monge–Ampère measure The determinant-Hessian measure of a continuous octonionic plurisubharmonic function on the octonionic plane.
Octonionic plurisubharmonic function An upper-semicontinuous function on the octonionic plane whose restriction to every affine octonionic line is subharmonic.
Octonionic Radon transform The injective transform integrating a function on the octonionic plane over affine octonionic lines.
Open mapping theorem for holomorphic functions A nonconstant holomorphic function on a domain sends open sets to open sets.
Order of a zero or pole The integer exponent in the local factorization of a meromorphic function.
Pluriharmonic function A function whose restriction to every affine complex line is harmonic.
Plurisubharmonic function An upper-semicontinuous function whose restriction to every affine complex line is subharmonic.
Power of a holomorphic function using a chosen logarithm The branch-dependent expression exp(a L(z)) when L is a holomorphic logarithm of f.
Projective connections form an affine space Holomorphic projective connections, when they exist, form an affine space over holomorphic quadratic differentials.
Quaternionic Błocki formula An inclusion–exclusion identity for quaternionic Monge–Ampère measures under maxima and minima.
Quaternionic Hessian The hyperhermitian matrix of mixed Cauchy–Fueter derivatives of a real-valued function.
Quaternionic Monge–Ampère equation A nonlinear elliptic equation prescribing the Moore determinant of a quaternionic Hessian.
Quaternionic Monge–Ampère measure The Moore-determinant Hessian measure of a continuous quaternionic plurisubharmonic function.
Quaternionic plurisubharmonic function An upper-semicontinuous function on quaternionic space whose restriction to every right quaternionic line is subharmonic.
Rational function A quotient of complex polynomials, equivalently a holomorphic self-map of the Riemann sphere.
Relations among harmonic and plurisubharmonic functions The inclusion and intersection relations among H, SH, PSH, and PH on a complex domain.
Residue The coefficient of the inverse-linear term in a Laurent expansion.
Residue theorem A contour integral of a meromorphic function is the winding-number-weighted sum of its residues.
Riemann mapping theorem Every nonempty proper simply connected plane domain is biholomorphic to the unit disc.
Riemann sphere The extended complex plane as a compact Riemann surface, analytically equal to the complex projective line.
Rouché's theorem A strict boundary perturbation does not change the number of zeros inside a contour.
Schwarzian chain rule The Schwarzian derivative obeys a quadratic-differential cocycle rule under composition.
Schwarzian derivative A third-order differential invariant of a locally univalent holomorphic function.
Schwarzian derivative and second-order linear ODEs Ratios of independent solutions of a second-order linear equation have prescribed Schwarzian.
Sharp three-transitivity of the Möbius group A unique Möbius transformation carries any ordered triple of distinct sphere points to any other.
Strictly plurisubharmonic function A plurisubharmonic function whose Levi form has a locally positive lower bound.
Strictly quaternionic plurisubharmonic function A smooth quaternionic plurisubharmonic function with positive-definite quaternionic Hessian.
Strictly quaternionically pseudoconvex domain A smooth bounded quaternionic domain admitting local strictly quaternionic PSH defining functions.
Subharmonic function An upper-semicontinuous function dominated at each point by its local spherical averages.
Subharmonicity of the logarithmic modulus The logarithm of the modulus of a nonzero holomorphic function is subharmonic, with Laplacian equal to its zero-counting measure.
Upper-semicontinuous function A function whose value at a limit point is at least the limsup of nearby values.
Winding number The integer measuring how a closed contour winds around a point.

Convex analysis178 knowls

A nonnegative real below every epsilon is zero A nonnegative real number smaller than every positive real number must be zero.
Absorbing a subquadratic power A power below two can be bounded by any positive multiple of the square plus a controlled constant.
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 A nonempty set is affine exactly when it is a translate of a linear subspace.
Algebra of limits in normed spaces Limits commute with addition and scalar multiplication
Algebraic Interior (Core) Points from which every vector-space direction remains in the set for a sufficiently short segment.
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 set is a basis if and only if it is maximal among the linearly independent subsets.
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 Fenchel conjugate of a function's conjugate, central to lower-semicontinuous convex relaxation.
Bounded Linear Functional and Its Norm A linear functional is bounded exactly when it is continuous; its norm is the supremum of its absolute value on the unit ball.
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 Every Cauchy sequence in a metric space is bounded.
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 Every closed ball in a metric space is a closed subset.
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 a metric space, a point lies in a set's closure exactly when a sequence from the set converges to it.
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.
Conical combination A finite linear combination with nonnegative coefficients, without a normalization on their sum.
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 body A nonempty compact convex subset of a finite-dimensional real vector space.
Convex combination A weighted average of finitely many points with nonnegative weights summing to one
Convex cone A nonempty set closed under nonnegative linear combinations of two of its elements.
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 of a nonempty set consists of its finite convex combinations.
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 A jointly convex objective minimized over a convex feasible graph has a convex marginal value function.
Convexity on a convex subset via extension A function on a convex set is convex exactly when its extension by positive infinity is convex.
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 of an extended-real-valued 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 Assuming the axiom of choice, every 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 The ordered real line with positive and negative infinity, together with standard infimum and supremum conventions.
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) The integral of |fg| is bounded by the product of the conjugate L^p and L^q norms.
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 zero on a set and positive infinity outside it.
Interior The largest open set contained in a given set
Interior and closure of a convex set are convex In a normed space, the interior and closure of a convex set are convex.
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 containing zero and closed under addition and scalar multiplication.
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) An extended-real-valued functional measuring the dilation needed for a set to contain a vector.
Minkowski sum The set obtained by adding every point of one subset of a vector space to every point of another.
Mixed volume The symmetric multilinear polarization of volume with respect to Minkowski addition.
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
Octonionic pseudovolume A Spin(9)-invariant continuous valuation obtained from the octonionic Hessian measure of a support function.
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.
Planar cone defined by a quadratic inequality A positive longitudinal component and a quadratic transverse bound describe an open planar wedge.
Pluripotential construction of convex-body valuations Complex and quaternionic Hessian measures of support functions define continuous translation-invariant valuations.
Positive span The set of all finite nonnegative combinations of a specified family of vectors.
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 whose value on a line segment never exceeds the larger endpoint value.
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 a subset to each point and is convex when its graph is convex.
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 Disjoint convex sets can be separated strictly on the side of an open set.
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
Support function of a convex body The sublinear function recording the maximal value of each linear functional on a convex body.
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
Two-generator cone test An invertible pair of planar generators converts cone membership into two coordinate inequalities.
Uniform directional margin inside a cone Normalized vectors in a compact subset of an open cone stay a positive distance from its boundary even when their magnitudes vanish.
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
Valuation on convex bodies A finitely additive functional on convex bodies, with continuity and symmetry as additional properties.
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 equations23 knowls

Continuation criterion for a finite-dimensional ODE A solution can be extended past a finite endpoint if its time-state graph stays in a compact subset of the equation domain.
Continuous-dependence estimate for ODEs A Lipschitz coefficient bounds the effect of initial-data and forcing errors by Gronwall amplification.
Differential equation for the Gamma average An integration-by-parts identity gives a second-order equation for the normalized Gamma integral profile.
Differential equations Ordinary differential equations, evolution, stability and parameter dependence.
Duhamel estimate with a common envelope A propagator ratio bound preserves a source envelope at the cost of the interval length.
Duhamel formula for a linear ODE The initial state is propagated and every source contribution is propagated from its insertion time.
Factoring scalar damping from a matrix evolution A scalar damping term can be removed exactly by an integrating factor even when matrix coefficients do not commute.
Fundamental matrix and principal propagator An invertible matrix of homogeneous solutions that transports data between two times.
Gronwall inequality An integral inequality bounds accumulated amplification by an exponential.
Holomorphic dependence of local ODE solutions A holomorphic vector field gives a local solution holomorphic in time, initial data and parameters.
Initial-value problem for an ODE A differential equation together with the prescribed value of its state at one time.
Instability of a linear equilibrium Failure of uniform smallness of solutions from arbitrarily small initial data in a linear evolution.
Integrating factor for a scalar linear ODE Multiplication by exp of an integral turns y prime plus a(t)y into one derivative.
Linear evolution under a moving normal constraint The projection term needed to preserve orthogonality to a time-dependent normal.
Linear ordinary differential system A system y prime equals A(t)y plus a prescribed forcing term.
Ordinary differential equation An equation relating a function of one independent variable to its derivatives.
Peano–Baker series The time-ordered matrix series for the propagator of a nonautonomous linear ODE.
Picard iteration Successive substitution into the integral equation for an initial-value problem.
Picard–Lindelöf local existence and uniqueness theorem Continuity in time and local uniform Lipschitz control in the state give a unique local solution.
Regular inverse of Y d2/dY2 plus nu d/dY The normalized solution of YG double prime plus nu G prime equals F, selected by regularity at zero.
Regular singular point of a second-order linear ODE A point where the first- and zeroth-order coefficients have at most first- and second-order poles after normalization.
Scalar comparison barriers for an ODE Subsolutions and supersolutions bound a scalar solution when the vector field is locally Lipschitz.
Smooth dependence of ODE solutions on parameters Smooth coefficients and initial data yield solutions smooth in parameters on a common local domain.

Differential geometry323 knowls

Action-angle coordinates Canonical coordinates adapted to invariant tori in which an integrable Hamiltonian depends only on action variables.
Algebra of smooth functions The unital commutative algebra of scalar-valued smooth functions on a smooth manifold under pointwise operations.
Almost symplectic manifold A smooth manifold equipped with a pointwise nondegenerate two-form, without a closedness requirement.
Almost-complex map A smooth map whose tangent map intertwines the almost-complex structures.
Almost-complex structure A smooth tangent-bundle endomorphism whose square is minus the identity.
Almost-Hermitian manifold A smooth manifold with an almost-complex structure and a compatible Riemannian metric.
Almost-quaternionic manifold A real manifold whose tangent bundle carries a rank-three family of endomorphisms locally satisfying the quaternion relations.
Ambient isotopy An ambient isotopy is a smooth family of diffeomorphisms of a manifold beginning at the identity.
Anosov flow A flow whose tangent dynamics split uniformly into flow, exponentially contracting, and exponentially expanding directions.
Associated graded Clifford algebra The Clifford filtration has exterior algebra as its associated graded algebra.
Atiyah–Guillemin–Sternberg convexity theorem The moment-map image of a compact connected Hamiltonian torus space is the convex hull of its fixed-point images, and its fibers are connected.
Bianchi cusps and ideal classes The cusp orbits of a Bianchi group correspond bijectively to ideal classes.
Bianchi orbifold The hyperbolic orbifold obtained by quotienting three-space by a Bianchi group.
Biholomorphism A bijective holomorphic map whose inverse is holomorphic.
Biholomorphism group The automorphism group of a complex manifold in the category of complex manifolds and holomorphic maps.
Bordism group The abelian group of bordism classes of closed manifolds under disjoint union.
Boundary and interior of a manifold with boundary The boundary and interior of a manifold with boundary are the points represented respectively on and off the bounding hyperplane in half-space coordinates.
Boundary chart A boundary chart gives local coordinates on a manifold with boundary in a relatively open subset of a Euclidean half-space.
Boundary defining function A smooth nonnegative function that vanishes simply and exactly on the boundary of a manifold.
Boundary face of a manifold with corners A globally chosen connected boundary hypersurface obtained by continuing one local boundary component through a manifold with corners.
Boundary orientation The orientation on a manifold boundary determined by placing an outward normal before a positive boundary basis.
Bump function A smooth function with compact support, often chosen to equal one on a prescribed closed set.
C-infinity ring An algebraic object carrying an operation for every smooth map between finite-dimensional real affine spaces.
Calabi–Yau manifold A compact connected Kähler manifold whose holomorphic canonical bundle is trivial.
Calabi–Yau theorem Each Kähler class on a compact Kähler manifold contains a unique metric with any prescribed Ricci form representing the first Chern class.
Canonical bundle of a complex manifold The holomorphic line bundle of holomorphic differential forms of top degree on a complex manifold.
Canonical moment map for a cotangent-lifted action The cotangent lift of a Lie group action is Hamiltonian with moment map given by pairing covectors with infinitesimal generators.
Canonical symplectic form on a cotangent bundle The symplectic form obtained as minus the exterior derivative of the tautological one-form on a cotangent bundle.
Category of complex manifolds The category whose objects are finite-dimensional complex manifolds and whose morphisms are holomorphic maps.
Category of smooth manifolds The category whose objects are smooth manifolds and whose morphisms are smooth maps.
Category of symplectic manifolds and symplectic maps The category whose objects are symplectic manifolds and whose morphisms preserve the symplectic forms by pullback.
Cauchy hypersurface A subset of a spacetime met exactly once by every inextendible timelike curve.
Causal curve A curve in a time-oriented Lorentzian manifold whose tangent vectors are everywhere future-directed or everywhere past-directed and nonspacelike.
Causal diamond The events causally after one event and causally before another.
Celestial sphere The sphere of future null directions, naturally identified with CP¹.
Characteristic distribution of a coisotropic submanifold The null distribution of the restricted symplectic form on a coisotropic submanifold.
Characteristic foliation The foliation integrating the null distribution of the symplectic form restricted to a coisotropic submanifold.
Chirality operator The normalized complex Clifford volume element acting on an even-dimensional spinor module.
Chronological and causal future The sets reachable from an event by future-directed timelike or causal curves.
Classical phase space A symplectic manifold whose points represent the instantaneous states of a finite-dimensional classical system.
Classification of complex Clifford modules The irreducible ungraded modules of a complex Clifford algebra are determined by the parity of the dimension.
Clean intersection An intersection of submanifolds whose set and tangent spaces fit together without a tangent-space jump.
Clifford algebra The universal associative algebra generated by a quadratic module subject to the Clifford relation.
Clifford algebra as a super-enveloping quotient A Clifford algebra is obtained by fixing the central character in the enveloping algebra of a quadratic Lie superalgebra.
Clifford module A module carrying a compatible representation of a Clifford algebra.
Coadjoint orbit An orbit of a Lie group acting on the dual of its Lie algebra by the coadjoint action.
Cobordism A compact manifold whose boundary is identified with a pair of closed manifolds.
Cobordism category A category whose objects are closed manifolds and whose morphisms are cobordisms composed by gluing.
Cobordism equivalence relation The equivalence relation on closed manifolds generated by the existence of a cobordism between them.
Codifferential The degree-lowering formal adjoint of the exterior derivative on an oriented Riemannian manifold.
Coisotropic reduction A smooth quotient of a coisotropic submanifold by its characteristic leaves inherits a unique symplectic form.
Coisotropic submanifold A submanifold whose tangent spaces contain their symplectic orthogonal complements.
Coisotropic subspace A subspace of a symplectic vector space that contains its symplectic orthogonal complement.
Collar neighborhood theorem Every smooth manifold with boundary has a neighborhood of its boundary diffeomorphic to a product with a half-open interval.
Comoment map A Lie algebra homomorphism assigning to each infinitesimal symmetry its Hamiltonian function.
Compactly supported de Rham cohomology The cohomology of the de Rham complex restricted to compactly supported differential forms.
Compactly supported differential form A differential form whose nonzero locus has compact closure.
Compatible almost-complex structure An almost-complex structure that preserves a symplectic form and makes its associated bilinear form positive definite.
Completely integrable Hamiltonian system A Hamiltonian system in dimension two n with n independent first integrals in pairwise involution.
Complex atlas A covering family of complex coordinate charts whose transition maps are holomorphic.
Complex coordinate chart A homeomorphism from an open manifold neighborhood to an open subset of complex Euclidean space.
Complex manifold A space locally modeled on complex Euclidean space with holomorphic transition maps.
Complex projective space The compact complex manifold of complex lines in complex Euclidean space.
Complex submanifold A subset of a complex manifold that is locally a complex coordinate subspace.
Complex torus A quotient of a finite-dimensional complex vector space by a full lattice.
Conformal map A smooth map that pulls the target Riemannian metric back to a positive pointwise multiple of the source metric.
Connection Laplacian The negative metric trace of the second covariant derivative on sections of a vector bundle.
Conormal bundle is Lagrangian The conormal bundle of an embedded submanifold is Lagrangian in the ambient cotangent bundle.
Constant rank theorem A smooth map of locally constant rank has coordinates in which it is a coordinate projection followed by a coordinate inclusion.
Coordinate chart as an isomorphism of smooth manifolds The coordinate map of a smooth chart is a diffeomorphism between two open smooth manifolds.
Coordinate transformation law for a Riemannian metric Metric coefficient matrices transform by inverse congruence under a change of coordinates.
Corner stratification The corner stratification partitions a manifold with corners according to the number of vanishing local boundary coordinates.
Cotangent lift The contragredient diffeomorphism of cotangent bundles induced by a diffeomorphism of base manifolds.
Critical point of a smooth map A point where the differential of a smooth map is not surjective onto the target tangent space.
Critical value of a smooth map A target point whose fiber contains a critical point of the smooth map.
Crossing form The quadratic form that records the infinitesimal crossing of a path of Lagrangian subspaces through a fixed one.
Cusp of a finite-volume hyperbolic three-orbifold An end represented by a horoball modulo its boundary-point stabilizer.
Darboux theorem for symplectic manifolds Every symplectic form is locally equivalent to the standard constant symplectic form.
de Rham complex The cochain complex of smooth differential forms with the exterior derivative.
de Rham homotopy operator The degree-minus-one operator obtained by integrating a pulled-back form along a smooth homotopy, yielding a chain homotopy between endpoint pullbacks.
de Rham theorem Integration identifies de Rham cohomology naturally with singular cohomology with real coefficients.
Degree of a smooth map An integer measuring the signed number of preimages of a regular value under a proper map of oriented manifolds.
Depth of a point in a manifold with corners The number of local boundary coordinates that vanish at a point of a manifold with corners.
Diagonal transversality criterion Two smooth maps are transverse exactly when their product map is transverse to the diagonal.
Diffeomorphism group The group of all smooth self-diffeomorphisms of a smooth manifold under composition.
Diffeomorphism groupoid of smooth manifolds The maximal subgroupoid of the smooth-manifold category, retaining every manifold but only diffeomorphisms.
Differentiable flow A C1 action of the real line on a manifold by differentiable time maps.
Differential form of type (p,q) A complex differential form whose covector factors have a fixed holomorphic and antiholomorphic bidegree.
Differential Geometry Definitions and results in differential geometry, including manifolds, tangent spaces, and related structures.
Differential operator between vector bundles A linear map on smooth bundle sections whose value at a point depends on only finitely many derivatives of the section there.
Dirac spinor An element or field valued in a chosen complex spinor module.
Disjoint union of smooth manifolds A finite or countable disjoint union inherits a smooth structure component by component and is the categorical coproduct of its components.
Dolbeault cohomology The cohomology of the d-bar complex of smooth differential forms on a complex manifold.
Dolbeault complex The cochain complex of smooth forms of fixed holomorphic degree with differential given by the d-bar operator.
Dolbeault operators The two bidegree components of the exterior derivative on a complex manifold.
Double of a manifold with boundary The boundaryless smooth manifold formed by gluing two copies of a manifold along their common boundary.
Einstein manifold A Riemannian manifold whose Ricci tensor is a constant multiple of the metric.
Elliptic differential operator A differential operator whose principal symbol is invertible at every nonzero cotangent vector.
Embedded submanifold A subset of a smooth manifold that is locally a coordinate plane in ambient smooth charts.
Equivariant moment map A moment map that intertwines a Lie group action with the coadjoint action on the dual Lie algebra.
Exact symplectic manifold A symplectic manifold whose symplectic form is the exterior derivative of a global one-form.
Fine sheaf A sheaf admitting endomorphism-valued partitions of unity subordinate to locally finite open covers.
First integral A smooth function that is constant along every trajectory of a dynamical system.
Flux group The subgroup of first cohomology realized as fluxes of loops of symplectomorphisms.
Flux homomorphism The homomorphism sending a symplectic-isotopy class to the integrated cohomology class of its generating contraction forms.
Formal adjoint of a differential operator The formal adjoint is the differential operator obtained by transferring derivatives across an integral pairing without boundary terms.
Frölicher spectral sequence The spectral sequence of the type filtration on the complex de Rham complex of a complex manifold.
Fubini–Study metric The canonical unitary-invariant Kähler metric on complex projective space.
Full faithfulness of the smooth-function functor Pullback identifies smooth maps with C-infinity-ring morphisms in the opposite direction.
Functions in involution A family of smooth functions whose pairwise Poisson brackets vanish.
Fundamental 2-form of an almost-Hermitian manifold The nondegenerate real 2-form obtained by pairing an almost-complex structure with its compatible metric.
Gaussian curvature The intrinsic sectional curvature of a Riemannian surface at each point.
Geodesic A constant-speed curve whose covariant acceleration vanishes on a Riemannian manifold.
Geodesic flow The flow on the unit tangent bundle that advances each unit tangent vector along its geodesic.
Global hyperbolicity and Cauchy hypersurfaces A time-oriented Lorentzian manifold is globally hyperbolic if and only if it admits a Cauchy hypersurface.
Globally hyperbolic spacetime A strongly causal spacetime in which every causal diamond is compact.
Grassmannian The parameter space of all subspaces of a fixed dimension in a finite-dimensional vector space.
Hamilton's equations The canonical-coordinate differential equations determined by a Hamiltonian function and the standard symplectic form.
Hamiltonian diffeomorphism A symplectomorphism obtained as the time-one map of a Hamiltonian isotopy.
Hamiltonian flow The local or global flow generated by a Hamiltonian vector field.
Hamiltonian function A Hamiltonian function is a smooth real-valued function on a symplectic manifold that determines a Hamiltonian vector field.
Hamiltonian isotopy A symplectic isotopy generated by a smooth time-dependent Hamiltonian.
Hamiltonian Lie group action A symplectic Lie group action equipped with a compatible equivariant moment map.
Hamiltonian system A symplectic phase space together with a Hamiltonian function specifying its autonomous dynamics.
Hamiltonian torus action A Hamiltonian torus action is a symplectic action of a compact torus equipped with a torus-invariant moment map.
Hamiltonian vector field The vector field obtained from the differential of a Hamiltonian function through a symplectic form.
Hard Lefschetz theorem Powers of a Kähler class give isomorphisms between complementary cohomological degrees on a compact Kähler manifold.
Harmonic differential form A differential form annihilated by the Hodge Laplacian of a Riemannian metric.
Hermitian manifold A complex manifold equipped with a Riemannian metric invariant under its complex structure.
HKT metric A hyper-Hermitian metric whose three Hermitian structures share a metric connection with skew torsion.
Hodge decomposition of a compact Kähler manifold The decomposition of complex de Rham cohomology into Dolbeault cohomology groups of fixed bidegree.
Hodge decomposition on a compact Riemannian manifold Smooth forms on a compact oriented Riemannian manifold split orthogonally into harmonic, exact, and coexact parts.
Hodge filtration The decreasing filtration obtained by collecting the Hodge components whose first bidegree is at least a prescribed integer.
Hodge Laplacian The Hodge Laplacian is the degree-preserving elliptic operator obtained by anticommuting the exterior derivative and codifferential.
Hodge numbers The dimensions of the bidegree components in the Hodge decomposition of a compact Kähler manifold.
Hodge star operator The Hodge star is the degree-complementing operator on differential forms determined by a metric and an orientation.
Hodge theorem Every de Rham cohomology class on a compact oriented Riemannian manifold has a unique harmonic representative.
Hodge–Riemann bilinear relations The positivity theorem for the Hermitian intersection form on primitive cohomology of a compact Kähler manifold.
Holomorphic cotangent bundle The holomorphic dual of the holomorphic tangent bundle of a complex manifold.
Holomorphic isometric immersion A holomorphic map between Hermitian manifolds that pulls the target metric back to the source metric.
Holomorphic line bundle A holomorphic vector bundle whose fibers have complex dimension one.
Holomorphic map A complex-differentiable map between open subsets or complex manifolds.
Holomorphic section A section of a holomorphic bundle that is holomorphic as a map into the total space.
Holomorphic symplectic form A closed holomorphic two-form that is nondegenerate at every point of a complex manifold.
Holomorphic symplectic manifold A complex manifold equipped with a closed nondegenerate holomorphic two-form.
Holomorphic tangent bundle The holomorphic vector bundle whose fibers are the complex tangent spaces of a complex manifold.
Holomorphic vector bundle A holomorphic vector bundle is a complex vector bundle whose local trivializations have holomorphic transition functions.
Holomorphic vector-bundle morphism A holomorphic map of vector-bundle total spaces that is linear on fibers and covers a holomorphic base map.
Homotopy invariance of de Rham cohomology Smoothly homotopic maps induce the same pullback homomorphism on de Rham cohomology.
Horoball in hyperbolic three-space The region on the ideal-center side of a horosphere.
Horocycle flow The stable or unstable unipotent flow on the unit tangent bundle of a hyperbolic surface.
Horosphere in hyperbolic three-space A surface centered at one ideal boundary point, modeled by a horizontal plane.
Hyper-Hermitian manifold A hypercomplex manifold with a Riemannian metric Hermitian for all three complex structures.
Hyperbolic three-orbifold A three-orbifold locally modeled on hyperbolic space with isometric changes of charts.
Hyperbolic three-space The simply connected complete three-dimensional Riemannian manifold of constant sectional curvature −1.
Hypercomplex manifold A smooth manifold carrying three integrable complex structures that satisfy the quaternion relations.
Hyperkähler isometric immersion A triholomorphic map of hyperkähler manifolds that pulls the target metric back to the source metric.
Hyperkähler isometry A diffeomorphism of hyperkähler manifolds that preserves the metric and each member of the specified quaternionic triple.
Hyperkähler manifold A Riemannian manifold carrying a parallel quaternionic triple of complex structures.
Immersed submanifold An immersed submanifold is a manifold equipped with an injective immersion into an ambient smooth manifold.
Infinitesimal generator of a Lie group action An infinitesimal generator is the vector field obtained by differentiating a one-parameter subgroup acting on a manifold.
Integrable almost-complex structure An almost-complex structure induced locally by holomorphic coordinate charts.
Integration of differential forms Integration assigns a real number to a compactly supported top-degree form on an oriented manifold.
Inverse function theorem for manifolds A smooth map whose differential is invertible at a point is a diffeomorphism between suitable neighborhoods of that point and its image.
Irreducible holomorphic symplectic manifold A compact simply connected Kähler manifold whose holomorphic two-forms are spanned by an everywhere nondegenerate form.
Isotopy extension theorem A smooth isotopy of a compact embedded manifold extends to a smooth ambient isotopy.
Isotropic Grassmannian The parameter space of fixed-dimensional isotropic subspaces of a symplectic vector space.
Isotropic submanifold An immersed submanifold on which the ambient symplectic form pulls back to zero.
Isotropic subspace A linear subspace on which the ambient symplectic form vanishes identically.
Iterated tangent bundle A tangent bundle obtained by repeatedly applying the tangent functor to a smooth manifold.
Jet of a smooth map A finite-order jet records the derivatives of a smooth map at one point through a specified order.
Kirillov–Kostant–Souriau symplectic form The canonical invariant symplectic form on a coadjoint orbit.
Kähler class The degree-two real de Rham cohomology class represented by a Kähler form.
Kähler form A closed real positive form of type one-one on a complex manifold.
Kähler identities Commutator formulas relating the Dolbeault operators, their formal adjoints, and the Lefschetz operators of a Kähler metric.
Kähler manifold A Hermitian manifold whose fundamental two-form is closed.
Kähler metric A Hermitian metric whose associated fundamental two-form is closed.
Lagrangian Gauss map The tangent-plane map of a Lagrangian immersion, valued intrinsically in the Lagrangian-Grassmannian bundle.
Lagrangian Grassmannian The smooth parameter space of all Lagrangian subspaces of a finite-dimensional symplectic vector space.
Lagrangian neighborhood theorem A neighborhood of a Lagrangian submanifold has the cotangent bundle as its canonical local symplectic model.
Lagrangian submanifold A half-dimensional submanifold on which the ambient symplectic form vanishes.
Lagrangian subspace A subspace equal to its symplectic orthogonal, equivalently a maximal isotropic subspace.
Laplace–Beltrami operator The negative metric trace of the covariant Hessian, with a sign chosen to be nonnegative in Riemannian signature.
Lefschetz operator The degree-two operator given by wedging with a symplectic form or multiplying by its cohomology class.
Linear coisotropic reduction The symplectic quotient of a coisotropic subspace by its characteristic subspace.
Linear symplectomorphism An invertible linear map that preserves the symplectic forms on its source and target.
Liouville vector field The unique vector field dual to a chosen primitive of an exact symplectic form.
Liouville–Arnold theorem A compact connected regular fiber of a completely integrable Hamiltonian system is a torus with local action-angle coordinates.
Local C-infinity-ringed space A space with a sheaf of C-infinity rings whose stalks are local, with morphisms preserving all smooth operations.
Local HKT potential theorem Smooth strictly quaternionic PSH functions are exactly the local potentials of HKT metrics.
Local Kähler potential A real smooth function whose complex Hessian is a given Kähler form on a coordinate neighborhood.
Lorentzian Dirac operator The first-order hyperbolic-type operator obtained by Clifford contraction of the Lorentzian spin connection.
Lorentzian manifold A pseudo-Riemannian manifold with one timelike direction at every point.
Lorentzian spin structure A lift of the oriented, time-oriented Lorentz frame bundle through the spin double cover.
Lorentzian spinor bundle The Clifford module bundle associated to a Lorentzian spin structure and a spin representation.
Majorana spinor A complex spinor fixed by a compatible Spin-equivariant real structure.
Majorana–Weyl spinor A chiral spinor fixed by a compatible real structure that preserves chirality.
Map transverse to a submanifold A smooth map whose differential and the submanifold tangent space span the target at every inverse-image point.
Mapping class group of a smooth manifold The group of smooth self-diffeomorphisms of a manifold modulo smooth isotopy.
Marsden–Weinstein–Meyer reduction theorem A free proper Hamiltonian action reduces a regular zero level to a symplectic manifold of dimension lowered by twice the group dimension.
Maslov class of a Lagrangian submanifold The pullback of the universal Maslov class along the Lagrangian Gauss map of a Lagrangian immersion.
Maslov class of the Lagrangian Grassmannian The canonical generator of the first integral cohomology of the real Lagrangian Grassmannian.
Maslov cycle The singular incidence hypersurface of Lagrangian planes that fail to be transverse to a fixed Lagrangian plane.
Maslov index The integer measuring the winding of a loop of Lagrangian subspaces, equivalently its signed intersection with a Maslov cycle.
Mayer–Vietoris sequence for de Rham cohomology An open two-set cover produces a long exact sequence relating the de Rham cohomology of the manifold, the two open sets, and their intersection.
Moment maps as Poisson maps An equivariant moment map is a Poisson map to the dual Lie algebra with its compatible Lie–Poisson bracket.
Moment polytope The convex-polytope image of the moment map for a compact connected Hamiltonian torus space.
Moment-map component The scalar Hamiltonian obtained by pairing a moment map with one Lie algebra element.
Moser stability theorem A cohomologically constant smooth family of symplectic forms on a compact manifold is trivialized by an isotopy.
Neat embedding A smooth embedding between manifolds with boundary that is exactly boundary preserving and transverse to the target boundary.
Neat submanifold A boundary-compatible embedded submanifold that meets the ambient boundary transversely and has no other boundary.
Newlander–Nirenberg theorem A smooth almost-complex structure is integrable exactly when its Nijenhuis tensor vanishes.
Nijenhuis tensor of an almost-complex structure The tensor measuring the failure of an almost-complex structure to be integrable.
Normal bundle The quotient of the ambient tangent bundle along an embedded submanifold by its tangent bundle.
Obata connection The unique torsion-free connection preserving the three complex structures of a hypercomplex manifold.
Octonionic projective line The eight-sphere obtained as the projective quotient of the unit sphere in the octonionic plane.
Open submanifold An open subset of a smooth manifold with the smooth structure obtained by restricting charts.
Orientability of a smooth manifold A smooth manifold is orientable when its tangent bundle admits an orientation.
Orientation of a smooth manifold An orientation of a smooth manifold is a continuous choice of orientation for all of its tangent spaces.
Oriented cobordism A cobordism whose orientation induces the reversed incoming and given outgoing boundary orientations.
Oriented Grassmannian The manifold whose points are fixed-dimensional real subspaces equipped with orientations.
Parametric transversality theorem A smooth family transverse to a submanifold has transverse members for almost every parameter.
Poincaré disk model The unit disk with its complete metric of constant curvature minus one, a model of the hyperbolic plane.
Poincaré lemma Every closed differential form of positive degree is locally exact.
Poisson algebra of smooth functions The commutative algebra of smooth functions equipped with the Poisson bracket induced by a symplectic form.
Poisson bracket on a symplectic manifold The Lie bracket on smooth functions induced by the inverse of a symplectic form.
Poisson manifold A smooth manifold whose smooth functions carry a Lie bracket that is a derivation in each argument.
Poisson map A Poisson map is a smooth map whose pullback preserves Poisson brackets.
Polarized Hodge structure A pure Hodge structure equipped with a parity-compatible bilinear form satisfying Hodge orthogonality and positivity.
Primitive cohomology The kernel of the appropriate power of the Lefschetz operator on the cohomology of a compact Kähler manifold.
Principal symbol of a differential operator The homogeneous bundle map on the cotangent bundle obtained from the highest-order part of a differential operator.
Product manifold The Cartesian product of smooth manifolds with the smooth structure generated by product charts.
Proper injective immersion is an embedding A proper injective smooth immersion between manifolds is a smooth embedding.
Proper smooth map A smooth map for which the inverse image of every compact set is compact.
Pseudo-Riemannian manifold A smooth manifold equipped with a nondegenerate symmetric metric whose signature is constant on each connected component.
Pure Hodge structure A finite-dimensional real vector space whose complexification decomposes into conjugate bidegree pieces of fixed total weight.
Quaternion-Kähler manifold A Riemannian manifold of dimension at least eight whose holonomy is contained in Sp(n)Sp(1).
Quaternion-Kähler manifolds are Einstein Every quaternion-Kähler manifold of real dimension at least eight has Ricci tensor proportional to its metric.
Quaternionic manifold An almost-quaternionic manifold whose rank-three quaternionic structure is preserved by a torsion-free connection.
Quaternionic plurisubharmonic function on a hypercomplex manifold A function whose del-del-J form is a nonnegative real (2,0)-form on a hypercomplex manifold.
Quaternionic-Hermitian manifold An almost-quaternionic manifold with a Riemannian metric invariant under every admissible local complex structure.
Rank of a smooth map The dimension of the image of a smooth map's differential at a specified point.
Real projective space The smooth manifold of real lines through the origin in real Euclidean space.
Regular level set The preimage of a regular value of a smooth map between manifolds.
Relative Maslov index The Maslov index of a pair of moving Lagrangian subspaces, including a crossing-form definition for nontransverse endpoints.
Ricci curvature The symmetric covariant two-tensor obtained by tracing the Riemann curvature tensor.
Ricci form The real closed (1,1)-form obtained from the Ricci curvature of a Kähler metric.
Riemann curvature tensor The Riemann curvature tensor measures the failure of Levi–Civita covariant derivatives to commute.
Riemann surface A connected complex manifold of complex dimension one.
Riemannian isometric immersion A smooth map whose pullback of the target Riemannian metric is the source metric.
Riemannian manifold A smooth manifold equipped with a smoothly varying positive-definite inner product on its tangent spaces.
Rotating hyperkähler isometry A Riemannian isometry that carries one specified hyperkähler triple to a constant SO(3)-rotation of the other.
Sard's theorem Sard's theorem states that the critical values of a smooth map between finite-dimensional manifolds form a measure-zero set.
Scalar curvature The metric trace of the Ricci curvature.
Sectional curvature The curvature assigned by a Riemannian metric to each two-dimensional tangent plane.
Sheaf of holomorphic functions The sheaf assigning to each open subset of a complex manifold its algebra of holomorphic functions.
Sheaf of smooth functions The sheaf assigning to each open subset of a smooth manifold its algebra of smooth real-valued functions.
Shifting trick A construction that turns reduction at a coadjoint value into zero-level reduction using the opposite coadjoint orbit.
Singular symplectic reduction Symplectic reduction without regularity or freeness, producing a stratified quotient rather than generally a manifold.
Smooth effective orbifold A space with compatible smooth local charts that are quotients by finite effective group actions.
Smooth exhaustion function A proper smooth nonnegative function whose compact sublevel sets exhaust a smooth manifold.
Smooth fiber product The manifold of pairs having the same image under two transverse smooth maps.
Smooth homotopy A smooth homotopy is a smoothly parameterized one-parameter deformation between smooth maps.
Smooth isotopy A smooth one-parameter family of smooth embeddings joining two given embeddings.
Smooth manifold with boundary A smooth manifold with boundary is locally modeled on open subsets of a closed Euclidean half-space.
Smooth manifold with corners A smooth space locally modeled on Euclidean orthants of possibly varying codimension.
Smooth map between manifolds with boundary A smooth map between manifolds with boundary has coordinate representatives that extend smoothly across the bounding hyperplanes.
Smooth map between manifolds with corners In Joyce's convention, a coordinate-smooth map that pulls each target boundary defining function back either to zero or to one source boundary defining function.
Smooth splitting of globally hyperbolic spacetimes Every globally hyperbolic spacetime has a smooth Cauchy temporal function and splits smoothly into time times a Cauchy hypersurface.
Smooth Urysohn lemma Disjoint closed subsets of a smooth manifold can be separated by a smooth function valued between zero and one.
Space of smooth maps The space of smooth maps from one smooth manifold to another, with topology and smooth structure specified by the chosen mapping-space framework.
Spinor bundle The vector bundle associated to a spin structure through a spin representation.
Spinor module A chosen Clifford module whose restriction to the spin group is a spin representation.
Stokes' theorem Generalization of the fundamental theorem of calculus to differential forms on oriented manifolds with boundary.
Strong causality The condition that every event has arbitrarily small neighborhoods which causal curves cannot leave and then re-enter.
Submanifold chart An ambient smooth chart in which an embedded submanifold becomes a coordinate plane.
Symmetric monoidal structure on the cobordism category The symmetric monoidal structure on a cobordism category whose tensor product is disjoint union and whose unit is the empty manifold.
Symplectic basis An ordered basis in which a symplectic form has its standard block matrix.
Symplectic basis theorem Every finite-dimensional symplectic vector space has even dimension and a basis in standard symplectic form.
Symplectic embedding A smooth embedding that preserves the symplectic forms by pullback.
Symplectic isotopy A smooth path of symplectomorphisms beginning at the identity, generated by vector fields with closed contraction one-forms.
Symplectic leaf A symplectic leaf is a maximal connected integral manifold of the characteristic distribution of a Poisson manifold.
Symplectic Lie group action A smooth Lie group action whose transformations preserve the symplectic form.
Symplectic linear map A linear map that pulls the target symplectic form back to the source symplectic form.
Symplectic manifold A smooth manifold equipped with a closed, nondegenerate 2-form.
Symplectic map A smooth map between symplectic manifolds that pulls the target symplectic form back to the source symplectic form.
Symplectic orthogonal complement The subspace of vectors symplectically orthogonal to every vector in a specified subspace.
Symplectic quotient The orbit space of a moment-map level set by the stabilizer of its value.
Symplectic reduction at nonzero momentum Regular reduction at a nonzero moment-map value uses the coadjoint stabilizer of that value.
Symplectic submanifold An embedded submanifold on which the ambient symplectic form restricts nondegenerately.
Symplectic subspace A linear subspace on which the ambient symplectic form remains nondegenerate.
Symplectic vector field A vector field whose local flow preserves the symplectic form.
Symplectic vector space A finite-dimensional real vector space equipped with a nondegenerate alternating bilinear form.
Symplectomorphism A diffeomorphism between symplectic manifolds that preserves their symplectic forms by pullback.
Symplectomorphism group The group of all self-symplectomorphisms of a fixed symplectic manifold.
Tangent functor The covariant endofunctor on smooth manifolds that assigns tangent bundles to manifolds and differentials to smooth maps.
Tangent Space The vector space of derivations at a point of a smooth manifold.
Tangent space of a Grassmannian The tangent space at a subspace in a Grassmannian is canonically the space of linear maps from that subspace to its quotient.
Tautological bundle on a Grassmannian The rank-k vector bundle over a Grassmannian whose fiber over a k-plane is that plane itself.
Tautological one-form on a cotangent bundle The canonical one-form that evaluates a cotangent vector on the projection of a tangent vector to the base.
The d-c operator A real differential operator on a complex manifold obtained from the difference of the two Dolbeault operators.
The del-J operator The first-order differential operator obtained by conjugating d-bar with a second complex structure on a hypercomplex manifold.
The h-principle Every formal solution can be deformed through formal solutions to an actual jet solution.
The ∂∂̄-lemma On a compact Kähler manifold, a pure-type closed form that is exact in one standard complex is ∂∂̄-exact.
Thom transversality theorem Smooth maps whose jet extensions are transverse to a fixed jet-space submanifold form a residual set.
Time orientation A continuous choice of one component of the timelike cone on a Lorentzian manifold.
Time-dependent Hamiltonian system A symplectic phase space with a smoothly time-varying Hamiltonian that generates nonautonomous dynamics.
Transverse intersection theorem Transverse embedded submanifolds intersect in an embedded submanifold of the expected codimension.
Transverse preimage theorem The inverse image of an embedded submanifold under a transverse map is an embedded submanifold of the same codimension.
Transverse smooth maps Smooth maps to a common target whose differentials jointly span the target tangent space over every coincidence.
Transverse submanifolds Submanifolds whose tangent spaces together span the ambient tangent space at every intersection point.
Triholomorphic map A smooth map between hypercomplex manifolds that preserves each member of the ordered quaternionic triple.
Tubular neighborhood A neighborhood of an embedded submanifold modeled by a neighborhood of the zero section in its normal bundle.
Tubular neighborhood theorem Every embedded submanifold has a neighborhood modeled on a neighborhood of the zero section in its normal bundle.
Twistor space of a quaternionic manifold The sphere bundle whose fiber consists of the compatible complex structures in a quaternionic structure.
Type decomposition of the complexified tangent bundle The splitting of a complexified tangent bundle into the positive and negative imaginary eigenspaces of an almost-complex structure.
Uniqueness and ambiguity of moment maps Two moment maps for the same action differ by a locally constant dual-Lie-algebra-valued function, constrained by equivariance.
Variation of Hodge structure A local system whose fibers carry Hodge structures varying holomorphically and satisfying Griffiths transversality.
Volume form A nowhere-vanishing top-degree differential form that specifies oriented volume on a smooth manifold.
Volume formula for a Bianchi orbifold The curvature-minus-one volume is the discriminant factor times a Dedekind zeta value.
Weyl spinor A complex spinor lying in one of the two half-spin representations in even dimension.
Whitney C-infinity topology The strong topology on smooth mapping spaces defined by locally varying control of all finite jets.
Whitney embedding theorem Every positive-dimensional smooth manifold embeds smoothly in Euclidean space of twice its dimension.

Discrete structures16 knowls

Boundary of a finite region Standard notions of boundary for a finite subset of a graph or lattice.
Countable graph A graph whose vertex set is finite or countably infinite.
Directed acyclic graph A directed graph containing no directed cycles.
Directed cycle A positive-length closed directed path with no repetitions except its common initial and final vertex.
Directed graph A vertex set equipped with a specified set of ordered edges.
Directed path A finite sequence of vertices joined by edges in their prescribed direction.
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.
Greedy coloring of a countable bounded-degree graph A countable graph of degree at most Delta has a proper coloring with Delta plus one colors.
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.
Proper graph coloring Assignment of colors to vertices so adjacent vertices receive different colors.
Support intersection graph The graph connecting distinct labels whose assigned support sets intersect.
Vertex degree The number of neighbors of a vertex in a simple undirected graph.

Fiber bundles359 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.
Anti-self-dual moduli space The gauge-equivalence classes of anti-self-dual connections on a fixed principal bundle over an oriented Riemannian four-manifold.
ASD deformation complex The elliptic complex whose cohomology records infinitesimal gauge symmetries, deformations, and obstructions of an anti-self-dual connection.
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.
Atiyah–Bott symplectic form The gauge-invariant symplectic form on the affine space of connections over an oriented closed surface.
Based gauge group The subgroup of gauge transformations that fixes a chosen point in a principal-bundle fiber.
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.
Bogomolny monopole equation A first-order gauge equation equating the curvature of a connection on a three-manifold with the Hodge dual of the covariant derivative of a Higgs field.
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.
Category of vector bundles over a manifold The fixed-base category of finite-rank vector bundles whose morphisms cover the identity of the base.
Change of connection formula for Chern Weil characteristic forms Exact formula relating characteristic forms computed from two different principal connections
Character variety The affine geometric-invariant-theory quotient of a representation variety by conjugation.
Characteristic class A de Rham cohomology class of a principal bundle defined from curvature via the Chern–Weil construction.
Characteristic number A scalar obtained by evaluating a top-degree product of characteristic classes on a fundamental class.
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 connection The unique connection on a Hermitian holomorphic vector bundle compatible with both structures.
Chern number An integer obtained by evaluating a top-degree monomial in Chern classes on an oriented fundamental class.
Chern roots Formal degree-two classes that express the Chern classes of a complex vector bundle as elementary symmetric polynomials.
Chern–Gauss–Bonnet theorem The integral of the Euler curvature form of a closed oriented even-dimensional Riemannian manifold equals its Euler characteristic.
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 homomorphism The graded-algebra map from invariant polynomials on a Lie algebra to de Rham cohomology.
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 whose exterior derivative vanishes.
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.
Compactly supported section A vector bundle section whose support is a compact subset of the base.
Complete flag bundle The smooth bundle of complete flags in the fibers of a complex vector 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 compatible with a reduction A principal connection whose horizontal spaces restrict to a chosen reduction of the structure group.
Connection on a Hom bundle The connection on a homomorphism bundle induced by connections on its source and target bundles.
Connection on a vector bundle A rule for differentiating sections along vector fields, linear over constants and satisfying a Leibniz rule.
Connection on exterior and symmetric powers The connections induced on exterior and symmetric power bundles by differentiating one tensor factor at a time.
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.
Coordinate chart A homeomorphism from an open subset of a manifold to an open subset of Euclidean space, providing local coordinates.
Coordinate transition map The change-of-coordinates map between the Euclidean images of two overlapping manifold charts.
Cotangent bundle The smooth vector bundle whose fiber at each point is the dual of the tangent space.
Coulomb gauge Coulomb gauge imposes a covariant divergence-free condition on the difference between a connection and a reference connection.
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.
Covariant Hodge Laplacian The covariant Hodge Laplacian is the second-order operator obtained by anticommuting a covariant exterior derivative with its formal adjoint.
Curvature The infinitesimal obstruction to integrability of a connection's horizontal distribution.
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 as a moment map The moment-map interpretation of curvature for the gauge action on connections over an oriented closed surface.
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.
Curvature of the Chern connection The End(E)-valued curvature two-form of a Hermitian holomorphic vector bundle's Chern connection.
de Rham cohomology group The quotient of closed differential forms by exact forms.
Deformation complex of a flat connection The twisted de Rham complex whose cohomology records infinitesimal symmetries, deformations, and obstructions of a flat connection.
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.
Donaldson–Uhlenbeck–Yau theorem The equivalence between slope-polystability of a holomorphic vector bundle and existence of a Hermitian Yang-Mills metric.
Dual connection The unique connection on a dual vector bundle for which differentiation obeys the natural evaluation pairing.
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 as pullback of the Thom class The topological Euler class obtained by pulling a vector bundle's Thom class back along its zero section.
Euler class obstruction to a nowhere-zero section A nowhere-zero section forces the Euler class to vanish, and in base dimension equal to the bundle rank this is the complete obstruction.
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 a form of one lower degree.
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 covariant derivative on a vector bundle The degree-one extension of a vector-bundle connection to bundle-valued differential forms.
Exterior derivative The differential operator on differential forms that squares to zero and satisfies 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 coordinates Coordinates in the typical-fiber factor supplied by a local bundle trivialization.
Fiber of a map The subset of the domain mapping to a fixed point in the codomain, also called a preimage fiber.
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.
Formal adjoint of the covariant exterior derivative The covariant codifferential is the formal adjoint of the covariant exterior derivative with respect to the bundle-valued L2 pairing.
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.
G-structure A reduction of the frame bundle of a smooth manifold to a specified Lie subgroup of the general linear group.
G-torsor A nonempty set or smooth space with a free and transitive action of a group G.
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 Pullback gives a right action of the gauge group on principal connections.
Gauge theory A field theory whose configurations carry local symmetries represented geometrically by gauge transformations.
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.
Gauge-fixing condition A supplementary condition used to choose local representatives of gauge-equivalence classes of fields.
Good cover An open cover whose nonempty finite intersections are contractible.
Hermitian connection A connection on a Hermitian vector bundle that preserves its fiberwise Hermitian inner product.
Hermitian metric A smoothly varying Hermitian inner product on the fibers of a complex vector bundle.
Hermitian Yang–Mills connection A unitary integrable connection whose curvature has constant central contraction with the Hermitian form.
Holonomy algebra The Lie algebra of the holonomy group of a connection at a chosen point.
Holonomy correspondence for flat connections The classification of flat principal bundles by conjugacy classes of fundamental-group representations.
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 closed Lie 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.
Instanton number For an SU(r)-connection on a closed oriented four-manifold, the instanton number is the integral second Chern number represented by curvature.
Integrable horizontal distribution A horizontal distribution closed under Lie brackets, equivalently tangent to a foliation transverse to the fibers.
Integral Chern classes The canonical integral characteristic classes of finite-rank complex vector bundles, characterized by the standard normalization and axioms.
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.
Intrinsic torsion of a G-structure The connection-independent torsion class that obstructs a G-structure from admitting a torsion-free compatible connection.
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.
Invariant polynomial on a Lie algebra A polynomial on a Lie algebra that is unchanged by the adjoint action of its Lie group.
Irreducible connection A connection whose stabilizer under the full gauge group is exactly the unavoidable central subgroup.
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.
Kernel and image bundles of a constant-rank morphism The smooth subbundles formed by the pointwise kernels and images of a constant-rank vector bundle morphism.
L2 inner product on bundle-valued forms The integral inner product on bundle-valued differential forms induced by metrics on the base and coefficient bundle.
Left translation on a Lie group The diffeomorphism of a Lie group given by multiplication by a fixed element on the left.
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 of a differential form along the flow of a vector field.
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.
Line bundle A real or complex vector bundle whose fibers have dimension one over the chosen scalar field.
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 system A locally constant sheaf, equivalently on a suitable space a representation of its fundamental groupoid.
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.
Maximal smooth atlas A smooth atlas containing every coordinate chart smoothly compatible with all of its charts.
Module of smooth sections The module of all smooth sections of a vector bundle under pointwise operations by smooth functions.
Moduli space of flat connections The quotient of the space of flat connections on a fixed principal bundle by gauge equivalence.
Moduli stack of connections The quotient stack of the space of connections by gauge transformations, retaining the automorphisms of each connection.
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.
Nonabelian Čech H1 and principal bundles Nonabelian Čech 1-cohomology classifies principal G-bundles as a pointed set.
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.
Nowhere-vanishing section A vector bundle section whose value is nonzero in every fiber.
Numerable principal bundle A principal bundle with a trivializing cover admitting a locally finite continuous partition of unity.
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 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 Integral characteristic classes of a real vector bundle defined from the Chern classes of its complexification.
Pontryagin number An integer obtained by evaluating a top-degree monomial in Pontryagin classes on an oriented fundamental class.
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 connection on a vector bundle The canonical connection on a pullback vector bundle induced by a connection on the original bundle.
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.
Quaternionic vector bundle A vector bundle locally modeled on a finite-dimensional right module over the quaternion division algebra.
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.
Quotient vector bundle The vector bundle whose fibers are the quotients of a vector bundle by a smooth vector subbundle.
Rank of a vector bundle The fiber dimension of a real or complex vector bundle, which is locally constant on the base.
Reconstructing a vector bundle from a projective module The vector bundle whose fibers are obtained by evaluating a finitely generated projective smooth-function module at each point.
Reducible connection A connection whose gauge stabilizer is larger than the unavoidable central subgroup.
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 ASD connection An anti-self-dual connection whose deformation complex has vanishing obstruction space.
Regular value A point in the target such that the differential is surjective along its fiber.
Representation variety The affine algebraic set parameterizing homomorphisms from a finitely presented group into an affine algebraic group.
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 module is finitely generated projective The theorem that smooth sections of a finite-rank vector bundle over a finite-dimensional manifold form a finite projective module.
Section of a fiber bundle A smooth right inverse to a fiber-bundle projection, selecting one point in every fiber.
Section of a vector bundle A smooth choice of one vector in every fiber of a smooth vector bundle.
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.
Self-dual and anti-self-dual connection A connection on an oriented Riemannian four-manifold whose curvature lies entirely in the self-dual or anti-self-dual summand of two-forms.
Serre–Swan theorem On a finite-dimensional smooth manifold, taking smooth sections gives an equivalence between vector bundles and finitely generated projective modules.
Sheaf of smooth sections The locally free sheaf assigning smooth local sections of a vector bundle to each open subset of its base.
Short exact sequence of vector bundles A pair of bundle morphisms whose induced sequence is exact in every fiber.
Slice theorem for the gauge action A local normal-form theorem modeling a neighborhood of a gauge orbit by a Coulomb slice modulo the connection stabilizer.
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 compatibility of charts and atlases The condition that overlapping coordinate charts have smooth transition maps, extended to compatibility of smooth atlases.
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 G-valued Čech 1-cocycle Smooth transition functions on double overlaps satisfying the identity, inverse, and triple-overlap cocycle laws.
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 structure A choice of mutually compatible smooth coordinate charts on a topological manifold, represented by their maximal smooth atlas.
Smooth submersion A smooth map whose differential is surjective at every point.
Sobolev completion of connections and gauge transformations The Banach configuration space obtained by completing connections and gauge transformations in compatible Sobolev norms.
Solder form on the frame bundle The canonical R^n-valued 1-form expressing projected tangent vectors in their moving frame.
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 of unitary frames calibrated by a chosen unit determinant trivialization.
Sphere bundle The fiber bundle of unit vectors in a positive-rank vector bundle equipped with a bundle metric.
Spin connection The lift of the Levi–Civita connection from the oriented orthonormal frame bundle to a chosen spin structure.
Spin structure A lift of the oriented orthonormal frame bundle through the double covering Spin(n) to SO(n).
Splitting of the Atiyah sequence A right inverse TM to TP/G that is equivalent to choosing a principal connection.
Splitting principle A complex vector bundle pulls back to a sum of line bundles on a flag bundle without losing cohomological information.
Splitting theorem for vector bundles Every short exact sequence of finite-rank vector bundles over a paracompact base admits a generally noncanonical bundle splitting.
Stabilizer of a connection The subgroup of gauge transformations that leave a connection fixed.
Stiefel–Whitney class A canonical mod-2 characteristic class of a real vector bundle.
Support of a section The closure of the set of base points where a vector bundle section is nonzero.
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.
Tangent space to flat-connection moduli The first cohomology of the flat-connection deformation complex is the infinitesimal moduli space, with zeroth and second cohomology controlling stabilizers and obstructions.
Tangent-bundle cocycle from coordinate changes Differentiating coordinate transitions produces the GL(n,R)-valued cocycle of the tangent bundle.
Tensor product connection The connection on a tensor product bundle obtained by differentiating each factor.
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.
Thom class The relative cohomology class that restricts to the orientation generator in every fiber of an oriented vector bundle.
Thom isomorphism theorem Cup product with the Thom class shifts the cohomology of a bundle base into the relative cohomology of its total space.
Topological complex vector bundle A topological bundle whose fibers are complex vector spaces and whose local trivializations are fiberwise complex linear.
Topological principal bundle A locally trivial bundle with a continuous right group action modeled equivariantly on a product.
Topological real vector bundle A topological bundle whose fibers are real vector spaces and whose local trivializations are fiberwise real linear.
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.
Uhlenbeck compactness theorem A compactness theorem giving subsequential convergence of bounded-energy instantons modulo gauge away from finitely many bubbling points.
Uhlenbeck removable singularity theorem A finite-energy Yang–Mills connection on a punctured four-ball extends smoothly across the puncture after a gauge transformation.
Unit tangent bundle The bundle of unit tangent vectors of a Riemannian manifold.
Unitary frame bundle The principal U(n)-bundle of unitary frames determined by a Hermitian metric on a complex rank-n bundle.
Universal characteristic class A cohomology class on a classifying space whose pullbacks assign a natural characteristic class to every principal 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 from an idempotent matrix The vector bundle formed by the pointwise images of a smooth idempotent matrix.
Vector bundle morphism A smooth map between total spaces of vector bundles that covers a base map and is linear on each fiber.
Vector bundles and locally free sheaves Taking local smooth sections is an equivalence from fixed-base vector bundles to finite-rank locally free smooth-module sheaves.
Vector field A smooth section of the tangent bundle; equivalently, an assignment of a tangent vector to each point varying smoothly.
Vector subbundle A smoothly varying family of linear subspaces inside the fibers of a vector bundle.
Vector-bundle-valued differential form A differential form whose value at each point lies in a specified vector-bundle fiber.
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 k-form and an ℓ-form into a (k+ℓ)-form.
Whitney sum formula Total characteristic classes multiply when vector bundles are combined by direct sum.
Wilson loop The trace of connection holonomy around a closed loop in a chosen representation.
Yang–Mills connection A principal connection satisfying the Yang–Mills equation.
Yang–Mills energy identity The four-dimensional decomposition of Yang–Mills energy into a topological curvature integral and a nonnegative self-dual or anti-self-dual term.
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.
Yang–Mills instanton A finite-action self-dual or anti-self-dual connection on an oriented Riemannian four-manifold.
Yang–Mills moduli space The gauge-equivalence classes of Yang–Mills connections on a fixed principal bundle over a Riemannian manifold.
Zero section The canonical smooth section assigning the zero vector to every fiber of a vector bundle.
Čech cocycle groupoid The groupoid whose objects are Čech cocycles and whose morphisms are changes of local trivialization.

Fields and Galois theory90 knowls

Absolute norm of an ideal The cardinality of the quotient by a nonzero integral ideal in a number ring.
Adeles and restricted products The locally compact restricted product of all completions of a global field.
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.
Algebraic integer An algebraic number satisfying a monic polynomial with integer coefficients.
Archimedean local field A local field isomorphic to the real or complex numbers.
Artin's theorem on fixed fields A finite group of field automorphisms yields a finite Galois extension with degree equal to the group order.
Chebotarev density theorem Frobenius conjugacy classes of unramified primes are equidistributed in a finite Galois group.
Choosing an embedding $\overline{\mathbb Q}\hookrightarrow\overline{\mathbb Q}_p$ How a p-adic embedding selects a place and a decomposition subgroup, with changes acting by conjugacy.
Class number of a number field The finite number of ideal classes in its ring of integers.
Completion of a field at a place The complete topological field obtained from a field and one of its places.
Cyclotomic extension A field extension obtained by adjoining a primitive nth root of unity.
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.
Decomposition group The subgroup of a Galois group that stabilizes a chosen prime above a prime of the base field.
Dedekind independence lemma Distinct K-embeddings of a field are linearly independent as functions.
Dedekind zeta function The Dirichlet series summing inverse powers of nonzero integral-ideal norms.
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) The determinant of the trace-pairing matrix associated with a basis of a finite field extension.
Discriminant of a number field The trace-pairing discriminant of an integral basis, independent of that basis.
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 p-power map is a ring endomorphism and induces the absolute Frobenius morphism of a scheme.
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 extension and Galois group A normal separable field extension and its group of base-field automorphisms.
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.
Global field A number field or a one-variable function field over a finite field.
Global function field A finite extension of a rational function field over a finite field.
Ideal class group of a number field Nonzero fractional ideals modulo multiplication by a nonzero field element.
Ideles, Hecke characters, and Artin reciprocity The idele class group, its continuous quasicharacters, and the global reciprocity map to the abelianized Galois group.
Imaginary quadratic field A degree-two number field with no real embedding.
Inertia subgroup The kernel of the residue-field action of a decomposition group.
Inseparable extension An algebraic extension that is not separable; it contains an element with a repeated-root minimal polynomial.
Integral basis of a number field A basis of its ring of integers as a free abelian group.
Intermediate field A subfield K with F ⊆ K ⊆ E inside a given field extension E/F.
Local field A nondiscrete locally compact topological field.
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.
Non-Archimedean absolute value A multiplicative absolute value satisfying the strong triangle inequality.
Nonabelian H1 and Galois 1-cocycles The pointed set of continuous Galois 1-cocycles modulo twisted conjugacy.
Nonarchimedean local field A complete discretely valued field with finite residue field.
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.
Number field A finite field extension of the rational numbers.
p-adic field A finite extension of Q_p with a discrete valuation, compact valuation ring, and finite residue 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.
Place of a global field An equivalence class of nontrivial absolute values on a global field.
Primitive element theorem Every finite separable extension is generated by a single element.
Primitive root of unity An element whose multiplicative order is exactly a prescribed positive integer.
Ring of integers of a number field The subring consisting of all algebraic integers in a number field.
Ring of integers of an imaginary quadratic field The explicit integral basis according to the square-free parameter modulo four.
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 field extensions, trace and norm are transitive.
Transcendental element An element is transcendental over a field if it satisfies no nonzero polynomial with coefficients in that field.
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).
Unramified extension of a nonarchimedean local field A finite local-field extension with ramification index one and separable residue extension.
Unramified prime and Frobenius element Arithmetic and geometric Frobenius conjugacy classes at an unramified place.
Valuation on a field A function into an ordered abelian group that converts products to sums and satisfies the ultrametric inequality.
Valuation ring of a valued field The local subring of elements having nonnegative valuation.
Value group of a valuation The ordered abelian subgroup consisting of the finite values of a field valuation.

Fluid dynamics83 knowls

Active scalar equation A transport equation whose advecting velocity is determined by the transported scalar.
Advective transport time scale The characteristic time L/U to move across length L at speed U.
Affine velocity field A velocity field affine in position, with a time-dependent matrix and translation vector.
Angular average in cylindrical components The normalized average in the angular variable, with a stated convention for vector components.
Angular velocity about an axis The azimuthal linear velocity divided by the distance from the axis.
Angular-velocity equation in axisymmetric flow The swirl equation after dividing azimuthal velocity by radius, with the radial heat operator of four transverse dimensions.
Averaged Navier–Stokes model Replace the Euler bilinear term by an average of transformed Euler trilinear forms.
Axial velocity The velocity component parallel to a chosen axis.
Axisymmetric Navier–Stokes component equations The incompressible momentum equations for radial, azimuthal, and axial velocity components independent of angle.
Axisymmetric vector field A vector field equivariant under rotations about a fixed axis, equivalently with angle-independent cylindrical components.
Caffarelli–Kohn–Nirenberg partial regularity theorem The interior singular set of an unforced suitable three-dimensional solution has zero parabolic one-dimensional measure.
Canonical pressure from an integrable tensor A bounded Fourier symbol assigns an H minus s pressure to an L1 momentum tensor.
Centripetal acceleration for circular motion The inward normal acceleration v²/r required by motion along a circle.
Changing viscosity by spatial rescaling A spatial dilation and matched amplitude change transport a solution between positive viscosities without changing time.
Classical uniqueness from difference energy An integrable reference gradient controls the squared L2 difference of regular solutions with common forcing.
Conservative incompressible momentum equation The Navier–Stokes momentum equation written as the divergence of a total momentum flux.
Curl completion of a transverse oscillatory wave A vector potential producing an exactly divergence-free wave with a lower-order curl correction.
Cutoff pressure-flux estimate for a velocity difference The pressure flux at a large radius is bounded by subquadratic powers of a localized L6 norm.
Cylindrical swirl diffusion operator The radial diffusion of an azimuthal vector component includes a negative inverse-square term.
Delayed parabolic rescaling at a fixed terminal time Parabolic scaling combined with a time shift shrinks spatial support while retaining a chosen terminal time.
Difference equation for two incompressible flows The common force cancels and the nonlinear difference splits into transport and reference-gradient terms.
Endpoint L-infinity-in-time L3 regularity criterion An unforced three-dimensional Leray–Hopf solution bounded in L3 uniformly in time is regular at positive times.
Energy bound from a time-integrable L2 force The accumulated L2 size of the force bounds velocity and total dissipation without dividing by a possibly zero norm.
Euler subsolution with prescribed energy A linear momentum solution whose velocity and trace-free flux lie below a prescribed energy matrix.
External force in a normalized fluid equation A prescribed vector-valued source representing acceleration due to forces outside the modeled internal stresses.
Fluid dynamics Velocity, transport, incompressible momentum equations, and cylindrical fluid geometry.
Fluid velocity field The vector assigning the instantaneous fluid velocity to each point of space and time.
Global kinetic-energy identity Smooth incompressible flow balances energy change and viscous dissipation against work by the force.
Hypodissipative Navier–Stokes equations Incompressible momentum with a positive fractional dissipation order less than the usual Laplacian.
Identifying pressure gradients without a pressure growth assumption Time averaging the momentum equation puts a harmonic pressure-gradient difference in a global negative Sobolev space.
Incompressible Euler equations The constant-density inviscid fluid equations, with pressure, advection, forcing, and zero divergence.
Incompressible Navier–Stokes equations The constant-density momentum equation with quadratic advection, pressure, positive viscosity, forcing, and zero divergence.
Incompressible porous media equation Density transport coupled to a Darcy velocity and zero divergence.
Incompressible velocity field A differentiable fluid velocity field with zero spatial divergence.
Integrability of the subtracted swirl heat tail The Gamma-integral heat profile differs from its terminal power by two extra powers of radial decay.
Integral profile for radial swirl heat flow A positive Gamma-integral profile solves the swirl heat equation away from the axis up to a specified terminal time.
Integration cancellation for divergence-free transport and pressure Divergence-free fields remove transport and pressure terms from unweighted whole-domain energy integrals.
Kinetic energy of an incompressible velocity field Half the squared spatial L2 norm when density is normalized to one.
Kinetic-energy inequality An integrated upper bound allowing energy loss beyond the explicitly recorded viscous dissipation.
Leray projection on Euclidean space The orthogonal L2 projection onto distributionally divergence-free vector fields.
Leray–Hopf solution An energy-class weak solution with weak time continuity, strong initial trace, and the global energy inequality.
Linearized Navier–Stokes operator The first variation of the momentum residual with respect to velocity and pressure.
Local energy inequality The distributional kinetic-energy balance with a nonnegative dissipation defect.
Local kinetic-energy balance for smooth incompressible flow The pointwise energy equation includes transport, pressure flux, diffusion, dissipation and forcing.
Localized difference-energy identity A compact spatial weight exposes the transport and pressure fluxes in the energy of two flows.
Material derivative The time derivative of a field as observed along motion generated by a velocity field.
Meridional velocity The radial and axial components of a cylindrical velocity field, excluding its azimuthal component.
Momentum flux tensor in an incompressible fluid The tensor whose i,j entry is the flux of the ith momentum component in the jth spatial direction.
Navier–Stokes energy class Velocity with essentially bounded kinetic energy and finite integrated squared spatial gradient.
Navier–Stokes momentum residual The defect obtained by evaluating the momentum equation on a trial velocity and pressure.
Parallel shear flow A velocity in one fixed direction whose magnitude varies in a transverse coordinate.
Particle trajectory of a velocity field A curve whose tangent velocity equals the fluid velocity at its current position.
Periodization of separated incompressible flows Disjoint translated supports preserve the nonlinear momentum equation under lattice periodization.
Phase transport defect The material derivative of a phase, measuring failure of exact transport by a chosen velocity.
Pressure in incompressible flow The scalar field whose negative spatial gradient supplies the pressure force per unit mass.
Pure swirl solution from a radial heat flow A radial swirl heat profile together with its centrifugal pressure gives an exact unforced Navier–Stokes field away from the axis.
Quadratic advection of a velocity field The nonlinear term (u·∇)u and its exact expansion around a background field.
Radial velocity in cylindrical coordinates The component of velocity pointing away from the chosen symmetry axis.
Radial-tangential velocity covariance The pair of averaged products transporting azimuthal and axial momentum in the radial direction.
Rate-of-strain tensor The symmetric part of the spatial velocity gradient.
Relative kinetic energy Half the squared L2 distance between two velocity fields.
Rescaling unit viscosity to positive viscosity A spatial and amplitude change that converts a unit-viscosity solution to viscosity nu.
Reynolds averaging stress The covariance of a fluctuating velocity that appears when averaging its quadratic transport.
Reynolds defect system Momentum balance with the divergence of a symmetric error tensor.
Reynolds number The dimensionless ratio UL/ν comparing advective and viscous scales.
Rotating flow about a fixed axis A flow with an azimuthal component, which may coexist with radial and axial motion.
Scaling symmetry of Navier–Stokes The parabolic dilation preserving the incompressible Navier–Stokes equations at fixed viscosity.
Shearing wave on an affine flow An exact transverse plane-wave perturbation with a wavevector transported by an affine background.
Singular set of a weak velocity The space-time points with no neighborhood on which the velocity is essentially bounded.
Specific angular momentum about an axis The axial component of position cross velocity, equal to r times the azimuthal velocity.
Suitable weak solution A local energy-class Navier–Stokes pair satisfying the local energy inequality.
Swirl component of a cylindrical velocity The azimuthal velocity component tangent to circles around the chosen axis.
Transport operator along a vector field The first-order spatial differential operator b·∇.
Uniformly bounded kinetic energy A uniform-in-time bound on the spatial square-integral of velocity.
Uniqueness against a smooth compactly supported reference flow A smooth bounded-energy competitor agrees with a smooth compact reference solution with the same data and force.
Viscosity in the incompressible Newtonian model The coefficient relating viscous stress to the rate of strain, with kinematic viscosity obtained by dividing by density.
Viscous damping of a Fourier mode The exponential decay produced by viscosity at a specified spatial wavevector.
Viscous diffusion time scale The characteristic time L²/ν for viscosity to act across a length L.
Viscous dissipation in the incompressible energy balance Viscosity times the spatial square-integral of the velocity gradient.
Vortex line A curve tangent to the vorticity vector field at a fixed time.
Vorticity equation for incompressible flow The curl of the momentum equation, with transport, vortex stretching, diffusion, and curl of forcing.
Vorticity of a velocity field The curl of a three-dimensional velocity field.
Weak Navier–Stokes solution Locally square-integrable velocity and locally integrable pressure satisfying momentum and incompressibility against tests.

Formal groups19 knowls

Additive and multiplicative formal group laws The standard one-dimensional laws F_a(X,Y)=X+Y and F_m(X,Y)=X+Y+XY.
Affine formal groups and complete Hopf algebras Formal spectrum gives a contravariant equivalence between affine adic formal groups and admissible complete commutative Hopf algebras.
BCH group of a complete filtered Lie algebra The convergent BCH series turns a complete pronilpotent Lie algebra into a filtered group.
Coordinate Hopf algebra of an affine formal group The complete commutative topological Hopf algebra of functions on an affine formal group.
Distribution algebra and universal enveloping algebra In characteristic zero, the distribution algebra of a smooth formal group is canonically its tangent Lie algebra's universal enveloping algebra.
Distribution algebra of a formal group The filtered Hopf algebra of finite-order functionals supported at the identity of a formal group.
Equivalence between Lie algebras and formal groups Over a characteristic-zero field, finite-dimensional Lie algebras are equivalent to formally smooth formal groups on finite formal discs.
Failure of tangent classification in positive characteristic In characteristic p, nonisomorphic formal groups can have isomorphic tangent Lie algebras.
Formal affine space The formal n-disc with its origin, coordinates, and n-dimensional tangent space.
Formal completion of a group at the identity The formal group obtained by retaining every infinitesimal neighborhood of the identity in a group scheme.
Formal group A group object in formal schemes; in finite-dimensional formal Lie theory its pointed formal scheme is a formal disc.
Formal group law An n-dimensional, not necessarily commutative group multiplication encoded by tuples of formal power series.
Formal group laws as coordinates on formal groups Group structures on a formal disc, expressed in chosen parameters, are exactly formal group laws.
Formal group logarithm Over a Q-algebra, a commutative formal group law has a unique strict isomorphism to the additive law.
Formal groups and formal group laws Formal power series, formal geometry, formal group laws, and the characteristic-zero formal Lie correspondence.
Height of a one-dimensional formal group law The exponent of the first nonzero term of the p-series of a one-dimensional commutative formal group law in characteristic p.
Morphism of formal group laws A pointed tuple of power series intertwining two formal group laws, with isomorphisms and strict isomorphisms distinguished by their linear terms.
One-dimensional commutative formal group law The classical one-variable commutative formal group law F(X,Y) over a commutative ring.
Tangent Lie algebra of a formal group The tangent space at the identity of a formal group, with bracket induced by invariant formal vector fields.

Foundations83 knowls

Axiom of Choice Every family of nonempty sets has a choice function.
Bijective function A function that is both injective and surjective.
Binary operation A function that combines two elements of a set to produce another element of the same set
Binomial coefficient The number of k-element subsets of an n-element set and its factorial formula.
Cardinality The size of a set, understood up to bijection
Cartesian product The set of all ordered pairs from two sets.
Ceiling function The least integer not less than a real number.
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
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
Descent of a function through a surjection A function factors uniquely through a surjection exactly when it is constant on its fibers.
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.
Finite permutation A bijection from a finite initial segment of the natural numbers to itself.
Finite product An ordered finite multiplication, with the identity as the empty product.
Finite set A set admitting a bijection with an initial segment of the natural numbers.
Finite sum Addition over a finite ordered list, with empty sum zero and order independence in an abelian group.
First-order logic and inference rules Primitive logical syntax, quantifiers, equality, and rules for classical mathematical inference.
Floor function The greatest integer not exceeding a real number.
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
Lattice A partially ordered set in which every pair has a join and a meet.
Lower bound An element that is less than or equal to every element of a subset in an ordered set.
Mathematical axiom A nonlogical statement adopted as part of a mathematical theory.
Mathematical induction A principle for proving statements for all natural numbers.
Mathematical proof A deduction from stated assumptions and axioms using specified inference rules.
Mathematical proposition A statement considered under specified hypotheses, with a truth value in an interpretation.
Mathematical theorem A statement established from the axioms and hypotheses of a theory.
Natural numbers The set of nonnegative integers used for counting and indexing.
Nearest integer An integer minimizing distance to a given real number, with a stated tie convention.
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.
Permutation sign The sign of a finite permutation, determined by the parity of its inversions.
Pointed set A set equipped with a distinguished element called its basepoint.
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.
Recursion on the natural numbers An initial value and a specified next-step rule determine a unique sequence.
Relation A set of ordered pairs encoding which elements are related.
Restriction of a function A function obtained by limiting another function's domain to a subset.
Rising factorial The product a(a+1)...(a+n-1), including the empty product at n=0.
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
Square-free integer An integer not divisible by the square of any prime.
Strict partial order An irreflexive and transitive binary relation.
Subset A set contained in another set, element by element.
Support of a function The closure of the set on which a function is nonzero.
Surjective function A function whose outputs cover the entire codomain
Symmetric difference The elements that lie in exactly one of two sets.
Telescoping sum Cancellation of successive differences in a finite sum and the additional limit needed for an infinite sum.
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.

Functional analysis128 knowls

A mixed-derivative estimate after radial inversion A radial inverse controls a product containing one parameter derivative and one dilation derivative.
Absolutely convergent series in a Banach space Summability of the norms ensures convergence of a vector-valued series in a complete normed space.
Adjoint of a bounded operator The unique bounded operator obtained by transferring a Hilbert-space operator across the inner product.
Adjoint of a densely defined operator The closed operator determined by the vectors on which an unbounded operator's inner-product pairing is bounded.
Banach algebra A complete normed algebra whose norm is submultiplicative.
Banach completion of a seminorm quotient The Banach space obtained by completing a seminorm quotient of a vector space.
Banach–Alaoglu theorem The closed unit ball of the continuous dual of a normed space is compact in the weak-star topology.
Barreled space A locally convex space in which every closed, convex, balanced, absorbing set is a zero-neighborhood.
Borel functional calculus for self-adjoint and normal operators The functional calculus that assigns a possibly unbounded operator to each Borel function through a normal operator's spectral measure.
Bornological space A locally convex space whose topology is determined by its bounded subsets.
Bounded bilinear map between normed spaces A bilinear map satisfying a uniform product bound on its output norm.
Bounded commutator A commutator on a dense domain that extends uniquely to a bounded operator on the ambient Hilbert space.
Bounded linear operator between normed spaces A linear map between normed spaces whose output norm is bounded by a fixed multiple of the input norm.
Bounded subset of a topological vector space A subset that is eventually contained in every sufficiently large scalar multiple of each zero-neighborhood.
Closable operator A linear operator whose graph closure is still the graph of an operator.
Closed graph theorem An everywhere-defined linear operator between Banach spaces is bounded when its graph is closed.
Closed linear operator A possibly unbounded linear operator whose graph is closed in the product space.
Closed quadratic form A lower-semibounded symmetric form whose domain is complete in its shifted form norm.
Closure of a closable operator The minimal closed extension obtained by closing the graph of a closable operator.
Commutator of possibly unbounded operators The difference of two operator products, defined only where both products make sense.
Compactly supported distribution A distribution whose distributional support is compact in its open domain.
Complex Structure Λ on K A bounded real-linear operator Λ with Λ² = −I; when orthogonal, it equips a real Hilbert space with a compatible complex Hilbert structure.
Continuous Banach-valued functions on a compact space Continuous functions into a Banach space form a Banach space under the uniform norm.
Continuous linear map between topological vector spaces A linear map that is continuous for the given vector-space topologies.
Contraction criterion on a closed norm ball Separate quantitative conditions ensuring that a map preserves a complete ball and contracts it.
Convolution of distributions Distributional convolution extends ordinary convolution under support hypotheses and includes convolution with test functions.
Core of a closed operator A subdomain whose graph closure recovers the entire closed operator.
Decomposable operator A bounded operator on direct integrals that is given almost everywhere by a measurable essentially bounded field of fiber operators.
Deficiency indices The dimensions of the two nonreal eigenspaces of the adjoint of a symmetric operator.
Densely defined operator A linear operator whose specified domain is dense in its ambient Hilbert space.
Derivative bound between Fourier Sobolev spaces Each distributional derivative lowers the Sobolev order by its degree.
Derivative loss in an estimate An estimate requiring more derivatives of the input than it controls on the output.
Derivative requirements on a finite computation graph A finite acyclic composition of operations with finite derivative requirements has a finite input requirement at each output order.
Determinant on I + Trace-Class The Fredholm determinant on identity-plus-trace-class operators.
Dirac delta distribution The distribution that evaluates a test function at one specified point.
Direct integral of Hilbert spaces The Hilbert space of square-integrable measurable sections of a measurable field of Hilbert spaces.
Direct integral of operators A direct integral operator acts fiberwise by a measurable family of closed operators on a direct integral Hilbert space.
Distribution A distribution on an open Euclidean set is a continuous linear functional on its space of compactly supported smooth test functions.
Distributional derivative The distributional derivative transfers differentiation to test functions with the sign dictated by integration by parts.
Equicontinuous family of continuous linear maps A family of continuous linear maps that satisfies one continuity estimate uniformly across all its members.
Essentially self-adjoint operator A densely defined symmetric operator whose closure is self-adjoint.
Finite-rank operator A bounded linear operator whose range is finite-dimensional.
Fourier inversion on Schwartz space Every Schwartz function is recovered pointwise and in the Schwartz topology from its Fourier transform.
Fourier Sobolev space on Euclidean space A tempered distribution whose Fourier transform is square integrable with a prescribed polynomial weight.
Fourier transform of tempered distributions The Fourier transform of a tempered distribution is defined by transposing the Fourier automorphism of Schwartz space.
Fourier transform on Schwartz space The Fourier transform restricts to a continuous linear automorphism of the Schwartz space.
Fredholm operator A bounded operator with closed range and finite-dimensional kernel and cokernel.
Friedrichs extension The canonical self-adjoint extension obtained by closing the quadratic form of a semibounded symmetric operator.
Fréchet space A Hausdorff locally convex space whose topology is induced by a complete metric.
Functional Analysis Locally convex spaces, distributions, direct integrals, and bounded and unbounded operators.
Graph norm The norm on an operator domain that controls both a vector and its image.
Graph of a linear operator The subspace of a product space consisting of each domain vector paired with its image.
Hilbert–Schmidt operator A Hilbert-space operator whose squared norms on an orthonormal basis have finite sum.
Holomorphic functional calculus in a Banach algebra A contour-integral calculus that evaluates functions holomorphic near the spectrum at a Banach-algebra element.
Holomorphic realization of weighted analytic coefficients The factorial-binomial coefficient bound gives a convergent radial series and a common complex parameter radius.
Inductive limit of locally convex spaces The algebraic direct limit equipped with the finest locally convex topology making its structure maps continuous.
Integrable functions in negative Sobolev spaces An integrable function belongs to H minus s whenever s exceeds half the dimension.
Integral operator with a measurable kernel A linear operator defined by integration against a kernel K(x,y).
Invertible element in a Banach algebra An element admitting a two-sided multiplicative inverse in a unital Banach algebra.
LF-space A Hausdorff locally convex inductive limit of a countable sequence of Fréchet spaces.
Local Sobolev space Sobolev regularity on every relatively compact subdomain.
Locally convex space A topological vector space whose origin has a neighborhood basis of convex sets.
Measurable field of Hilbert spaces A family of Hilbert spaces equipped with a countably generated measurable structure on its sections.
Montel space A barreled locally convex space in which every closed bounded subset is compact.
Multiplication operator The linear operator that multiplies a function by a fixed coefficient.
Nuclear operator between Banach spaces A bounded operator represented by an absolutely summable series of rank-one operators.
Nuclear space A locally convex space whose defining Banach-space transition maps are nuclear.
Open mapping theorem A surjective bounded linear operator between Banach spaces maps open sets to open sets.
Operator inversion from summable bounds on powers The series sum of alternating powers inverts I plus T whenever the operator powers are norm-summable.
Operator with compact resolvent A closed operator whose resolvent is compact at one, equivalently every, resolvent point.
Order of a distribution The order of a distribution is the least derivative degree needed in uniform seminorm estimates on compact supports.
Partial isometry A bounded Hilbert-space operator that is isometric on the orthogonal complement of its kernel.
Point, continuous, and residual spectrum The standard partition of an operator spectrum according to injectivity and density of the range.
Polar Decomposition Writing T as a unitary/orthogonal part times a positive part
Product estimate in a two-index analytic coefficient space Coefficient convolution and the Leibniz rule are bounded by the factorial-binomial weights.
Projection-valued measure A normalized, strongly countably additive measure whose values are orthogonal projections on a Hilbert space.
Projective limit of locally convex spaces The compatible-tuple subspace of a product of locally convex spaces.
Radial operator bounds in an analytic coefficient norm Raising radial degree yields bounds for integration and for a parameter derivative followed by integration.
Rank-one operator A bounded linear operator whose range is a one-dimensional vector space.
Reflexive Banach space A Banach space whose canonical embedding into its continuous bidual is surjective.
Regular tempered distribution A regular tempered distribution is a tempered distribution represented by integration against a locally integrable function.
Residual update with a fixed linear inverse A quadratic equation admits an exact residual formula when corrections use the derivative at one fixed reference point.
Resolvent identity An algebraic identity relating resolvent operators at different parameters or for different operators.
Resolvent of an element in a Banach algebra The inverse of the scalar shift of a Banach-algebra element, defined on its resolvent set.
Resolvent set of a closed operator The scalars for which a closed operator has an everywhere-defined bounded inverse.
Riesz representation theorem for Hilbert spaces Every continuous linear functional on a Hilbert space is inner product with a unique vector.
Rigged Hilbert space A dense nuclear test space inside a Hilbert space, together with its continuous anti-dual.
Schatten-class operator A compact Hilbert-space operator whose singular-value sequence is summable to a specified power.
Schwartz kernel theorem Every continuous operator from test functions to distributions is represented uniquely by a distributional kernel.
Schwartz space on Euclidean space The smooth functions whose derivatives decay faster than every inverse polynomial.
Self-adjoint extension A self-adjoint operator that extends a given symmetric operator without changing its action on the original domain.
Self-adjoint unbounded operator A densely defined Hilbert-space operator is self-adjoint when it equals its adjoint both in action and in domain.
Singular support of a distribution The singular support records the points near which a distribution cannot be represented by a smooth function.
Singular values of a compact operator The decreasing eigenvalue sequence of the absolute value of a compact Hilbert-space operator.
Small solution of an invertible linear equation with quadratic error A contraction estimate solves an invertible linear equation perturbed by a bounded quadratic term.
Smooth compact-support approximation in Euclidean Sobolev space Every finite-exponent integer-order Sobolev function on the whole space can be approximated by test functions.
Sobolev embedding into bounded continuous derivatives More than half a dimension of additional Sobolev order gives bounded continuous derivatives.
Sobolev estimate with a scaled cutoff A localized L6 norm is controlled by the localized gradient and a scale-weighted L2 norm.
Sobolev space A function space whose weak derivatives through a specified order are integrable to a fixed power.
Spectral integral An operator obtained by integrating a measurable scalar function against a projection-valued measure.
Spectral projection of a self-adjoint or normal operator The projection assigned to a Borel subset of the spectrum by an operator's projection-valued spectral measure.
Spectral radius formula for a Banach algebra The spectral radius of a Banach-algebra element is the asymptotic growth rate of the norms of its powers.
Spectral radius in a Banach algebra The largest modulus of a spectral value of a Banach-algebra element.
Spectral Theorem for Compact Selfadjoint Operators A compact self-adjoint operator on a separable Hilbert space has an orthonormal eigenbasis, with nonzero eigenvalues tending to zero.
Spectral theorem for unbounded self-adjoint operators Every self-adjoint operator is uniquely represented as a spectral integral against a projection-valued measure on the real line.
Spectrum of a closed operator The complement of the resolvent set for a closed, possibly unbounded operator.
Spectrum of an element in a Banach algebra The set of complex scalars for which an element shifted by that scalar fails to be invertible.
Strong dual of a locally convex space The continuous dual equipped with the topology of uniform convergence on bounded subsets.
Support of a distribution The support of a distribution is the complement of the largest open set on which the distribution vanishes.
Symmetric operator A symmetric operator is a densely defined Hilbert-space operator whose inner products agree with those of its adjoint on its own domain.
Symmetric Tensor Product (·)_s The symmetrization of a pure tensor, obtained by averaging over all permutations of its factors.
Symplectic Hilbert Space (K,B) A real Hilbert space with a continuous skew form, weakly or strongly nondegenerate according to convention.
Tempered distribution A continuous linear functional on the Schwartz space of rapidly decreasing smooth functions.
Test-function space The LF-space of compactly supported smooth functions on an open subset of Euclidean space.
Three-dimensional Sobolev inequality The L6 norm of a whole-space H1 function is bounded by its L2 gradient norm.
Topological dual The vector space of continuous linear functionals on a topological vector space.
Topological vector space A vector space with a Hausdorff topology for which addition and scalar multiplication are continuous.
Topology generated by a family of seminorms The coarsest vector topology making every seminorm in a prescribed family continuous.
Trace-Class Operator A bounded operator on a Hilbert space whose singular values are summable.
Two-index analytic coefficient space A norm on radial coefficients and parameter derivatives coupling their orders through factorial and binomial weights.
Uniform boundedness principle A pointwise bounded family of bounded operators on a Banach space is uniformly bounded in operator norm.
Unitary operator A surjective linear isometry between complex Hilbert spaces.
Wavefront set The wavefront set records the points and nonzero cotangent directions in which a distribution is not microlocally smooth.
Weak derivative A weak derivative is a locally integrable function representing a function's distributional derivative.
Weak topology on a topological vector space The coarsest vector topology making every continuous linear functional remain continuous.
Weak-star topology The topology of pointwise convergence on the continuous dual of a topological vector space.
Weakly continuous path A time-dependent vector whose value under every continuous linear functional depends continuously on time.
Z/2-graded Hilbert space A Hilbert space decomposed orthogonally into closed even and odd subspaces.

Groups151 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
Alternating Group The subgroup of even permutations in the symmetric group, equivalently the kernel of the sign homomorphism.
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)×
Bianchi group The projective determinant-one matrix group over an imaginary quadratic ring of integers.
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 every automorphism.
Center of a Group The subgroup of elements that commute with every element of a group.
Central Extension An extension whose kernel lies in the center of the total group
Centralizer The subgroup of elements that commute with every element of a given subset.
Characteristic Subgroup A subgroup preserved by every automorphism of the ambient group.
Chief series A normal series with no intermediate normal subgroups between successive terms
Class Equation The order of a finite group is the order of its center plus the sizes of its noncentral conjugacy classes.
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.
Commensurable subgroups Two subgroups whose intersection has finite index in each.
Commutative monoid A monoid whose multiplication is commutative.
Commutator The element g⁻¹h⁻¹gh, which is trivial exactly when g and h 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 in a group have the same order.
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 The subgroup consisting of all integer powers of one group 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 a prime p, every integer a satisfies a^p congruent to a modulo p.
Finite cyclic group is isomorphic to ℤ/nℤ A cyclic group of order n is isomorphic to the additive group ℤ/nℤ.
Finite p-Group Has Nontrivial Center The center of a nontrivial finite p-group has order divisible by p.
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 group homomorphism induces an isomorphism from the quotient by its kernel to its image.
Fixed-Point Set The subset of points fixed by every group element 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 A group freely generated by a set, characterized by its universal mapping property.
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 decomposition Ways to describe a group in terms of simpler subgroups, factors, or extensions.
Group epimorphism A surjective group homomorphism
Group Extension A group fitting into a short exact sequence from a normal subgroup to the group and its quotient.
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 group specified as a quotient of a free group by the normal closure of a set of relations.
Groups of order p^2 are abelian Every finite group of order p², for 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 subgroup 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 and isomorphism.
Lagrange's Theorem In a finite group, the order of a subgroup divides the order of the group
Lattice-ordered abelian group An abelian group with a translation-invariant lattice order.
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)
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
Norm-one group of a quaternion order The units in a quaternion order whose reduced norm is one.
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 The orbits of a group action form a partition of the underlying set.
Orbit–Stabilizer Theorem For a group action, the orbit of a point is in bijection with the coset space of its stabilizer.
Order of Element Divides Order of Group In a finite group, the order of any element divides the order of the group.
Ordered abelian group An abelian group with a translation-invariant total order.
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
PGL–PSL comparison Determinant modulo nth powers measures the difference between PGL_n(k) and PSL_n(k).
Product of normal subgroups is normal If N and M are normal subgroups of G, then NM is a normal subgroup and NM = MN.
Projective general linear group The quotient of a general linear group by its subgroup of nonzero scalar maps.
Projective special linear group The special linear group modulo its scalar center.
Projective special linear group over a ring The determinant-one matrix group modulo its scalar determinant-one matrices.
Proper Subgroup A subgroup that is strictly smaller than the whole group
Quasigroup A magma where division is always possible
Quaternion group The nonabelian group of eight elements generated by i and j with i²=j²=(ij)²=-1.
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 is generated by Schreier generators obtained from a transversal and a symmetric generating set.
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 proper normal subgroups.
Solvable Group A group whose derived series terminates at the trivial subgroup
Special linear group over a ring The group of determinant-one matrices over a commutative ring.
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 H of a group is a subgroup exactly when xy⁻¹ belongs to H for all x,y in H.
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 from the trivial subgroup to a group in which each term 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 is contained in K and both are normal in G, then (G/N)/(K/N) is canonically isomorphic to 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 A homomorphism that kills a normal subgroup factors uniquely through the corresponding quotient group.
Upper central series The ascending series built from successive centers of quotients

Harmonic analysis150 knowls

Admissible representation of a p-adic group A smooth representation whose compact-open fixed spaces are finite-dimensional.
Amenability via weak containment of the trivial representation The representation-theoretic criterion that a locally compact group is amenable exactly when its regular representation weakly contains the trivial representation.
Amenable locally compact group A locally compact group admitting a normalized positive mean invariant under left translation.
Analytic Beurling–Malliavin proposition A controlled plurisubharmonic function is majorized by the logarithmic modulus of a nonvanishing entire L2 function with matching growth.
Approximate identity in L1(G) A bounded net in the group convolution algebra that converges to the identity in its left and right actions.
Bernstein center The algebra of natural endomorphisms of the identity functor on smooth representations of a reductive p-adic group.
Bernstein decomposition The category of smooth representations of a reductive p-adic group decomposes into blocks indexed by inertial supercuspidal support.
Beurling–Malliavin multiplier theorem A Lipschitz negative weight with finite logarithmic integral majorizes the modulus of a nonzero band-limited L2 function.
Bochner's theorem for locally compact abelian groups Bochner's theorem identifies continuous positive-definite functions on a locally compact abelian group with Fourier transforms of finite positive measures on its dual.
Bounded Fourier support The spectral localization condition that a Fourier transform vanish outside a bounded frequency set.
Calderón–Zygmund kernel An off-diagonal kernel with inverse-volume size and Hölder regularity in each variable.
Calderón–Zygmund operator An L2-bounded operator represented off the support of its input by a standard singular kernel.
Carrier frequency parameter A selected base oscillation rate whose integer multiples label the harmonics of an ansatz.
Conjugate and contragredient unitary representations The conjugate action on the conjugate Hilbert space and the corresponding dual action on continuous linear functionals.
Continuous unitary character A continuous homomorphism from a topological group to the circle group.
Convolution on a locally compact group The product of functions obtained by integrating one function against left translates of another.
Damping function for a frequency set A band-limited L2 function with nontrivial local mass, global polynomial decay, and enhanced decay on a prescribed set.
Damping functions from line porosity A bounded set porous on every line admits band-limited damping functions with enhanced decay on that set.
Damping-function fractal uncertainty theorem Uniform damping functions for affine rescalings of a frequency set imply a fractal uncertainty principle against porous physical sets.
Decay of smooth periodic Fourier coefficients Integration by parts gives decay faster than every inverse power of frequency.
Diophantine direction A direction whose nonzero integer dot products have a specified reciprocal polynomial lower bound.
Direct integral of unitary representations A unitary representation assembled fiberwise from a measurable field of representations on a direct-integral Hilbert space.
Directional inverse with a Diophantine bound A zero-mean smooth periodic function has a unique zero-mean directional primitive with finite derivative loss under a polynomial divisor bound.
Dirichlet-to-Neumann operator on the upper half-plane The boundary operator sending Dirichlet data to the normal derivative of its harmonic Poisson extension.
Discrete and continuous spectrum of a unitary representation The discrete part of a unitary representation is generated by its irreducible subrepresentations, while the continuous part is its orthogonal complement.
Distribution on a local group A continuous linear functional on the archimedean or nonarchimedean test-function space of a local group.
Dual Haar measure The dual Haar measure is the uniquely normalized Haar measure on the Pontryagin dual that makes Fourier transformation unitary on L2.
Euclidean L2 Fourier multiplier A bounded measurable frequency function defines a bounded operator on Euclidean L2.
Exact plurisubharmonic Beurling–Malliavin extension Uniform Hilbert-transform and transverse Hessian bounds make a linewise Poisson extension plurisubharmonic after adding a multiple of the imaginary norm.
Fell topology on the unitary dual The Fell topology is the hull-kernel topology on equivalence classes of irreducible unitary representations.
Fourier algebra of a locally compact group The Banach algebra of coefficient functions of a group's left regular representation.
Fourier character on the standard torus An integer-frequency periodic complex exponential.
Fourier convolution theorem on a locally compact abelian group On a locally compact abelian group, the Fourier transform changes convolution into pointwise multiplication on the Pontryagin dual.
Fourier criterion for descent under a torus covering Smooth functions descending through an integer torus covering have Fourier frequencies in the transposed image lattice.
Fourier harmonic A single integer-multiple frequency in a periodic Fourier expansion.
Fourier inversion theorem for locally compact abelian groups The Fourier inversion theorem reconstructs a function on a locally compact abelian group from an integrable Fourier transform.
Fourier transform on a locally compact abelian group The transform integrating an integrable function against the characters of a locally compact abelian group.
Fourier–Stieltjes algebra The Banach algebra of coefficient functions of continuous unitary representations of a locally compact group.
Fractal uncertainty principle A power-saving estimate forbidding simultaneous concentration on porous or fractal sets in physical and Fourier space.
Fractional Laplacian on Euclidean space The Fourier operator with symbol (2 pi times frequency magnitude) to the power twice alpha.
Følner condition A small-boundary condition requiring finite-measure sets that are nearly invariant under translation by prescribed compact subsets.
Gaussian bound from a decreasing net growth rate A growth rate with uniformly negative slope yields a Gaussian envelope about its zero.
GNS construction for a positive-definite function The canonical cyclic unitary representation whose distinguished coefficient is a continuous positive-definite function.
Haar integral A nonzero positive linear functional on compactly supported continuous functions that is invariant under left translation.
Haar measure A nonzero regular Borel measure on a locally compact group that is invariant under translation on one chosen side.
Harish-Chandra Schwartz algebra The Harish-Chandra Schwartz space equipped with group convolution, involution, and its natural Fréchet topology.
Harish-Chandra Schwartz space A Fréchet space of smooth functions on a real reductive group whose invariant derivatives decay relative to the Harish-Chandra Xi-function.
Harish-Chandra Xi-function The normalized positive spherical matrix coefficient that controls decay on a real semisimple Lie group.
Harish–Chandra character of a p-adic representation The invariant distribution character of an admissible p-adic representation and its regular-semisimple representing function.
Harmonic Analysis Haar measure, convolution, Fourier analysis, positive-definite functions, and harmonic analysis on locally compact groups.
Hausdorff–Young inequality on a locally compact abelian group The Fourier transform maps Lp of a locally compact abelian group boundedly into Lq of its dual when 1 is at most p and p is at most 2.
Hecke algebra of a locally compact group pair The Hecke algebra of a locally compact group and compact subgroup is the convolution algebra of compactly supported continuous bi-invariant functions.
Higher-dimensional Beurling–Malliavin multiplier theorem Regular negative weights with summable radial-line growth admit nontrivial band-limited multipliers with controlled decay.
Higher-dimensional line-porous fractal uncertainty principle Ball porosity in physical space and line porosity in Fourier space imply an L2 power-saving uncertainty estimate in every dimension.
Hilbert direct sum of unitary representations The unitary representation obtained by acting coordinatewise on the Hilbert direct sum of representation spaces.
Hilbert transform The singular integral operator with principal-value kernel 1/(pi x), equivalently the Fourier multiplier -i sign(xi).
Induced unitary representation A unitary representation built from a closed-subgroup representation on square-integrable sections over a homogeneous space.
Induction in stages for unitary representations Unitary induction through an intermediate closed subgroup is equivalent to direct induction.
Integrated form of a unitary representation The representation of a group convolution algebra obtained by integrating a strongly continuous unitary representation.
Integrated operator of a continuous representation The operator obtained by averaging a strongly continuous Banach- or Fréchet-space representation against a compactly supported function.
Invariant mean A normalized positive linear functional on a translation-invariant function space that is fixed by translations.
Involution on a group convolution algebra The convolution involution reverses a function by group inversion, complex conjugation, and the modular correction.
Jacquet module The unipotent coinvariants of a smooth representation, normalized to be adjoint to parabolic induction.
Kernel cancellation in a multiplication commutator Commuting a singular integral with multiplication introduces a difference of coefficients.
Kernel description of Fell topology The kernel description of Fell topology expresses closure of irreducible representations through containment of intersections of their kernels.
Kernel of a double Riesz transform in three dimensions The Hessian of the Newtonian kernel represents a double Riesz transform with a diagonal identity term.
L1 group algebra The Banach star-algebra of integrable functions on a locally compact group under convolution.
Langlands classification for p-adic groups Every irreducible admissible representation is the unique irreducible quotient of a standard module induced from tempered data.
Left and right regular representations of a locally compact group The regular representations act unitarily on square-integrable functions by left and modularly corrected right translation.
Linewise Poisson extension operator An operator extending a function on real Euclidean space by taking a Poisson average along the real line determined by the imaginary direction.
Local wavevector The spatial gradient of the full angular phase of an oscillatory field.
Locally compact homogeneous space A quotient G/H of a locally compact group by a closed subgroup, equipped with its transitive continuous G-action.
Lp boundedness of Calderón–Zygmund operators A standard singular kernel together with L2 boundedness gives boundedness for every interior Lebesgue exponent.
Mackey imprimitivity theorem Transitive systems of imprimitivity are exactly those obtained by inducing representations from stabilizer subgroups.
Mackey machine for semidirect products A classification of irreducible unitary representations of regular semidirect products with an abelian normal factor.
Matrix coefficient of a unitary representation A scalar-valued function obtained by pairing the orbit of one Hilbert-space vector with another.
Maximal compact and hyperspecial subgroups Maximal compact subgroups and the more restrictive hyperspecial compact opens arising from reductive integral models.
Mean of a product of periodic Fourier sums Only pairs of opposite frequencies contribute to the average of a product without conjugation.
Measure algebra of a locally compact group The measure algebra of a locally compact group is the Banach algebra of bounded regular complex Borel measures under convolution.
Modular function of a locally compact group The positive continuous homomorphism measuring how a left Haar measure scales under right translation.
Multiplicity function of a representation A multiplicity function records the dimension of the multiplicity space attached to almost every irreducible fiber in a direct-integral decomposition.
Nearly resonant Fourier mode A nonresonant directional Fourier mode whose eigenvalue is small relative to a chosen tolerance.
Nonabelian group Fourier transform The operator-valued transform obtained by integrating a function against each irreducible unitary representation of a locally compact group.
Normalized parabolic induction for a p-adic group Smooth parabolic induction with the half-modulus normalization.
Normalized torus average Integration over a lattice cell divided by its volume gives a translation-invariant probability average.
Order-zero Fourier multiplier A bounded smooth-away-from-zero symbol with scale-invariant derivative bounds of every order.
Orthogonality of periodic Fourier characters Distinct integer-frequency characters have zero averaged Hermitian product.
Oscillatory integral An integral with a rapidly varying phase whose cancellation, rather than absolute size, governs its asymptotics.
Oscillatory modulation Multiplication of a coefficient by a rapidly varying complex phase factor.
Oscillatory phase A real-valued function used as the argument of a complex exponential.
Oscillatory pulse A wave with an amplitude localized in a finite time or path-coordinate interval.
Paley–Wiener theorem for bounded Fourier support An L2 function is band limited exactly when it extends to an entire function of the corresponding exponential type.
Parabolic modulus character The positive character measuring how a parabolic Levi acts on its unipotent radical.
Periodic antiderivative A periodic primitive, whose existence requires zero mean over a period.
Periodic Fourier coefficient The normalized integral of a periodic function against a conjugate Fourier character.
Periodic Fourier multiplier An operator defined by multiplying each periodic Fourier coefficient by a prescribed symbol.
Periodic Fourier series The series of characters weighted by the Fourier coefficients of a periodic function.
Periodization of a compactly supported smooth function Summing lattice translates creates a smooth periodic function.
Plancherel measure for a type I locally compact group The spectral measure on the unitary dual that makes the nonabelian Fourier transform an L2 isometry.
Plancherel theorem for locally compact abelian groups The Plancherel theorem extends Fourier transformation to a unitary operator between the L2 spaces of a locally compact abelian group and its dual.
Plancherel theorem for unimodular type I groups The nonabelian Fourier transform is an L2 isometry onto a direct integral of Hilbert–Schmidt operator spaces.
Plurisubharmonic Beurling–Malliavin proposition A regular weight with controlled radial-line growth has a Lipschitz plurisubharmonic minorant with linear growth in imaginary directions.
Poisson extension to the upper half-plane Boundary data on the real line convolved with the Poisson kernel to produce a harmonic function on the upper half-plane.
Poisson kernel of the upper half-plane The positive unit-mass kernel that harmonically extends boundary data from the real line to the upper half-plane.
Pontryagin dual The locally compact abelian group of continuous circle-valued characters of a locally compact abelian group.
Pontryagin duality theorem Every locally compact abelian group is canonically isomorphic as a topological group to the Pontryagin dual of its dual.
Positive-definite function on a group A positive-definite function is a continuous scalar function whose group-difference matrices are positive semidefinite.
Principal-value singular integral An integral operator defined by removing a neighborhood of its diagonal and taking a limit.
Pulse envelope A positive scalar function used to bound the amplitude of a localized pulse.
Quasi-invariant measure under a locally compact group action A quasi-invariant measure has the same null sets as each of its translates under the group action.
Quasi-regular representation A quasi-regular representation is the unitary action of a locally compact group on square-integrable functions over a homogeneous space.
Radial-line growth functional of a weight A maximal averaged magnitude along radial lines used as the growth condition in a higher-dimensional multiplier theorem.
Rapid decay of a shifted torus integral A zero integrated mean and a Diophantine drift give any fixed inverse power of a large shift, at a finite derivative cost.
Reiter condition An analytic approximation condition asserting the existence of normalized nonnegative L1 functions that are uniformly almost invariant on compact sets.
Resonant mode of a directional derivative A Fourier mode on which a specified constant directional derivative vanishes.
Restricted tensor product of local test-function spaces The tensor product of local test-function spaces in which almost every factor equals a fixed standard test function.
Riesz transform The order-zero Euclidean multiplier i times a frequency coordinate divided by frequency length.
Scaled cutoff commutator estimate A scale-R Lipschitz cutoff gives an L1 to Lq commutator bound below the first-order endpoint.
Schwartz–Bruhat space on a locally compact abelian group The canonical test-function space on a locally compact abelian group, assembled from Schwartz functions on elementary quotients.
Schwartz–Bruhat space on the adeles The restricted tensor product of the local Schwartz–Bruhat spaces over all places of a global field.
Schwartz–Bruhat space over a local field The test-function space on a local field, given by Schwartz functions at archimedean places and locally constant compactly supported functions otherwise.
Small divisor A small nonzero spectral denominator that amplifies coefficients when a linear equation is inverted.
Smooth periodic Fourier reconstruction The Fourier series of a smooth periodic function converges uniformly with every derivative.
Smooth representation of a totally disconnected group A representation in which every vector is fixed by an open subgroup.
Spherical Fourier transform The scalar transform of a bi-invariant function obtained by integrating it against elementary spherical functions.
Spherical function on a Gelfand pair A normalized bi-invariant function whose averaging product formula makes it a character of a Gelfand pair's convolution algebra.
Spherical Hecke algebra and Satake isomorphism The commutative unramified Hecke algebra and its normalized identification with the representation ring of the dual group.
Supercuspidal representation An irreducible admissible p-adic representation with no contribution from a proper parabolic subgroup.
System of imprimitivity A system of imprimitivity is a unitary representation together with a projection-valued measure covariant for a group action.
Tempered dual of a locally compact group The part of a group's unitary dual represented by irreducible unitary representations weakly contained in the regular representation.
Tempered representation of a p-adic group An irreducible admissible representation whose matrix coefficients have almost square-integrable decay.
Tempered unitary representation A unitary representation weakly contained in the regular representation of its locally compact group.
Tensor product of unitary representations The unitary representation acting diagonally on the completed Hilbert tensor product of two representation spaces.
Test-function space on a local group The compactly supported smooth functions on an archimedean Lie group or locally constant compactly supported functions on a locally profinite group.
Torus coverings preserve normalized averages Pulling a smooth function back by a nonsingular integer torus map preserves its average.
Transverse polarization of a vector wave An amplitude lying in the plane perpendicular to the local wavevector.
Unimodular locally compact group A unimodular locally compact group has trivial modular function, so its left Haar measures are also right invariant.
Unitary dual of a locally compact group The space of unitary-equivalence classes of irreducible continuous unitary representations of a locally compact group.
Unitary Frobenius reciprocity For a compact group, bounded intertwiners into an induced representation correspond to subgroup intertwiners.
Unramified representation of a p-adic group An irreducible admissible representation with a nonzero hyperspecial-fixed vector.
Wave amplitude The scalar or vector coefficient multiplying an oscillatory phase factor.
Wavelength The spatial distance for a plane wave’s phase to advance by one full cycle.
Wavenumber The magnitude of an angular wavevector, measuring radians of phase change per unit distance.
Weak containment of unitary representations One unitary representation is weakly contained in another when its diagonal coefficients are locally approximable by finite sums from the other.
Weil integration formula The Weil integration formula decomposes integration on a locally compact group into integration along a closed subgroup and over its homogeneous space.
Whittaker model A realization of a generic representation using functions transforming by a nondegenerate character of a maximal unipotent subgroup.
Wiener Transform W A unitary transform on Gaussian L₂ intertwining T with T^{*-1}
X-ray transform The integral of a function over affine lines, parameterized by a base point and a direction.
Young's convolution inequality Young's convolution inequality bounds the Lp norm of a convolution by the product of the norms of its factors.
Zero Fourier mode The constant component of a periodic Fourier expansion, equal to its normalized mean.

Homological algebra32 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 five-term sequences, suitable isomorphism, epimorphism, and monomorphism hypotheses force the middle map to be an isomorphism.
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, exact columns and two exact rows force the remaining row to be exact.
Projective resolution An exact augmented chain complex of projective modules resolving a given module.
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).

Hyperstructures, semirings, and blueprints34 knowls

Algebra: Hyperstructures, semirings, and blueprints Hyperoperations, semirings, hyperrings, tracts, bands, pastures, and ordered blueprints.
Band A pointed commutative monoid equipped with an ideal of null formal sums and unique additive inverses.
Canonical hypergroup A commutative associative hyperoperation with zero, unique inverses, and reversibility.
Classification of stringent hyperfields Stringent hyperfields are tropical extensions whose residue layer is a field, the Krasner hyperfield, or the sign hyperfield.
Doubly distributive hyperfield A hyperfield in which the product of two binary hypersums equals the four-term hyper-sum of pairwise products.
Doubly distributive hyperfields are stringent Double distributivity forces every nonopposite binary hypersum to be a singleton.
Field embeddings into semifields and hyperfields Fields embed fully faithfully as semifields and as singleton-addition hyperfields.
Fusion band A band whose nullset is closed under the fusion rule.
Fusion rule for bands The cancellation rule that combines two null formal sums containing opposite terms.
Hyperfield A nontrivial hyperring whose nonzero elements form a multiplicative group.
Hyperfield of a field quotient The quotient hyperfield K/G formed from a field and a multiplicative subgroup.
Hyperoperation A binary operation whose value is a nonempty set of possible outputs.
Hyperring A canonical additive hypergroup with a single-valued distributive multiplication.
Hyperring homomorphism A unit-preserving multiplicative map that weakly or strongly preserves hyperaddition.
Idyll A band in which every nonzero element is multiplicatively invertible.
Integral hyperring A nontrivial commutative hyperring without multiplicative zero divisors.
Krasner hyperfield The two-element hyperfield with 1⊞1={0,1}.
Nullset of a band The ideal of formal sums declared null in a band.
Ordered blueprint associated with a tract A tract defines a purely positive ordered blueprint whose order is generated by its null sums.
Partial field A multiplicative group of ring units on which addition is only partially defined.
Partial hyperfield A multiplicative group selected inside an integral hyperring, with a possibly empty multivalued partial addition.
Pasture An idyll whose nullset is the fusion ideal generated by its three-term null relations.
Phase hyperfield The quotient hyperfield of the complex numbers by positive real scaling.
Quotient hyperring The hyperring of multiplicative orbits R/G of a ring under a subgroup of units.
Sign hyperfield The three-element hyperfield recording only whether a real number is negative, zero, or positive.
Stringent hyperfield A hyperfield whose sum is single-valued unless the summands are additive inverses.
Tract An abelian multiplicative group equipped with a null set of formal sums.
Tract associated with a hyperfield The tract whose null formal sums are precisely the hypersums containing zero.
Tract associated with a partial field The tract whose null sums are the formal sums vanishing in the ambient ring of a partial field.
Tract associated with a partial hyperfield The tract obtained from the ambient null hypersums of a partial hyperfield.
Tropical hyperfield The max-plus hyperfield whose tied sum is the full lower interval.
Tropical semifield and hyperfield tied-sum distinction The tropical semifield and hyperfield share their carrier and multiplication but differ exactly at tied addition.
Valuations as tropical hyperfield morphisms Negating an additive non-Archimedean valuation gives a weak morphism to a max tropical hyperfield.
Valuative hyperfield The ordered-group generalization of the tropical hyperfield.

Langlands125 knowls

$L$ -packet The finite family of irreducible admissible representations sharing one local Langlands parameter.
Affine Grassmannian The ind-projective loop-group quotient parametrizing a G-bundle on a formal disc with a punctured-disc trivialization.
Affine Schubert variety The closure of a positive-loop-group orbit in the affine Grassmannian.
Affine Springer fiber The locus of affine-Grassmannian lattices on which a fixed loop-Lie-algebra element becomes integral.
Algebraic automorphic representation An automorphic representation with integral archimedean infinitesimal data, with the normalization stated explicitly.
Arthur multiplicity formula The component-group character formula selecting representations and multiplicities in a global Arthur packet.
Arthur packet A finite packet of irreducible representations attached to an Arthur parameter.
Arthur parameter A bounded Langlands parameter enlarged by an additional algebraic SL2 factor.
Arthur truncation An alternating subtraction of parabolic constant terms that makes automorphic kernels rapidly decreasing in cuspidal directions.
Arthur-Selberg trace formula An equality between geometric orbital distributions and spectral automorphic distributions.
Artin conductor A nonnegative integer measuring the tame and wild ramification of a local Galois or Weil representation.
Automorphic form A smooth, finite, moderate-growth function on an adelic automorphic quotient.
Automorphic form and Hecke eigenvalues The historical passage from an automorphic form to unramified spherical Hecke eigencharacters and Satake parameters.
Automorphic representation An irreducible representation occurring as a subquotient of a space of automorphic forms.
Automorphic-Galois correspondence The conjectural and partially proved association between algebraic automorphic representations and l-adic Galois representations.
Basic local Langlands correspondence The finite-to-one parameterization of irreducible admissible representations by relevant local L-parameters.
C-algebraic automorphic representation An automorphic representation whose archimedean parameter is integral after the half-sum-of-positive-roots shift.
Categorical geometric Langlands theorem In characteristic zero, the automorphic category of half-twisted D-modules on Bun_G is equivalent to the spectral category IndCoh_N on dual-group local systems.
Coalescence of shtuka legs The fusion isomorphism relating shtuka cohomology before and after several legs are identified.
Cohomological automorphic representation An automorphic representation whose archimedean components have nonzero relative Lie algebra cohomology with algebraic coefficients.
Compatibility properties of local Langlands The structural conditions relating representation operations, local factors, and local Langlands parameters.
Compatible system of Galois representations A family of l-adic Galois representations with coefficient-independent Frobenius polynomials at almost all places.
Component group of an $L$ -parameter The finite group of connected components of the dual-group centralizer of a Langlands parameter, with refinements used to index packets.
Constant term of an automorphic form The integral of an automorphic form along the unipotent radical of a parabolic subgroup.
Continuous automorphic spectrum The non-discrete part of the automorphic L2 spectrum, assembled from Eisenstein series induced from proper Levi subgroups.
Convolution of sheaves on the affine Grassmannian The sheaf obtained by external product on the convolution Grassmannian followed by proper pushforward along multiplication.
Crystalline Galois representation A p-adic Galois representation with the full expected space of crystalline periods.
Cuspidal automorphic representation An irreducible constituent generated by automorphic forms whose proper parabolic constant terms vanish.
De Rham Galois representation A p-adic Galois representation with the full expected space of de Rham periods.
Discrete automorphic spectrum The Hilbert direct sum of irreducible representations occurring discretely in an automorphic L2-space.
Dominant coweight A cocharacter pairing nonnegatively with every positive root determined by a Borel subgroup.
Drinfeld's lemma Partial Frobenius structures on a product over a finite field produce an action of a product of Weil or fundamental groups.
Eisenstein series on a reductive group An automorphic series formed from parabolically induced cuspidal data, with meromorphic continuation and intertwining-operator functional equations.
Endoscopic datum Dual-group data defining a reductive endoscopic group and its L-embedding into a given L-group.
Endoscopic transfer The matching of test functions and stable distributions between a group and an endoscopic group.
Endoscopic transfer factor A normalized scalar comparing orbital integrals of matching elements on an endoscopic group and the original group.
Euler product and determinant local L-factor Unramified and Weil-Deligne local L-factors and their incomplete global Euler product.
Excursion algebra The commutative algebra generated by all excursion operators on a cuspidal automorphic space.
Excursion operator An operator on cuspidal automorphic forms built by creating shtuka legs, moving them by Galois elements, and annihilating them.
Fargues-Fontaine curve A one-dimensional curve built from a local field and a perfectoid characteristic-p space, translating isocrystals into vector bundles.
Fargues-Scholze parameter map The theorem attaching a unique semisimple local L-parameter to an irreducible smooth representation.
Fargues-Scholze spectral action The action of perfect complexes on the local parameter stack on sheaves over the stack of G-bundles.
Fontaine period rings Galois-equivariant p-adic coefficient rings whose invariant periods define Hodge–Tate, de Rham, crystalline, and semistable representations.
Fundamental lemma of endoscopy The theorem that unramified unit Hecke functions have matching endoscopic orbital integrals.
G-bundle on the Fargues-Fontaine curve A principal G-bundle on a Fargues-Fontaine curve, classified on geometric points by a G-isocrystal.
G-local system A principal G-bundle with flat connection; over the complex numbers, Riemann-Hilbert comparison relates it to a G-valued monodromy representation.
G-shtuka A principal G-bundle on a curve with a Frobenius identification away from bounded modification legs.
Geometric Hecke functor A pull-push functor on automorphic sheaves defined from the Hecke correspondence and a dual-group representation.
Geometric Langlands index Index of geometric-Langlands foundations, Hecke structures, spectral parameters, and ramified examples.
Geometric Satake equivalence The tensor category of spherical perverse sheaves on Gr_G is equivalent to representations of the Langlands dual group.
Global Langlands parameter A conjectural global admissible homomorphism into an L-group whose localizations are local Langlands parameters.
Global Langlands reciprocity The conjectural relation between global arithmetic parameters and automorphic representations.
Global nilpotent cone for local systems The conical locus in the scheme of singularities of LocSys_G consisting of local systems equipped with a horizontal nilpotent coadjoint section.
Hecke correspondence The moduli correspondence relating two G-bundles by a modification at a point of the curve.
Hecke eigensheaf An automorphic sheaf on Bun_G whose geometric Hecke transforms are governed by a dual-group local system.
Hecke modification A change of a G-bundle at one point, given by an isomorphism away from that point.
Hitchin fibration The map sending a Higgs field to the invariant-polynomial coefficients of its characteristic data.
Hodge–Tate representation A p-adic Galois representation that splits over C_p into integral Tate twists.
Ind-coherent sheaf An object of the cocomplete category obtained by adjoining filtered colimits to coherent sheaves on a derived stack.
Ind-coherent sheaves with nilpotent singular support The spectral category IndCoh_Nilp on the stack of dual-group local systems.
Ind-scheme A functor presented as a filtered colimit of schemes, usually along closed immersions.
Intersection-cohomology complex The perverse middle extension of a shifted local system from a smooth dense stratum.
Irregular-singular connection A meromorphic flat connection whose singularity is not regular singular and whose local classification requires exponential and Stokes data.
Isobaric automorphic representation An automorphic representation of GL_n formed as the Langlands quotient of cuspidal data on a standard Levi subgroup.
Iwahori level structure A level structure defined by the inverse image of a Borel subgroup under evaluation of positive loops at the marked point.
Kappa-orbital integral A character-weighted sum of orbital integrals inside one stable conjugacy class.
Kottwitz set B(G) The set of Frobenius-conjugacy classes in a reductive group over the completed maximal unramified extension.
L-algebraic automorphic representation An automorphic representation whose archimedean Langlands-parameter exponents are integral cocharacters.
Langlands $L$ -group The complex dual group extended by the Weil or Galois action on its based root datum.
Langlands dual group The pinned connected complex reductive group whose based root datum is dual to that of a reductive group.
Langlands functoriality and L-homomorphisms An L-group homomorphism predicting compatible transfer of local parameters and global automorphic representations.
Langlands program Index of local and global Langlands correspondences, automorphic representations, endoscopy, function-field methods, and geometric Langlands.
Level structure on a G-bundle A reduction of the formal-frame torsor of a G-bundle at a marked point to a chosen subgroup of the positive loop group.
Local class field theory The reciprocity isomorphism between a local field's multiplicative group and the abelianization of its Weil group.
Local epsilon factor The normalization-sensitive local constant in the functional equation of a local L-factor.
Local gamma factor The local functional-equation factor combining epsilon and the ratio of dual L-factors.
Local Langlands correspondence for $\mathrm{GL}_n$ The canonical bijection between irreducible smooth representations of GL_n and n-dimensional Frobenius-semisimple Weil–Deligne representations.
Local Langlands parameter An admissible homomorphism from the local Langlands group to the L-group of a reductive group.
Local Shimura variety A tower of p-adic moduli spaces attached to local Shimura data and parametrizing bounded modifications of G-bundles.
Local shtuka A bounded Frobenius modification of G-bundles in local or Fargues-Fontaine geometry.
Local-global compatibility Agreement between the localization of a global Galois representation and the local Langlands parameter of an automorphic component.
Loop group The group-valued functor LG(R)=G(R((t))) associated with an affine algebraic group G.
Modern Langlands dependency closure index The reusable arithmetic, representation-theoretic, geometric, and categorical prerequisites added by the semantic closure audit of the modern Langlands collection.
Modern Langlands foundations index A guided index of local and global parameters, automorphic representations, endoscopy, function-field shtukas, and local geometrization.
Moduli stack of G-local systems The derived moduli stack LocSys_G(X) of principal G-bundles with flat connection.
Orbital integral The invariant distribution obtained by integrating a test function over a conjugacy orbit.
Parahoric level structure An extension of a punctured-disc G-bundle to a torsor for a Bruhat-Tits parahoric group scheme on the formal disc.
Partial Frobenius on shtukas The operation that applies Frobenius to selected shtuka legs and cyclically rotates the corresponding modification.
Perverse sheaf An object in the heart of the middle-perversity t-structure on a constructible derived category.
Positive loop group The subgroup-valued functor of G-valued loops that extend to the formal disc.
Ramification of a G-local system The obstruction to extending a G-local system across a puncture as an unramified object in a fixed Betti, de Rham, or etale category.
Ramified geometric Langlands A geometric Langlands problem with fixed local singularity data on the spectral side and matching level and equivariance data on the automorphic side.
Rapoport–Zink space A formal moduli space of p-divisible groups quasi-isogenous to a fixed framing object, often with additional structure.
Refined local Langlands correspondence The internal parametrization of local L-packets, simultaneously across rigid inner forms, by representations of a parameter centralizer.
Regular algebraic cuspidal automorphic representation A cuspidal automorphic representation of a general linear group with regular integral archimedean infinitesimal character.
Regular-singular connection A flat connection on a punctured curve that has at worst logarithmic poles after extension across the punctures.
Residual automorphic spectrum The noncuspidal discrete automorphic spectrum generated by residues of Eisenstein series.
Restricted tensor-product factorization of an automorphic representation The factorization of an irreducible admissible adelic representation into local components.
Riemann–Hilbert correspondence Regular-singular algebraic flat connections on a smooth complex variety correspond to finite-dimensional complex local systems on its analytic space.
Rigid inner twist An inner twist equipped with a cohomological rigidification used to normalize refined local Langlands packets and transfer factors.
Root number The unit-modulus constant in a normalized local or global L-function functional equation.
Satake parameter The semisimple dual-group conjugacy class classifying an unramified representation.
Semistable Galois representation A p-adic Galois representation whose semistable period module has full dimension.
Shtuka A vector bundle on a curve equipped with a Frobenius isomorphism away from finitely many moving legs.
Singular support of a coherent sheaf A conical subset of the scheme of singularities of a quasi-smooth derived stack that records the derived directions in which a coherent sheaf is singular.
Spectral Bernstein center The ring of global functions on the stack of local Langlands parameters.
Stable conjugacy Conjugacy over an algebraic closure, retaining the Galois descent class of the centralizer.
Stable distribution An invariant distribution that depends only on stable orbital-integral data.
Stable orbital integral The sum of orbital integrals over rational conjugacy classes in one stable conjugacy class.
Stable trace formula The decomposition of the invariant trace formula into stable distributions on a group and its endoscopic groups.
Stack of local L-parameters The quotient stack of continuous dual-group-valued Weil cocycles by dual-group conjugation.
Stokes data The sectorial transition data that, together with formal type, classify an irregular meromorphic connection analytically.
Strong multiplicity one theorem Cuspidal automorphic representations of GL_n are determined by their local components outside any finite set of places.
Strongly regular semisimple element A semisimple element of a reductive group whose centralizer is a maximal torus.
Tamagawa measure A canonical adelic Haar measure assembled from an invariant differential form and global convergence factors.
Tame ramification Ramification of an etale local system for which wild inertia acts trivially.
Tate–Nakayama duality Duality identifying Galois cohomology of a local torus with characters of the component group of its dual torus.
Twisted conjugacy Conjugacy modified by an automorphism, including Frobenius and sigma-conjugacy as basic cases.
Vincent Lafforgue global parameterization The canonical decomposition of cuspidal automorphic forms over a function field by semisimple global Langlands parameters.
Weighted orbital integral An orbital integral multiplied by Arthur's parabolic weight, forming the fine geometric terms of the trace formula.
Weil group A locally compact refinement of an absolute Galois group used in local and global class field theory and Langlands parameters.
Weil–Deligne group The local Weil group augmented by monodromy, presented either with an additive factor or an auxiliary SL_2.
Weil–Deligne representation A Weil-group representation together with a compatible nilpotent monodromy operator.
Whittaker datum A conjugacy class of a rational Borel subgroup together with a nondegenerate character of its unipotent radical.
Wild ramification Ramification of an etale local system on which wild inertia acts nontrivially.

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 groups348 knowls

(g,K)-module A compatible locally finite action of a complexified Lie algebra and a compact subgroup.
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.
Admissible representation of a real reductive group A representation whose restriction to a maximal compact subgroup has finite multiplicity for every irreducible type.
Ado’s theorem Every finite-dimensional Lie algebra over characteristic 0 has a faithful finite-dimensional representation.
Analytic vector of a Lie-group representation A representation vector whose orbit map is real analytic near the identity of the Lie group.
Arithmetic Kleinian group A Kleinian group commensurable with a projective norm-one group from a quaternion order with one complex place.
Arithmeticity of Bianchi groups A Bianchi group is the projective norm-one group of the full matrix order over an imaginary quadratic number ring.
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.
Bianchi groups are nonuniform lattices Bianchi groups are discrete and have finite covolume but noncompact quotient.
Borel–de Siebenthal theory The root-theoretic classification of connected maximal-rank subgroups of compact Lie groups.
Branching rule for Lie representations The decomposition of a representation after restriction from a Lie group or Lie algebra to a subgroup or subalgebra.
Cartan decomposition of a real reductive Lie algebra The decomposition of a real reductive Lie algebra into the two eigenspaces of a Cartan involution.
Cartan involution of a real Lie algebra An involutive automorphism whose compactly twisted invariant form is positive definite.
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 A maximal toral subalgebra of a complex semisimple Lie algebra equals its Lie-algebra normalizer.
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].
Casimir element and Casimir operator A basis-independent central element of an enveloping algebra and the operator through which it acts in a representation.
Casselman subrepresentation theorem Every irreducible Harish–Chandra module for a real reductive group embeds into the Harish–Chandra module of a minimal principal-series representation.
Casselman–Wallach globalization theorem Taking maximal-compact-subgroup-finite vectors is an equivalence from smooth admissible moderate-growth representations to Harish–Chandra modules.
Casselman–Wallach representation A smooth admissible Fréchet representation of a real reductive group whose action has moderate growth.
Celestial sphere and Möbius action The proper Lorentz action on future null directions becomes the PSL(2,C) Möbius action on CP¹.
Center of a Lie algebra The elements that bracket to zero with every element of a Lie algebra.
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.
Center of the universal enveloping algebra The commutative subalgebra of enveloping-algebra elements that commute with every element.
Central quotient of a Lie group A Lie-group quotient by a closed subgroup of the center.
Centralizer of a Lie subalgebra The Lie subalgebra of elements commuting with every element of a given Lie subalgebra.
Centralizer of a regular reductive subalgebra A root-space formula for the centralizer of a regular reductive subalgebra of a complex semisimple Lie algebra.
Chevalley basis A root-adapted basis of a split semisimple Lie algebra with normalized integral bracket constants.
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 exceptional Lie group E6 The compact simply connected exceptional Lie group of type E6, rank 6, dimension 78, and center of order 3.
Compact exceptional Lie group E7 The compact simply connected exceptional Lie group of type E7, rank 7, dimension 133, and center of order 2.
Compact exceptional Lie group E8 The compact connected exceptional Lie group of type E8, rank 8 and dimension 248; it is simply connected and centerless.
Compact exceptional Lie group F4 The compact connected simple 52-dimensional Lie group of rank 4 that is the automorphism group of the real Albert algebra.
Compact exceptional Lie group G2 The compact connected simple 14-dimensional Lie group of rank 2 that is the automorphism group of the real octonions.
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).
Compact real form A real form of a complex semisimple Lie algebra whose Killing form is negative definite.
Compact symplectic group The compact Lie group of quaternionic linear transformations preserving the standard quaternionic Hermitian form.
Compatible Cartan subalgebras Cartan subalgebras chosen along an inclusion so that the smaller one is obtained by intersection with the larger.
Complementary series representation A complementary series consists of unitary non-tempered members obtained by unitarizing principal-series representations away from the unitary parameter axis.
Complete filtered Lie algebra A Lie algebra complete for a bracket-compatible descending filtration.
Completely reducible Lie representation A representation that splits as a direct sum of irreducible subrepresentations.
Complex Lie algebra sl2(C) The three-dimensional simple complex Lie algebra of trace-zero 2 by 2 matrices, of rank 1 and Dynkin type A1.
Complex Lie algebra sl3(C) The eight-dimensional simple complex Lie algebra of rank 2 and Dynkin type A2.
Complex Lie algebra sl5(C) The 24-dimensional simple complex Lie algebra of rank 4 and Dynkin type A4.
Complex Lie algebra sl6(C) The 35-dimensional simple complex Lie algebra of rank 5 and Dynkin type A5.
Complex Lie algebra so10(C) The 45-dimensional simple complex orthogonal Lie algebra of rank 5 and Dynkin type D5.
Complex Lie algebra so12(C) The 66-dimensional simple complex orthogonal Lie algebra of rank 6 and Dynkin type D6.
Complex Lie group A complex manifold whose multiplication and inversion maps are holomorphic.
Complexification of a real Lie algebra The complex Lie algebra obtained by extending scalars from the real numbers to the complex numbers.
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.
Coroots and the character-cocharacter pairing Coroots and their integral pairing with characters, controlling dominance and dual root data.
Coset space The quotient space of left cosets G/H, which is a smooth manifold when H 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.
Cyclic vector and cyclic unitary representation A vector whose group orbit has dense linear span, and a representation possessing such a vector.
Defining representation of a classical Lie algebra The natural representation obtained from the matrix realization of a classical Lie algebra.
Derivation of a Lie algebra A linear map satisfying the Leibniz rule for the Lie bracket; derivations form a Lie algebra.
Derived representation on smooth vectors The Lie-algebra representation obtained by differentiating a strongly continuous unitary representation on its smooth vectors.
Derived series of a Lie algebra The descending sequence obtained by repeatedly taking derived subalgebras.
Derived subalgebra The Lie subalgebra spanned by commutators; it measures how far a Lie algebra is from being 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 series representation An irreducible unitary representation occurring as a closed subrepresentation of the left regular representation.
Discrete subgroup A subgroup that is discrete in the manifold topology; its Lie algebra is zero.
Distribution vectors of a Lie-group representation Continuous antilinear functionals on the Fréchet space of smooth vectors of a Lie-group representation.
Dixmier–Malliavin factorization theorem Every compactly supported smooth function on a Lie group is a finite sum of convolutions of such functions, and every smooth representation vector is a finite sum of smoothed vectors.
Dual (contragredient) representation The dual representation acts on linear functionals by precomposition with the inverse group action; infinitesimally, it is the negative transpose.
Dynkin diagram A graph encoding the angles and relative lengths among simple roots of a semisimple Lie algebra via the Cartan matrix.
Dynkin-diagram automorphisms and dual representations Diagram symmetries act on highest weights, while the opposition involution gives the highest weight of the dual representation.
E7 branching under A1 + D6 Under each generation sl2 plus so12 subalgebra, the adjoint module of e7 branches as (3,1) plus (1,66) plus (2,32).
E7 branching under A2 + A5 The adjoint E7 module branches under the mutually centralizing generation sl3 and standard sl6 as 8 + 35 + (3⊗15) + (3⊗15).
E7 root-projection trichotomy Projection of an E7 root to the generation plane is zero, a defining A2 weight up to sign, or a generation A2 root.
Effective action A Lie group action whose only element fixing every point is the identity.
En series of Lie algebras The extended E_n convention linking the exceptional types E6, E7, E8 to D5, A4, and A2+A1 by successive Dynkin-node deletion.
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.
Essential skew-adjointness of derived representation operators A derived operator on smooth vectors closes to the skew-adjoint generator of the corresponding one-parameter unitary group.
Even and odd exterior algebra The parity decomposition of an exterior algebra into its even-degree and odd-degree parts.
Example: $SU(2)$ and its Lie algebra is simply connected; is 3D with Pauli-matrix commutators, and is a 2-fold cover.
Example: SO(3) and rotations The Lie algebra of SO(3) consists of real skew-symmetric matrices, whose exponentials are rotations.
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 The three-dimensional nilpotent Lie algebra with basis X, Y, Z and bracket [X,Y]=Z.
Example: the sphere as a homogeneous space The sphere is a homogeneous space under the standard transitive action of the special orthogonal group.
Example: the torus $T^n$ The n-torus is a compact connected abelian Lie group with Lie algebra R^n and quotient exponential map.
Example: U(1) (the circle group) The circle group has Lie algebra iR and a surjective exponential map with kernel 2πiZ.
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.
Exceptional chain to the Standard Model Lie algebra A chain of regular semisimple subalgebras obtained by deleting simple roots, terminating in the semisimple part of the Standard Model Lie algebra.
Exceptional E-type root systems The three exceptional simply laced irreducible root systems E_6, E_7, and E_8.
Exceptional Lie algebra e6 The 78-dimensional simple complex Lie algebra of rank 6 and exceptional Dynkin type E6.
Exceptional Lie algebra e7 The 133-dimensional simple complex Lie algebra of rank 7 and exceptional Dynkin type E7.
Exceptional Lie algebra e8 The 248-dimensional simple complex Lie algebra of rank 8 and exceptional Dynkin type E8.
Exceptional Lie algebra f4 The 52-dimensional simple complex Lie algebra of rank 4 and exceptional Dynkin type F4.
Exceptional Lie algebra g2 The 14-dimensional simple complex Lie algebra of rank 2 and exceptional Dynkin type G2.
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.
Exterior algebra of a direct sum The canonical graded-algebra isomorphism between the exterior algebra of a direct sum and a graded tensor product.
Exterior-power representation The representation induced on an exterior power by applying each group element, or infinitesimally by a derivation formula.
Fixed-vector subspace The subspace of a representation on which every group or Lie-algebra element acts trivially.
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.
Furstenberg boundary of a semisimple Lie group The compact homogeneous space obtained by quotienting a semisimple Lie group by a minimal parabolic subgroup.
Gelfand pair A locally compact group and compact subgroup whose bi-invariant convolution algebra is commutative.
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.
Generation module as an even exterior algebra Adding two generation-root singlets to each 30-dimensional module gives Λeven C6, which restricts to ΛC5 and hence one full Standard Model generation.
Generation plane in e7 The real two-plane spanned by the A2 roots of the generation sl3 after choosing a compatible Cartan subalgebra.
Generation sl2 subalgebras The three regular sl2 subalgebras associated with the three root lines of the generation A2 root system.
Generation sl3 in e7 A good Standard Model subalgebra of e7 has centralizer sl3 plus a two-dimensional center, with a unique sl3 subalgebra.
Global Cartan decomposition The decomposition of a real reductive group as K exp(p), refined to the KAK decomposition.
Global character of an admissible representation The conjugation-invariant distribution obtained by tracing the integrated operators of an admissible representation.
Good Standard Model embedding in e7 The distinguished automorphism class of regular embeddings of the complexified Standard Model Lie algebra in e7 used in the three-generation construction.
Gårding density theorem The Gårding subspace, and therefore the smooth-vector space, is dense in every strongly continuous unitary representation of a Lie group.
Gårding subspace The subspace generated by applying compactly supported smooth convolution operators to vectors in a continuous Lie-group representation.
Half-spin representation Either irreducible chiral summand of the complex spin representation in even dimension.
Harish–Chandra admissibility theorem Every irreducible unitary representation of a real reductive group has finite multiplicity for each irreducible maximal-compact-subgroup type.
Harish–Chandra c-function The meromorphic coefficient governing spherical-function asymptotics and spherical Plancherel density.
Harish–Chandra isomorphism The theorem identifying the center of a semisimple enveloping algebra with Weyl-invariant polynomials on a Cartan subalgebra.
Harish–Chandra module An admissible (g,K)-module that is finitely generated over the universal enveloping algebra.
Harish–Chandra module of K-finite vectors The algebraic representation obtained by retaining the K-finite vectors of an admissible representation.
Harish–Chandra regularity theorem The theorem representing invariant eigendistributions by locally integrable functions analytic on regular semisimple elements.
Harish–Chandra's discrete-series criterion A connected semisimple real group with finite center has discrete series exactly when it has a compact Cartan subgroup.
Heisenberg group The basic nonabelian nilpotent Lie group, central extension of an abelian group.
Hermitian matrix model of Minkowski space Hermitian 2×2 matrices model Minkowski space, with determinant equal to the negative of the chosen (−+++) quadratic form.
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.
Identity component of a Lie group The connected component containing the identity; it is an open normal Lie subgroup with discrete quotient.
Indefinite pin group The Clifford group Pin(p,q) that double-covers the orthogonal group of a real quadratic space.
Indefinite spin group The even Clifford group Spin(p,q) associated with a nondegenerate real quadratic form of mixed signature.
Induced representation attached to a polarization The unitary representation obtained by inducing the character defined by a coadjoint functional from a polarizing subgroup.
Infinitesimal character An algebra character through which the center of an enveloping algebra acts on a representation.
Infinitesimal equivalence of admissible representations The relation in which two admissible representations have isomorphic maximal-compact-subgroup-finite modules.
Infinitesimal generator of a unitary representation The self-adjoint Stone generator associated with a one-parameter subgroup of a unitary Lie-group representation.
Inner derivation A derivation of the form ad_x(y) = [x,y].
Intersection of the three A1 + D6 subalgebras in e7 The three generation sl2 plus so12 subalgebras have common intersection sl6 plus the complexified generation plane; pairwise intersections already agree.
Intertwining operator between unitary representations A bounded linear operator commuting with the actions of a group on two Hilbert spaces.
Invariant closed subspace of a unitary representation A closed Hilbert subspace preserved by every operator in a unitary group representation.
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.
Irreducible unitary representation An irreducible unitary representation has no nonzero proper closed invariant subspace.
Iwasawa decomposition The canonical KAN factorization of a real reductive Lie group after choices of Cartan and positive restricted-root data.
K-finite vector A vector whose orbit under a compact subgroup spans a finite-dimensional subspace.
K-type An irreducible compact-group representation occurring in the restriction of a representation to K.
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 symmetric invariant bilinear form defined by the trace of the product of adjoint operators.
Killing form nondegeneracy criterion A finite-dimensional Lie algebra is semisimple iff its Killing form is nondegenerate.
Kirillov correspondence for nilpotent Lie groups The classification of irreducible unitary representations of a connected simply connected nilpotent Lie group by its coadjoint orbits.
Kleinian group A discrete subgroup of PSL(2,C), acting on hyperbolic three-space and its boundary.
Knapp–Stein intertwining operator A Knapp–Stein operator is the meromorphically continued and normalized standard integral intertwining two Weyl-related principal-series representations.
Langlands classification for real reductive groups Every irreducible admissible representation of a real reductive group is the unique quotient of a standard module.
Langlands decomposition of a parabolic subgroup A real parabolic subgroup decomposes into a reductive factor, a split abelian factor, and a nilpotent radical.
Langlands quotient The Langlands quotient is the unique irreducible quotient of a standard module with its inducing parameter in the chosen positive chamber.
Lattice in a Lie group A discrete subgroup whose homogeneous quotient has finite invariant measure.
Left Maurer–Cartan form The canonical Lie-algebra-valued one-form obtained by left-translating tangent vectors to the identity.
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.
Levi subalgebra A reductive subalgebra obtained from a subset of simple roots by retaining the full Cartan and the corresponding root spaces.
Lie algebra A vector space over a stated field with an alternating bilinear bracket satisfying 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 functor commutes with realification For a complex Lie group, taking the Lie algebra and forgetting complex scalars commute.
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.
Limit of discrete series representation A nonzero irreducible representation obtained by extending the discrete-series parameter construction to a singular dominant Harish-Chandra parameter.
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 compact subgroup of a real reductive group A compact subgroup maximal by inclusion, canonically associated up to conjugacy with a Cartan involution.
Maximal connected closed subgroup A proper connected closed subgroup maximal among connected closed subgroups.
Maximal Levi subalgebra A proper Levi subalgebra maximal among proper Levi subalgebras, obtained by deleting one simple root.
Maximal Lie subalgebra A proper Lie subalgebra not contained in any other proper Lie subalgebra of the ambient algebra.
Maximal torus theorem In a compact connected Lie group, maximal tori exist and are all conjugate.
Maximal-rank Lie subalgebra A subalgebra reductive in its ambient reductive Lie algebra and having the same rank as the ambient algebra.
Metaplectic group The connected double cover of the real symplectic group.
Metaplectic representation The canonical unitary representation of the metaplectic group on square-integrable functions.
Minimal parabolic subgroup A parabolic subgroup minimal under inclusion, constructed from a choice of positive restricted roots.
Multiplicity of an irreducible representation The number of copies of an irreducible representation in a semisimple decomposition.
Mutual centralizers in a Lie algebra Two subalgebras each equal to the full centralizer of the other in an ambient Lie algebra.
Nelson Laplacian The positive sum of squares of infinitesimal generators used to detect regular vectors in a unitary Lie-group representation.
Nelson's analytic vector theorem A symmetric operator with a dense set of analytic vectors is essentially self-adjoint.
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.
Noncocompact arithmetic Kleinian groups Every noncocompact arithmetic Kleinian group is commensurable up to conjugacy with a Bianchi group.
Normal Lie subgroup A Lie subgroup invariant under conjugation; infinitesimally, it corresponds to an ideal.
Normalized parabolic induction Parabolic induction with a half-modular correction that preserves unitarity on the unitary parameter axis.
Octonionic special linear group The Lie group generated by traceless octonionic two-by-two transformations, isomorphic to Spin(9,1).
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 method A program relating irreducible unitary representations of a Lie group to symplectic coadjoint orbits.
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 that does not arise by taking the bracket with a fixed Lie algebra element.
Parabolic isometry of hyperbolic three-space A nonidentity orientation-preserving isometry with exactly one ideal fixed point and no interior fixed point.
Parabolic subgroup of a real reductive group A subgroup whose Lie algebra contains a minimal parabolic subalgebra of a real reductive Lie algebra.
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.
Poincaré–Birkhoff–Witt theorem Ordered monomials in an ordered basis of a Lie algebra form a basis of its universal enveloping algebra.
Polarization at a coadjoint functional A maximal-dimensional Lie subalgebra on which the alternating form defined by a coadjoint functional vanishes.
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.
Principal series representation A representation of a real reductive group obtained by normalized induction from a minimal parabolic subgroup.
Product Lie group The Cartesian product of Lie groups, with componentwise multiplication, is again a Lie group.
Projective general linear Lie group The real or complex general linear Lie group modulo its closed central subgroup of scalar matrices.
Projective special linear Lie group The real or complex special linear Lie group modulo its finite scalar center.
Projective unitary representation A continuous group representation by unitary operators modulo scalar phases.
Proper action A smooth Lie group action is proper if the action graph map is proper; this guarantees good quotient behavior.
Proper orthochronous Lorentz group The identity component SO⁺(1,3) of the four-dimensional Lorentz group.
PSL(2,C) The center quotient of SL(2,C), viewed either as a complex Lie group or as a six-dimensional real Lie group.
PSL(2,C) action on hyperbolic three-space PSL(2,C) is the orientation-preserving isometry group of hyperbolic three-space.
PSL(2,C)–proper Lorentz isomorphism Quotienting the SL(2,C) spin cover gives PSL(2,C)ℝ≅SO⁺(1,3) as real Lie groups.
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.
Rank of a semisimple Lie algebra The dimension of a Cartan subalgebra, equivalently the rank of the associated root system.
Real rank of a reductive group The dimension of a maximal abelian subspace in the noncompact part of a Cartan decomposition.
Real reductive Lie group A real Lie group satisfying the reductivity and finiteness hypotheses used in Harish–Chandra theory.
Reductive Lie algebra A Lie algebra that decomposes as a direct sum of its center and a semisimple ideal.
Regular Lie subalgebra A Lie subalgebra normalized by a Cartan subalgebra of the ambient semisimple Lie algebra.
Relative Lie algebra cohomology Cohomology of a Lie algebra relative to a subalgebra, computed by equivariant alternating cochains on the quotient.
Removing a simple root Deleting a Dynkin-diagram vertex to obtain a full root subsystem, a regular semisimple subalgebra, and a Levi subalgebra.
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.
Restricted General Linear Group rGL(H) Invertible operators whose positive part differs from I by a Hilbert–Schmidt operator
Restricted root system The roots obtained from the adjoint action of a maximal abelian subspace of the noncompact Cartan summand.
Restricted spin group The full preimage Spin⁺(p,q) of the identity component SO⁺(p,q) under the indefinite spin covering.
Restricted Symplectic Group rSp(K) The implementable symplectic transformations in Shale's Fock representation
Restricted Weyl group The finite group obtained by normalizing a maximal split abelian subspace and quotienting by its centralizer.
Riemannian symmetric space of noncompact type A simply connected Riemannian symmetric space with nonpositive curvature and no Euclidean factor.
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 $\mathfrak{sl}_2$ -subalgebra The canonical rank-one subalgebra generated by the root spaces for a root and its negative.
Root lattice The integer span of the roots in a crystallographic root system.
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 subsystem A subset of a root system that is itself a root system in its real span.
Root system A finite set of vectors closed under reflections and satisfying integrality; the combinatorial data behind semisimple Lie theory.
Roots, Weyl group, and dominant weights The root system, Weyl action, positive chamber, and highest-weight classification attached to a maximal torus.
Schur orthogonality for compact Lie groups Matrix coefficients of distinct irreducible unitary representations are orthogonal in L²(G), with a sharp normalization.
Schur's lemma for unitary representations An irreducible complex unitary representation has only scalar bounded operators in its commutant.
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.
Simply laced root system A reduced root system whose Dynkin diagram has only single edges.
SL(2,C) as a real and complex Lie group The determinant-one complex matrix group has complex dimension three and underlying real dimension six.
SL(2,C) spin cover of the Lorentz group The Hermitian-matrix action gives a two-to-one covering SL(2,C)→SO⁺(1,3).
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.
Smooth vector of a Lie-group representation A Hilbert-space vector whose orbit map under a Lie-group representation is infinitely differentiable in norm.
Solvable Lie algebra A Lie algebra whose derived series eventually becomes zero; the Lie-algebra analogue of solvable groups.
Special block unitary group The block-diagonal subgroup of a special unitary group with unitary blocks and total determinant one.
Special linear group The determinant-one matrix group, with dimensions distinguished over the real and complex numbers.
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 SU(n): traceless skew-Hermitian matrices with the commutator bracket.
Spherical Plancherel theorem The spherical Fourier transform is a unitary decomposition of K-invariant L2-functions on a noncompact symmetric space.
Spherical principal series representation The spherical principal series is the normalized minimal-parabolic induction of a character of the split factor with trivial compact and nilpotent data.
Spherical representation An irreducible unitary representation of a reductive group containing a nonzero vector fixed by a maximal compact subgroup.
Spin group For n at least 2, the connected double cover Spin(n) of SO(n), constructed inside the even Clifford algebra.
Spin representation of Spin(9) The real 16-dimensional spin representation of Spin(9), realized on the octonionic plane.
Spin(8) triality The order-six outer symmetry of Spin(8) that permutes its vector and two half-spin representations.
Square-integrable representation modulo the center An irreducible unitary representation having a nonzero matrix coefficient that is square-integrable after quotienting by the center.
Stabilizer (isotropy subgroup) in a Lie group action For a smooth action , the stabilizer fixes a point and is a closed Lie subgroup.
Standard module of a real reductive group A standard module is a normalized parabolically induced representation with tempered Levi data and an inducing parameter in a chosen positive chamber.
Standard sl5 in e7 There is a unique sl5 subalgebra between a good Standard Model algebra and the standard sl6, and it is regular in e7.
Standard sl6 in e7 The centralizer of the generation sl3 in e7 is a distinguished sl6, and the two subalgebras are mutual centralizers.
Stone's theorem for one-parameter unitary groups Strongly continuous one-parameter unitary groups are exactly exponentials of self-adjoint operators.
Stone–von Neumann theorem Irreducible unitary representations of the real Heisenberg group with a fixed nontrivial central character are unitarily equivalent.
Strongly continuous unitary representation A unitary group representation whose orbit map at every vector is norm-continuous.
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 A subspace invariant under every operator in a Lie algebra representation.
Symplectic group The Lie group of linear transformations preserving a nondegenerate alternating form on .
Symplectic Group Sp(K) Bounded invertible real-linear maps preserving the symplectic form B
Symplectic Lie algebra The Lie algebra of the symplectic group, consisting of matrices satisfying XᵀJ + JX = 0.
Tensor product of representations The diagonal action on a tensor product, defined by tensoring group actions or by the Leibniz rule for Lie algebras.
The group Sp(n)Sp(1) The central quotient of the product of two compact symplectic groups acting on quaternionic Euclidean space.
The isomorphism Spin(6) ≅ SU(4) The low-rank isomorphism between the simply connected compact groups of types D3 and A3.
Thirty-dimensional generation module in e7 Each 30-root projection class in E7 is an sl6 module Λ²C⁶ plus Λ⁴C⁶ and restricts to one Standard Model generation without two neutrino singlets.
Three D6 subsystems in E7 Each of the three generation root lines has a 60-root orthogonal complement of type D6, yielding a copy of so12 in e7.
Three-generation decomposition of e7 After choosing generation root lines, e7 decomposes as sl6 plus a two-dimensional trivial module plus three 32-dimensional Standard Model generation modules.
Transitive Lie group action A smooth Lie group action is transitive when it has a single orbit, making the manifold a homogeneous space.
Transitivity on real Grassmannians The orthogonal and special orthogonal groups act transitively on real Grassmannians, with stabilizers determined by block orthogonal groups.
Type A root system The simply laced root system A_n formed by the differences e_i-e_j in the sum-zero hyperplane of R^{n+1}.
Type D root system The simply laced root system D_n consisting of the vectors ±e_i±e_j in R^n.
Type I locally compact group A locally compact group is type I when its full group C-algebra is a type I C-algebra.
Underlying real Lie algebra A complex Lie algebra regarded as a real Lie algebra by restricting scalars.
Underlying real Lie group The real Lie group obtained from a complex Lie group by forgetting its complex structure.
Uniform lattice A lattice with compact quotient; a nonuniform lattice has noncompact quotient.
Unitary dual of an abelian Lie group For a connected abelian Lie group, every irreducible unitary representation is a character and the dual is Euclidean-discrete.
Unitary equivalence of representations Two unitary representations are unitarily equivalent when a unitary operator intertwines their actions.
Unitary group $U(n)$ The compact Lie group of complex matrices preserving the standard Hermitian inner product.
Unitary Lie algebra The Lie algebra of the unitary group: 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.
Universal enveloping algebra action on smooth vectors The differentiated Lie-algebra action on smooth vectors extended uniquely to the complex universal enveloping algebra.
Universal enveloping algebra of a Lie algebra The associative algebra universally generated by a Lie algebra subject to its bracket relations.
Weak and strong continuity for unitary representations For a unitary representation, continuity of all matrix coefficients, of diagonal coefficients, and of all orbit maps are equivalent.
Weak Continuity of a Representation Continuity of matrix coefficients (π(g)x,y) in the group parameter g
Weakly continuous unitary representation A unitary representation whose matrix coefficients are continuous functions of the group element.
Weight lattice The lattice of vectors pairing integrally with every coroot.
Weight of a representation A functional occurring as a simultaneous eigenvalue for the action of a Cartan subalgebra.
Weight space The simultaneous eigenspace of a representation corresponding to a weight on an abelian subalgebra.
Weights in the dual Cartan Weights are elements of ; integrality conditions define weight lattices tied to maximal tori and characters.
Weights of the defining sl₃-representation The three multiplicity-one weights of the natural three-dimensional representation of sl3.
Weyl group The finite group N_G(T)/T associated with a maximal torus, or equivalently the reflection group of a semisimple root system.
Weyl’s theorem on complete reducibility Every finite-dimensional representation of a finite-dimensional semisimple Lie algebra in characteristic zero is completely reducible.

Linear algebra79 knowls

Alternating multilinear map A multilinear map that vanishes whenever two arguments agree.
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.
Block matrix A matrix partitioned into compatible rectangular submatrices.
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 The basis-independent determinant polynomial attached to a finite-dimensional 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.
Complexification The complex vector space obtained from a real vector space by extending scalars to the complex numbers.
Cross product in three-dimensional Euclidean space The oriented bilinear product of two real three-vectors.
Determinant A scalar invariant of a square matrix measuring volume scaling and invertibility.
Dual basis The coordinate linear functionals associated with a basis of a finite-dimensional vector space.
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.
Elementary row operation Swapping rows, rescaling one row by a nonzero scalar, or adding a multiple of another row.
Euclidean norm The norm induced by an inner product on a Euclidean space.
Euclidean space A finite-dimensional real inner product space.
Exterior power of a vector space The vector space representing alternating multilinear maps of a fixed degree.
Frobenius norm The Euclidean norm of the list of entries of a finite matrix.
Full-rank Euclidean lattice Integer linear combinations of a basis of a Euclidean space.
Fundamental domain of a Euclidean lattice A measurable set containing one representative of each lattice translation class.
Hadamard product Entrywise multiplication of vectors or matrices of the same shape.
Hermitian matrix A complex square matrix equal to its conjugate transpose.
Hilbert space A complete inner product space.
Hyperhermitian form A quaternionic Hermitian form, represented in a basis by a self-adjoint quaternionic matrix.
Index of an integer sublattice The quotient of the integer lattice by the image of a nonsingular integer matrix has order equal to its absolute determinant.
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.
Inverse matrix A square matrix whose product with a given matrix on either side is the identity.
Kronecker delta A two-index symbol equal to one for matching indices and zero otherwise.
Left inverse of a linear map A linear map L with LT equal to the identity on the domain of T.
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.
Linearized inverse for squared amplitudes A positive amplitude representation provides a linear inverse for signed target increments, with an exact quadratic remainder.
Matrix A rectangular array of numbers, symbols, or expressions arranged in rows and columns.
Matrix exponential The absolutely convergent power series sum A^n/n! for a square real or complex matrix.
Minimal polynomial Smallest-degree monic polynomial that annihilates a linear operator.
Minkowski vector space Four-dimensional real vector space with the Lorentzian quadratic form −t²+x²+y²+z².
Mixed discriminant The symmetric multilinear polarization of a determinant on self-adjoint matrices.
Moment pairing matrix A finite matrix of integrals pairing chosen weights with chosen correction profiles.
Moore determinant The real polynomial determinant of a hyperhermitian quaternionic matrix.
Moving frame for a family of subspaces A smoothly varying full-column-rank matrix whose columns span the subspace at each parameter.
Multilinear map A map of vector spaces that is linear in each argument separately.
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.
Orthogonal reflection across a hyperplane The orthogonal map x -> x - 2(n dot x)n for a unit normal n.
Orthogonality Condition that two vectors have inner product equal to zero.
Orthonormal basis A basis whose vectors have unit length and are pairwise orthogonal.
Outer product The rank-at-most-one matrix vw^T associated with two real vectors.
Positive realization by squared amplitudes An invertible linear combination of squared amplitudes realizes targets with positive inverse coordinates.
Positive semidefinite matrix A real symmetric or complex Hermitian matrix whose quadratic form is nonnegative.
Quadratic form A homogeneous degree-two scalar-valued function whose polarization is bilinear.
Quadratic map over the real numbers A map Q(v)=B(v,v) obtained from a symmetric bilinear map.
Quaternion division algebra The four-dimensional real associative division algebra generated by two anticommuting square roots of minus one.
Quaternionic vector space A quaternionic vector space is a left or right module over the quaternion division algebra, with the scalar side fixed as part of the convention.
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.
Realification of a complex vector space A complex vector space regarded as a real vector space by restriction of scalars.
Right inverse of a linear map A linear map R with TR equal to the identity on the target of T.
Semilinear map An additive map between vector spaces that twists scalar multiplication by a field homomorphism.
Signature of a symmetric bilinear form The numbers of negative and positive squares in a real symmetric bilinear form.
Surjective linear isometry A bijective linear map that preserves the norm exactly.
Symmetric matrix A square matrix equal to its transpose.
Symmetric positive-definite matrix A real symmetric matrix whose quadratic form is strictly positive on every nonzero vector.
Symplectic Form A nondegenerate skew-symmetric bilinear form B on a real vector space
Trace Sum of diagonal entries of a square matrix, invariant under change of basis.
Transpose of a matrix The matrix obtained by interchanging row and column indices.
Unit sphere of a normed space The set of vectors having norm exactly one.
Unit vector A vector of norm one and the normalization of a nonzero vector.
Vandermonde matrix The matrix of consecutive nonnegative integer powers of a list of nodes.
Vector space A set with addition and scalar multiplication satisfying the vector space axioms.

Mathematical physics46 knowls

Advanced and retarded Green operators Two-sided Green operators on compactly supported sections whose outputs have future- or past-causal support.
Cauchy problem for normally hyperbolic operators Compactly supported source and Cauchy data determine a unique smooth solution with finite propagation on a globally hyperbolic spacetime.
Cauchy problem for the Lorentzian Dirac operator Initial spinor data on a smooth spacelike Cauchy hypersurface determine a unique Lorentzian Dirac solution with causal propagation.
Clifford slash notation Notation for Clifford multiplication by a vector or metric-dual covector.
Compact hyperbolic surface A compact connected surface equipped with a complete Riemannian metric of constant curvature minus one.
Conformal coupling of a scalar field The scalar-curvature coupling that makes the massless wave equation conformally covariant.
Connection form of a normally hyperbolic operator Every normally hyperbolic operator is uniquely a connection wave operator plus a bundle endomorphism.
Creation and Annihilation Operators Operators adding/removing one symmetric tensor factor in bosonic Fock space
Dirac equation The relativistic first-order field equation for a spinor of prescribed mass.
Dirac–Klein–Gordon factorization The free Minkowski Dirac factors multiply to the scalar Klein–Gordon operator up to sign.
Duality Transform D (Segal) Unitary map identifying bosonic Fock space with Gaussian L₂(M,n)
d’Alembert operator The scalar normally hyperbolic operator obtained from a Lorentzian metric.
Existence of advanced and retarded Green operators A normally hyperbolic operator on a globally hyperbolic spacetime has unique future and past Green operators.
Fock–Cook Quantization The bosonic canonical-commutation-relation representation defined by field operators on symmetric Fock space.
Gamma matrices Matrices representing Clifford multiplication in a chosen basis of a spinor module.
Geometric quantization A method that constructs a quantum state space from prequantum line-bundle data and a polarization of a symplectic manifold.
Georgi–Glashow SU(5) embedding The Standard Model central quotient embedded block-diagonally in SU(5).
Groenewold–Van Hove theorem The obstruction to extending canonical quantization consistently to every polynomial observable on a finite-dimensional linear phase space.
Hyperbolic plane wave A generalized Laplace eigenfunction on the Poincaré disk obtained from a boundary Poisson kernel raised to a complex power.
Hyperbolic Poisson kernel on the disk The positive kernel comparing a point in the Poincaré disk with a point on its ideal boundary.
Klein–Gordon equation The relativistic field equation for a free massive scalar field.
Klein–Gordon operator The normally hyperbolic scalar operator obtained by adding mass and curvature coupling to the wave operator.
Laplace–Beltrami eigenfunction A nonzero function transformed into a scalar multiple of itself by the Laplace–Beltrami operator.
Mathematical physics Navigation for Lorentzian geometry, relativistic field equations, and their symmetry and spin structures.
Minkowski Dirac operator The flat Lorentzian first-order operator obtained by Clifford contraction of ordinary differentiation.
Minkowski spacetime The flat affine Lorentzian spacetime underlying special relativity.
Normally hyperbolic operator A second-order operator whose principal symbol is the negative Lorentzian inverse metric times the identity.
Poincaré algebra The semidirect-product Lie algebra of Lorentz transformations and spacetime translations.
Prequantization The geometric construction that represents all classical observables on sections of a line bundle before imposing a polarization.
Quantization map A map from a selected Poisson algebra of classical observables to operators on a quantum state space.
Right-handed neutrino gauge singlet A right-handed neutrino transforming in the trivial representation of the Standard Model internal gauge group.
Schrödinger representation The standard Weyl representation on L²(ℝⁿ) by translations and modulations.
Semiclassical measure A weak-star limit in phase space of quadratic observables evaluated on a bounded family of high-frequency states.
Single Particle Structure Σ(H) Segal's package (K,B) extracted from a complex Hilbert space H for CCR quantization
Standard Model exterior-algebra representation The 32-dimensional restriction of the SU(5) exterior-algebra representation to the Standard Model group.
Standard Model fermion generation One repeated family of quark and lepton multiplets with the same internal gauge quantum numbers.
Standard Model gauge group The compact internal-symmetry group of the Standard Model, with an important finite central-quotient convention.
Standard Model gauge group as an F4 stabilizer intersection The effective Standard Model group arises as the intersection of two stabilizers in compact F4 acting on the exceptional Jordan algebra.
Standard Model Lie algebra The real compact Lie algebra of Standard Model internal symmetries and its complexification.
Super-Poincaré algebra The Lie superalgebra obtained by adjoining Lorentz transformations to a supertranslation algebra.
Supertranslation algebra A two-step nilpotent Lie superalgebra whose odd spinors bracket into even spacetime translations.
Symmetric Fock Space S(H) The Hilbert direct sum of the symmetric tensor powers of a complex Hilbert space, with its vacuum vector.
Uniform mass lower bound for hyperbolic-surface eigenfunctions Every normalized Laplace eigenfunction on a compact hyperbolic surface has a uniformly positive amount of L2 mass in each fixed nonempty open set.
Wave equation The massless hyperbolic field equation determined by a Lorentzian metric.
Weak hypercharge The U(1) weight that combines with weak isospin to determine electric charge.
Weyl CCR Quantization A regular unitary representation of the Weyl relations for a real symplectic space.

Measure theory72 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.
Averaged second-moment matrix The average of the outer product of a square-integrable vector with itself.
Bochner integral The norm limit of integrals of simple Banach-valued functions approximating in integral norm.
Borel measure A measure defined on the Borel sigma-algebra of a topological space.
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.
Complete measure A measure for which every subset of a measurable null set is measurable.
Completeness of Lebesgue spaces Every Cauchy sequence in Lp converges in Lp for 1 <= p <= infinity.
Completion of a measure space The extension that makes all subsets of existing null sets measurable.
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.
Differentiation under an integral sign An integrable bound on parameter derivatives justifies passing a derivative through an integral.
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.
Hausdorff dimension The critical power at which Hausdorff measures become zero.
Hausdorff measure in a metric space An outer measure obtained from covers by sets of small diameter and a power cost.
Indicator function A function that equals 1 on a set and 0 outside it.
Integrable majorant for a family A single integrable function controlling every member of a parameterized family almost everywhere.
Interpolation between two Lebesgue norms The Lp norm at an intermediate reciprocal exponent is bounded by a geometric mean of endpoint norms.
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 L^p 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
Locally integrable function A measurable function integrable on every compact subset of its Euclidean open domain.
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 foundations of Lebesgue integration.
Minkowski inequality in Lp Triangle inequality for the Lp norm.
Mixed Lebesgue norm An iterated norm that measures space integrability first and time integrability second.
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.
Ordered integration simplex The region of ordered times in an interval, whose volume is the interval length to the nth power divided by n factorial.
Outer measure A monotone, countably subadditive set function defined on all subsets.
Parabolic Hausdorff measure Hausdorff measure calculated with parabolic space-time scaling.
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.
Product sigma-algebra The sigma-algebra generated by measurable rectangles in a product of measurable spaces.
Projection onto zero-mean functions The linear operation subtracting a function’s normalized average.
Pushforward measure The measure obtained by transporting a measure through a measurable map.
Radon–Nikodym Derivative The almost-everywhere unique density dν/dμ representing an absolutely continuous measure ν relative to μ.
Reference-subtracted integral An integral of a difference that may converge even when neither term is separately integrable.
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.
Sigma-algebra generated by a family The smallest sigma-algebra containing a specified family of subsets.
Sigma-finite measure A measure space covered by countably many measurable sets of finite measure.
Simple function A measurable function that takes only finitely many values.
Strongly measurable Banach-valued function A Banach-valued function approximable almost everywhere in norm by measurable simple functions.
Tonelli's theorem Interchange of integrals for nonnegative measurable functions on a product measure space.
Uniform integrability A uniform L1 bound together with uniformly vanishing large-value tails.
Zero-mean function An integrable function whose integral in the specified averaging variables vanishes.

Modules92 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 of modules that is linear in each variable; over a noncommutative ring, tensor products instead represent biadditive balanced maps.
Bimodule A module with commuting left and right actions by (possibly different) rings.
Category of finitely generated projective modules The fixed-ring category whose objects are finitely generated projective modules and whose morphisms are module homomorphisms.
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 lattice The integral dual Hom(L,Z) of a finite free abelian group and its role in dual root data.
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.
Exterior algebra The graded algebra universally generated by a module with every generator squaring to zero.
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.
Symmetric algebra The free commutative algebra generated by a module.
Tensor algebra The free unital associative algebra generated by a module.
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.

Nonassociative and Jordan algebras47 knowls

Alternative algebra A nonassociative algebra in which repeated adjacent factors associate.
Automorphism group of a Jordan algebra The group of invertible linear maps preserving a Jordan product.
Automorphism transitivity on Jordan frames The identity component of the automorphism group of a simple Euclidean Jordan algebra acts transitively on its ordered Jordan frames.
Complex-qubit Jordan algebra The four-dimensional Euclidean Jordan algebra of Hermitian two-by-two complex matrices.
Complex-qutrit Jordan algebra The nine-dimensional Euclidean Jordan algebra of Hermitian three-by-three complex matrices.
Complex-qutrit stabilizer in F4 The identity component of the F_4 stabilizer of an H_3(C) subalgebra is (SU(3) times SU(3))/Z_3.
Complex-subalgebra decomposition in the Albert algebra A framed H_3(C) subalgebra of H_3(O) is encoded by three compatible real 2-planes in the octonions.
Composition algebra A unital algebra with a nondegenerate quadratic norm that composes under multiplication.
Conjugated cross product on C³ The SU(3)-equivariant conjugate-bilinear cross product on complex three-space.
Derivation of a Jordan algebra A linear infinitesimal symmetry satisfying the Leibniz rule for the Jordan product.
Determinant of a two-by-two octonionic Hermitian matrix The quadratic determinant on the octonionic spin factor.
Euclidean Jordan algebra A finite-dimensional real Jordan algebra with a positive-definite inner product associative with the Jordan product.
Exceptional Jordan algebra The 27-dimensional Euclidean Jordan algebra of three-by-three Hermitian octonionic matrices, also called the Albert algebra.
F4 transitivity on compatible Jordan-subalgebra pairs Compact F_4 acts transitively on incident octonionic-spin-factor and complex-qutrit subalgebra pairs with complex-qubit intersection.
F4 transitivity on complex-qutrit subalgebras Compact F_4 acts transitively on the H_3(C) Jordan subalgebras of the Albert algebra.
Frame decomposition of Hermitian Jordan algebras The diagonal-line and off-diagonal-pair decomposition of H_3(K) determined by its standard Jordan frame.
Hermitian Jordan triple system A complex vector space with a symmetric outer, conjugate-linear middle triple product satisfying the Jordan triple identity.
Hermitian matrix Jordan algebra Self-adjoint matrices over a real normed division algebra, with the symmetrized matrix product in the degrees where the Jordan identity holds.
Hurwitz's theorem on normed division algebras The only finite-dimensional real normed division algebras are R, C, H, and O.
Ideal in a Jordan algebra A linear subspace stable under multiplication by every element of the ambient Jordan algebra.
Jordan algebra A commutative algebra satisfying the Jordan identity.
Jordan algebra homomorphism A linear map preserving the Jordan product; preservation of units is an additional condition.
Jordan frame A decomposition of the unit of a Euclidean Jordan algebra into pairwise orthogonal primitive idempotents.
Jordan idempotent An element e of a Jordan algebra satisfying e composed with e equals e.
Jordan subalgebra A linear subspace closed under the Jordan product, with unit preservation stated separately.
Nonassociative algebra A vector space with a bilinear multiplication, with no associative law assumed.
Nonassociative, division, and Jordan algebras Nonassociative, alternative, composition, octonion, and Jordan algebras.
Octonion algebra The eight-dimensional real alternative normed division algebra.
Octonion conjugation, norm, and inner product The canonical involution and Euclidean geometry carried by the octonions.
Octonionic spin factor The ten-dimensional Euclidean Jordan algebra of two-by-two Hermitian octonionic matrices.
Octonions as complex vectors The real-algebra model of the octonions on C ⊕ C³ with an SU(3)-equivariant product.
Orthogonal Jordan idempotents Jordan idempotents e and f are orthogonal when their Jordan product vanishes.
Peirce decomposition of a Jordan algebra The splitting of a Jordan algebra into the 0, one-half, and 1 eigenspaces of multiplication by an idempotent.
Peirce-one Jordan corner The Peirce 1-space of an idempotent, a unital Jordan subalgebra with that idempotent as its unit.
Power-associative algebra A possibly nonassociative algebra in which each subalgebra generated by one element is associative.
Primitive Jordan idempotent A nonzero Jordan idempotent admitting no decomposition into two nonzero orthogonal idempotents.
Real normed division algebra A finite-dimensional real unital algebra with a positive multiplicative Euclidean norm.
Simple Euclidean Jordan algebra A Euclidean Jordan algebra with no nonzero proper Jordan ideals.
Special and exceptional Jordan algebras The distinction between Jordan algebras realizable by symmetrizing associative multiplication and those that are not.
Spectral theorem for Euclidean Jordan algebras Every element of a Euclidean Jordan algebra is a real linear combination of a Jordan frame, with uniquely determined eigenvalues.
Spin(8) stabilizer of an Albert-algebra frame The pointwise stabilizer in compact F_4 of a labelled Albert-algebra Jordan frame is Spin(8).
Spin(9) stabilizer of an octonionic spin factor The stabilizer in compact F_4 of an H_2(O) corner of the Albert algebra is Spin(9).
Spin-factor Jordan algebra The rank-two Euclidean Jordan algebra built from a real inner-product space.
SU(3) stabilizer of a complex octonion subalgebra Fixing a unit imaginary octonion reduces the octonion automorphism group G₂ to SU(3).
Trace form of a Euclidean Jordan algebra The canonical positive-definite associative bilinear form obtained from the Jordan trace.
Tripotent in a Jordan triple system An element e of a Jordan triple system satisfying {e,e,e}=e, the triple-system analogue of an idempotent.
Unique octonionic spin-factor corner Every complex-qubit Jordan subalgebra of the Albert algebra lies in a unique octonionic spin-factor corner.

Noncommutative geometry56 knowls

Almost-commutative spectral triple A spectral triple obtained by multiplying the canonical spin geometry of a manifold by a finite spectral triple.
Analytic K-homology The two-periodic homology theory of a complex C*-algebra represented by stable homotopy classes of Fredholm modules.
Bounded transform of a spectral triple The bounded operator obtained from a spectral triple's Dirac operator by the function x divided by the square root of one plus x squared.
Canonical spectral triple of a compact spin manifold The spectral triple formed by smooth functions, square-integrable spinors, and the spin Dirac operator on a closed Riemannian spin manifold.
Chern character of a Fredholm module The periodic cyclic-cohomology class represented by trace cocycles constructed from a summable Fredholm module.
Commutative reconstruction theorem for spectral triples A sufficiently regular commutative spectral triple satisfying the geometric axioms is the spectral geometry of a compact smooth manifold.
Compact quantum metric space An order-unit space with a seminorm whose state-space metric induces the weak-star topology.
Connes spectral distance The extended distance between states obtained by maximizing their difference over the spectral Lipschitz unit ball.
Cyclic cohomology A cohomology theory of associative algebras built from multilinear cochains invariant under cyclic permutation.
Derivation δ(T) = [|D|, T] The closed commutator derivation with the absolute value of the Dirac operator in a spectral triple.
Dimension spectrum of a spectral triple The discrete set of possible poles of weighted spectral zeta functions associated with a regular spectral triple.
Dirac operator The spin-geometric first-order operator obtained by Clifford contraction of the spin connection.
Dirac-type operator A first-order differential operator whose principal symbol is Clifford multiplication.
Entire cyclic cohomology A Z/2-graded cyclic cohomology of Banach algebras whose infinite cochain sequences satisfy a factorial analytic-growth condition.
Even Fredholm module A Fredholm module on a graded Hilbert space whose representation is even and whose Fredholm operator is odd.
Even K-theory/K-homology index pairing The integer obtained by compressing the off-diagonal operator of an even Fredholm module by a K-zero projection.
Even spectral triple A spectral triple with a grading that commutes with the represented algebra and anticommutes with its self-adjoint operator.
Finitely summable spectral triple A spectral triple that is strictly p-summable for at least one finite exponent.
Finiteness and absolute-continuity axiom for a spectral triple The reconstruction axiom requiring the smooth spinor domain to be finite projective and its Hilbert product to arise from noncommutative integration.
First-order condition for a real spectral triple The double-commutator axiom requiring Dirac commutators to be linear for the opposite-algebra action.
Fredholm module A representation of a complex C*-algebra equipped with an operator whose commutators and local self-adjointness and involutivity defects are compact.
Hochschild chain complex The tensor chain complex of an associative algebra with coefficients in a bimodule and boundary given by adjacent multiplications.
Hochschild cycle A Hochschild chain annihilated by the Hochschild boundary.
Hochschild orientability axiom for a spectral triple The requirement that a Hochschild cycle be represented by the grading in even parity or the identity in odd parity.
Jaffe–Leśniewski–Osterwalder cocycle An entire cyclic cocycle constructed from heat kernels and Dirac commutators of a theta-summable spectral triple.
KO-dimension sign table for a real spectral triple The mod-eight rule assigning the commutation signs of the real structure with the Dirac operator and grading.
Local index formula in noncommutative geometry A residue formula representing the cyclic Chern character of suitable regular finitely summable spectral triples.
Metric dimension of a spectral triple The critical exponent at which powers of a spectral triple's regularized inverse Dirac operator become trace class.
Noncommutative Geometry Spectral triples, noncommutative metric geometry, cyclic theory, K-homology, and semifinite geometry.
Noncommutative integral of a spectral triple The Dixmier-trace functional that integrates represented algebra elements against a critical power of the inverse Dirac operator.
Nonunital spectral triple A nonunital spectral triple replaces global compact resolvent by compactness after multiplication by algebra elements.
Odd Fredholm module An ungraded Fredholm module representing an odd analytic K-homology class.
Odd K-theory/K-homology index pairing The Fredholm index of a unitary compressed by the positive projection of an odd Fredholm module.
Odd spectral triple An ungraded spectral triple representing the odd parity of analytic K-homology.
Order-zero condition for a real spectral triple The requirement that the represented algebra commute with the opposite-algebra action induced by the real structure.
p-summable Fredholm module A Fredholm module whose algebra commutators belong to the Schatten ideal of order p.
p-summable spectral triple A spectral triple whose regularized inverse Dirac operator belongs to a specified Schatten class.
Periodic cyclic cohomology The Z/2-graded stabilization of cyclic cohomology under Connes's degree-two periodicity operator.
Pre-C*-algebra A possibly incomplete normed involutive algebra satisfying the C*-identity.
Product of spectral triples The graded tensor product combines spectral triples using the Dirac operator D1 tensor 1 plus Gamma1 tensor D2.
Real spectral triple A spectral triple equipped with a KO-dimensional real structure satisfying the order-zero and first-order conditions.
Real structure on a spectral triple An antiunitary operator encoding charge conjugation, KO-dimensional signs, and the opposite-algebra action of a spectral triple.
Regular spectral triple A spectral triple whose algebra and Dirac commutators are smooth for every iterated commutator with the absolute Dirac operator.
Semifinite K-theory index pairing The pairing of a semifinite spectral triple with K-theory through a trace-valued Breuer index.
Semifinite local index formula A residue-cocycle formula computing trace-valued index pairings for regular summable spectral triples in semifinite von Neumann algebras.
Semifinite spectral triple A spectral triple represented in a semifinite von Neumann algebra with compactness measured by a semifinite trace.
Smooth domain of the Dirac derivation The algebra of bounded operators lying in the domain of every iterated commutator with the absolute Dirac operator.
Smooth subalgebra of a C*-algebra A dense Fréchet -subalgebra of a C-algebra whose finer topology records regularity and which is closed under holomorphic functional calculus.
Spectral flow The signed count of eigenvalues crossing zero along a continuous path of self-adjoint Fredholm operators.
Spectral Lipschitz seminorm The extended seminorm that measures an algebra element by the norm of its commutator with a spectral triple's Dirac operator.
Spectral triple An algebra represented on a Hilbert space together with a self-adjoint operator having compact resolvent and bounded algebra commutators.
Spectral zeta function of a spectral triple A Dirichlet-type trace function that records the spectral growth of the Dirac operator in a spectral triple.
Tau-summable semifinite spectral triple A semifinite spectral triple whose resolvent has finite noncommutative lp norm with respect to the chosen trace.
Theta-summable spectral triple A spectral triple whose Dirac heat operator has finite trace at every positive time.
Unbounded Kasparov module A Hilbert C-star module cycle with a regular self-adjoint operator whose commutators are bounded and whose resolvent is locally compact.
Unitary equivalence of spectral triples Two spectral triples are unitarily equivalent when a unitary identifies their algebra representations and Dirac operators.

Operator algebras275 knowls

*-automorphism A bijective star-homomorphism from a C-star algebra to itself.
*-homomorphism A complex-algebra homomorphism that preserves the involution.
*-isomorphism A star-isomorphism is a bijective star-homomorphism between C*-algebras.
Type I factor representation A factor representation whose generated von Neumann algebra is a type I factor.
Von Neumann algebra generated by a representation The bicommutant, equivalently the weak operator closure of the unital algebra, generated by the image of a C-star representation.
Abelian projection A projection whose compressed von Neumann algebra is commutative.
Adjointable operator on a Hilbert C*-module A module map between Hilbert C*-modules that admits an adjoint with respect to their module-valued inner products.
Amenability and equality of full and reduced group C*-algebras A locally compact group is amenable exactly when its canonical full-to-reduced group C*-algebra quotient is injective.
Amenable action on a C*-algebra A C*-dynamical action admitting an equivariant conditional mean after passage to the bidual.
Approximate identity in a C*-algebra A net of positive contractions that converges to an identity through multiplication on every algebra element.
Bicommutant The commutant of the commutant of a family of operators.
Bounded star-representation A representation of an involutive algebra by bounded operators on a Hilbert space.
Breuer–Fredholm operator An operator in a semifinite von Neumann algebra that is invertible modulo the ideal of trace-compact operators.
Brown–Green–Rieffel stabilization theorem For sigma-unital C-star algebras, strong Morita equivalence is equivalent to stable isomorphism.
C*-algebra A Banach -algebra whose norm and involution satisfy the C-identity.
C*-algebra by generators and relations A universal C*-algebra determined by specified generators, relations, and a uniformly bounded class of representations.
C*-algebra C_0(X) The C*-algebra of continuous complex-valued functions that vanish at infinity on a locally compact space.
C*-algebra of bounded operators The unital C*-algebra of all bounded linear operators on a Hilbert space.
C*-algebra of compact operators The norm-closed C*-algebra of compact operators on a Hilbert space.
C*-correspondence A right Hilbert C*-module equipped with a nondegenerate left action by adjointable operators.
C*-dynamical system A C*-algebra equipped with a point-norm continuous action of a locally compact group by automorphisms.
C*-identity The norm identity equating the squared norm of an element with the norm of its adjoint product.
C*-subalgebra A norm-closed adjoint-stable subalgebra of a C*-algebra.
C*-tensor norm A norm on the algebraic tensor product of two C-algebras whose completion is again a C-algebra.
C_0(X)-algebra A C*-algebra equipped with a central nondegenerate action of C_0(X).
C_0(X)-algebras and upper-semicontinuous C*-bundles C_0(X)-algebras are precisely section algebras of upper-semicontinuous C*-bundles over X.
Calkin algebra The quotient of the bounded operators on an infinite-dimensional Hilbert space by its compact operators.
Canonical conditional expectation on a reduced crossed product For a discrete-group action, the faithful conditional expectation that extracts the identity Fourier coefficient from the reduced crossed product.
Canonical map from full to reduced crossed product The surjective star-homomorphism obtained by completing the identity on the dense crossed-product convolution algebra in the reduced norm.
Canonical operator trace The basis-independent extended sum of diagonal matrix coefficients of a positive Hilbert-space operator.
Canonical quotient from full to reduced group C*-algebra The regular representation induces the canonical surjection from the full group C-algebra onto the reduced group C-algebra.
Cauchy–Schwarz inequality for a positive functional A positive functional on a C*-algebra induces a positive semidefinite form satisfying the scalar Cauchy–Schwarz inequality.
Cauchy–Schwarz inequality for Hilbert C*-modules The algebra-valued inner product of a Hilbert C-star-module satisfies a norm-valued Cauchy–Schwarz bound.
Center of a von Neumann algebra The commutative von Neumann subalgebra consisting of elements that commute with every element of the original algebra.
Central decomposition of a von Neumann algebra A direct-integral realization of a von Neumann algebra whose fibers are factors and whose center acts by scalar fields.
Central support of a projection The least central projection dominating a given projection in a von Neumann algebra.
Centralizer of a weight The centralizer is the fixed-point von Neumann subalgebra of the modular automorphism group of a faithful normal semifinite weight.
Character of a C*-algebra A nonzero multiplicative star-homomorphism from a C-star algebra to the complex numbers.
Character space of a commutative C*-algebra The locally compact space of nonzero characters of a commutative C-star algebra.
Closed two-sided ideal of a C*-algebra A norm-closed linear subspace of a C*-algebra stable under multiplication from both sides.
Commutant The algebra of bounded operators that commute with every operator in a specified set.
Commutative C*-algebra A C*-algebra in which every pair of elements commutes.
Commutative von Neumann algebra A von Neumann algebra whose elements commute, modeled by an algebra of essentially bounded functions.
Compact operator on a Hilbert C*-module An operator in the norm-closed span of rank-one operators between Hilbert C*-modules.
Comparison theorem for projections Two projections in a von Neumann algebra become subequivalent in opposite directions after a central decomposition.
Completely bounded map A linear map whose entrywise matrix amplifications have uniformly bounded operator norms.
Completely positive approximation property Approximation of the identity map of a C*-algebra in point-norm by completely positive contractions factoring through matrix algebras.
Completely positive contraction A completely positive linear map whose operator norm is at most one.
Completely positive map A linear map between C*-algebras whose every matrix amplification is positive.
Completely positive order-zero map A completely positive map that sends orthogonal positive elements to orthogonal positive elements.
Concrete C*-algebra A norm-closed operator algebra on a Hilbert space that is closed under adjoints.
Conditional expectation of C*-algebras A conditional expectation is a completely positive contractive bimodule retraction onto a C*-subalgebra.
Connes classification of type III factors The subdivision of type III factors into types III_0, III_lambda, and III_1 by the Connes S-invariant.
Connes cocycle derivative The Connes cocycle derivative is a unitary cocycle that compares the modular automorphism groups of two faithful normal semifinite weights.
Continuous C_0(X)-algebra A C_0(X)-algebra whose quotient-fiber norms are continuous functions of the base point.
Continuous core of a von Neumann algebra The semifinite crossed product of a von Neumann algebra by the modular automorphism group of a faithful normal semifinite weight.
Continuous field of C*-algebras A family of C*-algebras equipped with enough sections whose pointwise norms vary continuously.
Continuous functional calculus A normal element of a C*-algebra canonically determines a star-homomorphism from continuous functions on its spectrum.
Continuous-trace C*-algebra A C*-algebra whose irreducible-representation traces vary continuously and densely detect its positive elements.
Corona C*-algebra The quotient of the multiplier algebra of a C-star algebra by its canonical essential ideal.
Countably generated Hilbert C*-module A Hilbert C*-module generated densely by the right module span of a countable family.
Covariant representation of a C*-dynamical system A compatible pair consisting of a nondegenerate representation of the coefficient algebra and a strongly continuous unitary representation of the acting group.
Crossed-product convolution *-algebra C_c(G,A) The compactly supported A-valued convolution algebra associated with a C*-dynamical system.
Cuntz-Pimsner algebra The universal C*-algebra for representations of a correspondence satisfying Katsura's covariance relation.
Cyclic representation of a C*-algebra A representation of a C*-algebra is cyclic when the orbit of one vector has dense linear span in its Hilbert space.
Cyclic vector A vector whose orbit under a represented algebra has dense linear span in the representation space.
Cyclic-separating duality For a von Neumann algebra and its commutant, cyclicity for either algebra is equivalent to separation for the other.
Dimension function on projections An additive normal numerical invariant of Murray-von Neumann equivalence classes of projections in a factor.
Direct sum of C*-representations The representation acting coordinatewise on the Hilbert direct sum of a family of representation spaces.
Disjoint representations Representations of a C-star algebra having no nonzero unitarily equivalent subrepresentations.
Dixmier trace A singular trace on the Marcinkiewicz–Macaev ideal obtained by applying an invariant generalized limit to logarithmic singular-value means.
Dixmier–Douady class The degree-three integral cohomology class measuring the twisting of a continuous-trace C*-algebra over its spectrum.
Double centralizer of a C*-algebra A compatible pair of left and right multiplier maps on a C*-algebra.
Dual action on a crossed product The canonical action of the Pontryagin dual on a crossed product by a locally compact abelian group.
Dual action on a von Neumann crossed product The canonical action of the dual group on a von Neumann crossed product by an abelian group.
Elementary C*-algebra A C*-algebra isomorphic to the compact operators on a nonzero Hilbert space.
Entire analytic element for a one-parameter automorphism group An element whose orbit under a one-parameter automorphism group extends to an entire algebra-valued function.
Enveloping C*-algebra The universal C*-completion of an involutive algebra in the maximal norm detected by bounded representations.
Essential ideal of a C*-algebra A closed two-sided ideal that meets every nonzero closed two-sided ideal nontrivially.
Even and odd operators on a graded Hilbert space A homogeneous bounded operator on a graded Hilbert space is even when it preserves parity and odd when it reverses parity.
Exact C*-algebra A C-star algebra whose minimal tensor product preserves every short exact sequence of C-star algebras.
Extended positive cone of a von Neumann algebra A cone of possibly unbounded positive objects evaluated on normal positive functionals.
Extension of a nondegenerate homomorphism to multiplier algebras A nondegenerate homomorphism between C*-algebras extends uniquely to a unital strictly continuous homomorphism of their multiplier algebras.
Exterior tensor product of Hilbert C*-modules The Hilbert module obtained by tensoring two Hilbert modules over the minimal tensor product of their coefficient algebras.
Factorial representation A representation whose generated von Neumann algebra is a factor.
Faithful *-homomorphism A faithful star-homomorphism is an injective star-homomorphism between C*-algebras.
Faithful normal semifinite trace A tracial weight on a von Neumann algebra that is faithful, normal, and semifinite.
Faithful normal state A normal state that detects every nonzero positive element of a von Neumann algebra.
Faithful representation of a C*-algebra An injective representation of a C*-algebra by bounded operators on a Hilbert space.
Faithful state on a C*-algebra A state that is strictly positive on every nonzero positive element.
Faithful weight A weight whose zero set on the positive cone contains only the zero element.
Fell bundle A bundle over a group whose fibers multiply and take adjoints like the homogeneous pieces of a C*-algebra.
Fiber of a C_0(X)-algebra The quotient of a C_0(X)-algebra by the ideal of elements vanishing at a point.
Finite normal trace on a von Neumann algebra A normal tracial weight on a von Neumann algebra whose value at the identity is finite.
Finite projection A projection not Murray-von Neumann equivalent to any proper subprojection of itself.
Finite von Neumann algebra A von Neumann algebra whose identity is a finite projection.
Flow of weights The canonical flow induced by the dual action on the center of the continuous core of a von Neumann algebra.
Fock module of a C*-correspondence The Hilbert-module direct sum of all tensor powers of a C*-correspondence, including the coefficient algebra in degree zero.
Full crossed product The universal C-completion of the convolution algebra of a C-dynamical system.
Full group C*-algebra The full group C-algebra is the universal C-completion of the convolution algebra of a locally compact group.
Full hereditary C*-subalgebra A hereditary C*-subalgebra whose generated closed two-sided ideal is the whole ambient algebra.
Gelfand duality A contravariant equivalence between locally compact Hausdorff spaces and commutative C-star algebras.
Gelfand transform The map representing each element of a commutative C-star algebra as a function on its character space.
Gelfand–Naimark theorem Every abstract C*-algebra has a faithful concrete representation by bounded Hilbert-space operators.
Generated C*-subalgebra The smallest C-subalgebra of an ambient C-algebra containing a specified set.
Glimm's type I theorem Equivalent regularity criteria that characterize separable type I C*-algebras.
GNS construction The canonical construction of a cyclic Hilbert-space representation from a positive functional on a C*-algebra.
GNS construction for a weight The Hilbert-space representation obtained by completing the finite left ideal of a weight.
GNS uniqueness theorem Any two cyclic representations implementing the same positive functional are canonically unitarily equivalent.
Graded C*-algebra A C-star algebra decomposed into even and odd closed subspaces compatible with multiplication and involution.
Green imprimitivity theorem An induced dynamical system crossed by its ambient group is Morita equivalent to the original system crossed by the subgroup.
Group C*-algebras of a discrete group The full and reduced C*-completions of the complex group algebra of a discrete group.
Group von Neumann algebra The von Neumann algebra generated by the left regular representation of a locally compact group.
Hereditary C*-subalgebra A hereditary C*-subalgebra contains every positive ambient element dominated by one of its positive elements.
Hilbert C*-module A module over a C-star algebra that is complete for the norm induced by an algebra-valued inner product.
Hull–kernel topology The topology on the primitive ideals of a C*-algebra generated by containment of ideals.
Hyperfinite II₁ factor The separable type II₁ factor obtained as the strong closure of an increasing union of finite-dimensional algebras.
Imprimitivity bimodule A full Hilbert module whose compact operators are identified with a second C-star algebra by the left action.
Infinite projection A projection that is Murray–von Neumann equivalent to one of its proper subprojections.
Integrated form of a covariant representation The representation of a crossed-product convolution algebra obtained by integrating a covariant pair.
Internal tensor product of C*-correspondences The composite correspondence obtained by balancing two Hilbert modules, quotienting null vectors, and completing.
Involutive algebra A complex associative algebra equipped with a conjugate-linear involution reversing products.
Irreducible representation of a C*-algebra A nonzero representation of a C*-algebra is irreducible when it has no nontrivial closed invariant Hilbert subspaces.
K_0 of a C*-algebra The Grothendieck group built from stable equivalence classes of projections over a C*-algebra.
K_1 of a C*-algebra The stable homotopy group of unitary elements over a C*-algebra.
Kadison–Schwarz inequality A unital two-positive map bounds the product of an image by the image of the corresponding product.
Kasparov stabilization theorem A countably generated Hilbert C-star module is absorbed by the standard Hilbert module.
Kernel of a *-homomorphism The closed two-sided ideal sent to zero by a *-homomorphism.
Left ideal of a weight The elements whose positive square has finite value under a weight.
Liminal C*-algebra A C*-algebra whose every irreducible represented image is the algebra of compact operators.
Liminal C*-algebras are type I Every liminal C*-algebra is type I because each irreducible image equals the compact operators.
Linking algebra A block C-star algebra whose diagonal corners are Morita-equivalent algebras and whose off-diagonal corner is their imprimitivity bimodule.
Local lifting property A C*-algebraic property requiring completely positive quotient maps to lift on every finite-dimensional operator subsystem.
Local reflexivity for C*-algebras Finite-dimensional pieces of a C*-algebra's bidual can be approximated inside the algebra with asymptotically optimal matrix norm.
Lower semicontinuous trace An extended-valued tracial weight on a C-star algebra that is lower semicontinuous in the norm topology.
Marcinkiewicz–Macaev ideal The compact-operator ideal defined by logarithmic bounds on partial sums of singular values.
Matrix C*-algebra The C-star algebra of square matrices over a C-star algebra with its canonical operator norm.
Maximal C*-seminorm The maximal C*-seminorm of an involutive algebra is the supremum of the operator norms of an element over all bounded Hilbert-space representations.
Maximal C*-tensor product The universal C*-completion of an algebraic tensor product, defined by taking the supremum over representations with commuting ranges.
Measurable operator affiliated with a von Neumann algebra An affiliated closed operator whose domain contains an increasing family of almost-full projections with finite complements.
Minimal C*-tensor product The C-completion defined by representing two C-algebras faithfully on separate Hilbert spaces and tensoring those representations.
Minimal projection A nonzero projection in a von Neumann algebra that has no nonzero proper subprojection.
Modular automorphism group The modular automorphism group is the canonical one-parameter automorphism group associated with a faithful normal semifinite weight.
Modular conjugation Modular conjugation is the antiunitary involution in the polar decomposition of a Tomita operator.
Modular KMS condition The modular KMS condition is the analytic boundary relation that characterizes a faithful normal state's modular dynamics.
Modular operator The modular operator is the positive self-adjoint part of the polar decomposition of a Tomita operator.
Multiplier algebra The canonical unital C-algebra in which a C-algebra sits as an essential ideal.
Multiplier module The multiplier module of a Hilbert module consists of adjointable module maps from its coefficient algebra into the module.
Murray–von Neumann equivalence of projections The equivalence relation on projections implemented by partial isometries inside an operator algebra.
Murray–von Neumann subequivalence of projections The comparison relation saying that one projection is equivalent to a subprojection of another.
n-positive map A linear map between C*-algebras whose entrywise amplification to n-by-n matrices preserves positivity.
Natural positive cone The canonical self-dual cone obtained from the modular data of a cyclic separating representation.
Noncommutative L^p space Haagerup's weight-independent Lp space constructed from homogeneous measurable operators in a crossed product.
Nondegenerate *-homomorphism A C*-algebra homomorphism whose image acts densely on the target algebra or representation space.
Nondegenerate representation of a C*-algebra A representation whose image acts on a dense subspace of the whole representation Hilbert space.
Normal *-homomorphism A star homomorphism between von Neumann algebras that is continuous for their ultraweak topologies.
Normal completely positive map A completely positive map between von Neumann algebras that is ultraweakly continuous.
Normal conditional expectation An ultraweakly continuous conditional expectation from a von Neumann algebra onto a von Neumann subalgebra.
Normal functional A bounded linear functional on a von Neumann algebra that is continuous for the ultraweak topology.
Normal linear map A normal linear map between von Neumann algebras is a bounded linear map that is continuous for their ultraweak topologies.
Normal positive map A positive linear map between von Neumann algebras that is ultraweakly continuous.
Normal representation A representation of a von Neumann algebra on a Hilbert space that is ultraweakly continuous.
Normal semifinite faithful weight A weight on a von Neumann algebra that is simultaneously normal, semifinite, and faithful.
Normal state A state on a von Neumann algebra that belongs to its predual.
Normal unital completely positive map An ultraweakly continuous completely positive map that preserves the identity.
Normal weight A weight on a von Neumann algebra that preserves suprema of increasing positive nets.
Nuclear C*-algebra A C-algebra whose algebraic tensor product with every C-algebra has a unique C*-tensor norm.
Nuclear C*-algebras are exact Every nuclear C-star algebra is exact, although exact C-star algebras need not be nuclear.
Nuclearity of reduced group C*-algebras and amenability for discrete groups A discrete group is amenable exactly when its reduced group C*-algebra is nuclear.
Operator affiliated with a von Neumann algebra A closed densely defined operator whose domain and action are invariant under every unitary in the commutant.
Operator Algebras C*-algebras, von Neumann algebras, their maps, representations, states, weights, and structure theory.
Operator-valued weight An extended-positive-valued bimodular weight from a von Neumann algebra to a von Neumann subalgebra.
Opposite algebra The C*-algebra obtained by reversing multiplication while retaining the norm and involution.
Order on self-adjoint elements The partial order whose nonnegative elements are the positive elements of a C-star algebra.
Order-unit space A real ordered vector space with an Archimedean positive element that bounds every vector.
Positive and negative parts of a self-adjoint element The canonical orthogonal positive elements whose difference is a given self-adjoint element.
Positive cone of a C*-algebra The closed convex cone of positive self-adjoint elements in a C*-algebra.
Positive cone of the predual The positive cone of a von Neumann algebra predual consists of its normal positive functionals.
Positive element of a C*-algebra A positive element is a self-adjoint C*-algebra element whose spectrum is contained in the nonnegative real axis.
Positive linear functional A complex linear functional on a C*-algebra that is nonnegative on every positive element.
Positive linear map A linear map between C*-algebras that sends positive elements to positive elements.
Positive square root in a C*-algebra The unique positive element whose square is a prescribed positive element.
Predual of a von Neumann algebra The canonical Banach space whose continuous dual is a von Neumann algebra.
Primitive ideal of a C*-algebra A primitive ideal is the kernel of a nonzero irreducible representation of a C*-algebra.
Primitive ideal space The primitive ideal space is the set of primitive ideals equipped with the hull-kernel topology.
Projection in a C*-algebra A self-adjoint idempotent in a C*-algebra.
Properly infinite projection A projection that contains two orthogonal subprojections, each equivalent to the original projection.
Properly infinite von Neumann algebra A properly infinite von Neumann algebra is one whose identity is equivalent to two orthogonal subprojections of itself.
Pure state of a C*-algebra A state that is an extreme point of the C*-algebra's state space.
Pure states and irreducible GNS representations A state is pure exactly when its GNS representation is irreducible.
Quasi-equivalent representations Representations whose generated von Neumann algebras are normally isomorphic by an isomorphism agreeing on the represented C-star algebra.
Quasicentral approximate identity A quasicentral approximate identity for an ideal also asymptotically commutes with every element of the ambient C*-algebra.
Quotient C*-algebra The C-algebra obtained by dividing a C-algebra by a closed two-sided ideal.
Reduced crossed product The C*-completion of a dynamical system obtained from a faithful regular covariant representation.
Reduced group C*-algebra The operator-norm closure of the integrated left regular representation of a locally compact group.
Regular covariant representation The canonical covariant pair on L2(G,H) induced from a representation of the coefficient C*-algebra.
Regular operator on a Hilbert C*-module A closed densely defined Hilbert-module operator whose adjoint is densely defined and whose graph has the required complementability.
Representation of a C*-algebra A -homomorphism from a C-algebra to the bounded operators on a Hilbert space.
Rieffel induction The representation of one C-star algebra obtained by tensoring a correspondence with a representation of its coefficient algebra.
Sakai characterization theorem A C-algebra is a W-algebra exactly when it is isometrically a Banach dual space.
Section algebra of a C*-bundle The C*-algebra of continuous bundle sections that vanish at infinity.
Self-adjoint element of a C*-algebra An element of a C-star algebra that equals its involution.
Semifinite von Neumann algebra A von Neumann algebra in which every nonzero projection dominates a nonzero finite projection.
Semifinite weight A weight whose finite positive elements are order-dense in the positive cone.
Separating vector A vector on which no nonzero operator from the represented algebra vanishes.
Short exact sequence of C*-algebras An injective ideal inclusion followed by a surjective C*-quotient map with matching image and kernel.
Singular state on a von Neumann algebra A state on a von Neumann algebra that dominates no nonzero normal positive functional.
Spatial derivative Connes's positive self-adjoint Radon-Nikodym operator for a normal semifinite weight relative to a weight on the commutant.
Spectral characterization of affiliation Affiliation is equivalent to membership of the polar partial isometry and all absolute-value spectral projections in the von Neumann algebra.
Spectral mapping theorem Functional calculus carries the spectrum of an element to the image of that spectrum under the function.
Spectral permanence A C-star-subalgebra computes the same spectrum as its ambient C-star-algebra when units are treated consistently.
Standard form of a von Neumann algebra A faithful representation of a von Neumann algebra equipped with its canonical conjugation and self-dual positive cone.
Standard Hilbert C*-module The Hilbert C*-module of square-summable sequences with entries in a C-star algebra.
State on a C*-algebra A norm-one positive linear functional on a C*-algebra.
State space of a C*-algebra The convex set of all positive norm-one functionals on a C*-algebra.
States separate positive elements The theorem that states detect positivity and recover the norm of every positive element of a C-star algebra.
Stinespring dilation theorem Every completely positive map into bounded operators is a compression of a star-representation.
Strict topology on a multiplier algebra The strict topology records norm convergence after multiplication on either side by every algebra element.
Strictly positive element of a C*-algebra A strictly positive element is a positive element whose generated hereditary subalgebra is the whole C*-algebra.
Strong Morita equivalence of C*-algebras An equivalence of C*-algebras witnessed by a full imprimitivity bimodule.
Strong operator topology The topology of pointwise norm convergence for bounded operators on a Hilbert space.
Strong vs Weak Operator Topology The strong and weak operator topologies on the bounded operators of a Hilbert space.
Strong-* operator topology The operator topology of pointwise norm convergence for both operators and their adjoints.
Support projection of a normal positive functional The least projection on which a normal positive functional is concentrated.
Takai duality Crossing an abelian C*-dynamical system successively by an action and its dual recovers the original algebra up to stabilization.
Tau-compact operator An element of a semifinite von Neumann algebra lying in the norm-closed ideal generated by finite-trace projections.
Tau-measurable operator An affiliated closed operator whose domain is arbitrarily close to full in the measure determined by a semifinite trace.
Toeplitz representation of a C*-correspondence A compatible pair representing a C-correspondence and its coefficient algebra inside another C-algebra.
Toeplitz-Pimsner algebra The universal C-algebra generated by a Toeplitz representation of a C-correspondence.
Tomita operator The Tomita operator is the closed antilinear operator obtained from the adjoint operation relative to a cyclic separating vector.
Tomita–Takesaki theorem The Tomita–Takesaki theorem identifies the commutant through modular conjugation and proves invariance under modular evolution.
Trace on a C*-algebra A bounded positive linear functional on a C*-algebra that is invariant under cyclic permutation of two factors.
Tracial noncommutative L^p space The space of trace-measurable operators whose p-th absolute powers have finite trace.
Tracial state A normalized positive functional invariant under cyclic interchange of two factors.
Tracial weight An extended positive weight whose values are unchanged when the two factors in an operator square are reversed.
Transformation-group C*-algebra The crossed-product C*-algebra obtained from a continuous action of a locally compact group on a locally compact space.
Type decomposition of von Neumann algebras Every von Neumann algebra decomposes uniquely into central summands of types I, II, and III.
Type I composition series A transfinite ideal filtration whose successive quotients are continuous-trace C*-algebras.
Type I von Neumann algebra A von Neumann algebra in which every nonzero central summand contains a nonzero abelian projection.
Type I C*-algebra A type I C*-algebra is one whose every irreducible represented image contains all compact operators on its representation space.
Type I factor A von Neumann factor containing a nonzero abelian projection.
Type I stability under ideals and quotients Closed ideals and quotient algebras of a type I C*-algebra are again type I.
Type II von Neumann algebra A type II von Neumann algebra is semifinite and has no nonzero abelian projection.
Type III factor A von Neumann factor having no nonzero finite projection.
Type III von Neumann algebra A von Neumann algebra with no nonzero finite projections.
Type II₁ factor A finite type II von Neumann factor, equivalently a continuous factor with finite identity.
Type II∞ factor A type II von Neumann factor whose identity projection is infinite.
Type Iₙ factor A type I factor isomorphic to the algebra of complex n by n matrices for a positive integer n.
Type I∞ factor A type I-infinity factor is a type I von Neumann factor acting on an infinite-dimensional Hilbert space.
Ultrastrong operator topology The locally convex topology on bounded operators generated by square-sums of pointwise operator norms.
Ultrastrong-star topology The involution-compatible refinement of the ultrastrong topology on a von Neumann algebra.
Ultraweak operator topology The weak-star topology on the bounded operators induced by their trace-class predual.
Unique predual theorem The Banach-space predual of a von Neumann algebra is uniquely determined up to its canonical isometry.
Uniqueness of standard form A normal isomorphism between von Neumann algebras in standard form has a unique cone-preserving unitary implementation.
Unital *-homomorphism A star-homomorphism between unital C-star algebras that preserves the multiplicative identity.
Unital completely positive map A unital completely positive map preserves the identity and is positive at every matrix level.
Unital C*-algebra A C-star algebra containing a multiplicative identity.
Unitary element of a C*-algebra An element of a unital C*-algebra whose adjoint is its two-sided inverse.
Unitary equivalence of C*-representations The equivalence relation on Hilbert-space representations implemented by a unitary intertwining operator.
Unitary representations and representations of the full group C*-algebra Integrated form gives a natural correspondence between continuous unitary group representations and nondegenerate representations of the full group C*-algebra.
Unitization of a C*-algebra The canonical unital C-algebra formed by adjoining an identity to a nonunital C-algebra.
Universal C*-algebra A C*-algebra characterized by a universal mapping property for specified generators and operator relations.
Universal representation of a C*-algebra The faithful representation obtained by summing the GNS representations of all states.
Upper-semicontinuous C*-bundle A topological bundle of C*-algebras whose fiber norm is upper semicontinuous.
Vector functional on a von Neumann algebra A vector functional evaluates a concrete von Neumann algebra between two fixed Hilbert-space vectors.
Vector state A state obtained by evaluating a represented algebra against a unit vector.
Von Neumann algebra A unital self-adjoint algebra of bounded Hilbert-space operators that is closed in the weak operator topology.
Von Neumann bicommutant theorem A unital self-adjoint operator algebra has the same strong closure, weak closure, and bicommutant.
Von Neumann crossed product The von Neumann algebra generated by the standard covariant representation of an ultraweakly continuous group action.
Von Neumann factor A von Neumann algebra whose center consists only of scalar multiples of its identity.
Weak expectation property A unital C*-algebra has WEP when every faithful representation admits a weak expectation from all bounded operators into its bidual.
Weak operator topology The topology of pointwise convergence of all matrix coefficients on an algebra of Hilbert-space operators.
Weak Schatten ideal The operator ideal of compact operators whose singular values decay at the weak-lp rate.
Weight on a von Neumann algebra An extended nonnegative additive and positively homogeneous functional on the positive cone of a von Neumann algebra.
σ-unital C*-algebra A sigma-unital C*-algebra has an approximate identity indexed by the natural numbers.

Partial differential equations42 knowls

A harmonic global Sobolev distribution vanishes A harmonic distribution in an inhomogeneous Hs space on the whole Euclidean space is zero.
Backward uniqueness Coincidence at a later time determines equality at earlier times within a specified evolution class.
Cauchy problem for an evolution PDE An evolution equation together with prescribed initial data.
Classical solution of a PDE A function with the required classical derivatives that satisfies a PDE pointwise.
Conservation law in divergence form An evolution equation equating the change of a density plus the divergence of its flux to a source.
Convex integration A method using increasingly fine oscillations to realize a nonlinear differential constraint from relaxed data.
Difference estimate for a differential polynomial Subtracting products one factor at a time bounds the nonlinear change by the difference jet times a polynomial in the input jets.
Differential inclusion A derivative is required to belong to a prescribed set instead of equaling a single prescribed value.
Differential polynomial A polynomial expression in finitely many derivatives of an unknown, with prescribed coefficient functions.
Diffusion in a fixed direction The second directional derivative multiplied by a positive diffusivity.
Duhamel formula for the heat equation The forced heat equation is represented by evolving the initial data and integrating evolved source increments.
Euclidean heat kernel The normalized Gaussian that propagates the heat equation on Euclidean space.
Finite-time blowup in a specified norm Unbounded growth of a chosen solution norm as time approaches a finite endpoint.
First-order differential relation A subset of a first jet bundle that specifies allowed values and first derivatives.
Flat residual from asymptotic summation A realized field has an infinitely small residual when finite partial sums improve in order and the nonlinear comparison loses only fixed powers.
Formal and holonomic solutions A formal solution assigns allowed jets; a holonomic one is the actual jet of a section.
Harmonic distribution A distribution annihilated by the distributional Laplacian.
Heat equation Evolution whose time derivative is a positive constant times the spatial Laplacian.
Heat semigroup The family of Gaussian convolution operators with addition of times as its composition law.
Heat smoothing estimate Gaussian convolution gains spatial derivatives and improves integrability with explicit powers of time.
Initial datum for an evolution equation The prescribed state at the starting time, with a specified meaning of convergence to that state.
Linear partial differential operator A finite linear combination of partial derivatives with coefficient functions.
Local well-posedness of an evolution problem Local existence, uniqueness in a specified class, and continuous dependence on the data.
Localized characteristic primitive with an explicit defect Subtracting a transported total integral makes a primitive compactly supported and records the cutoff error exactly.
Maximal existence time in a solution class The endpoint beyond which a solution cannot be extended while remaining in its specified class.
Newtonian fundamental solution in three dimensions One over four pi times radius is a distributional fundamental solution of minus the Laplacian.
Newtonian potential in three dimensions Convolution with the fundamental solution gives a solution of a Poisson equation.
Nonuniqueness for an initial-value problem Two distinct solutions in the same stated class satisfy the same equation, data, and prescribed force.
Partial differential equation An equation relating an unknown function of several variables to its partial derivatives.
Partial differential equations PDE formulations, solution classes, evolution problems, and differential operators.
Partial regularity Regularity away from an exceptional set whose size is quantitatively controlled.
Poisson equation A Laplacian equation with a prescribed source.
Principal part of a differential operator The sum of the highest-order derivative terms in a linear differential operator.
Regularity criterion A specified additional condition that implies a stated regularity or continuation conclusion for a solution class.
Self-similar profile A fixed profile that determines an evolving field by dilation and amplitude rescaling.
Shifted characteristic primitive on a torus An integral along a constant torus drift inverts a transport derivative.
Similarity variables Rescaled space, time, and unknowns adapted to selected concentration exponents.
Solution of the heat equation by Gaussian convolution Convolving Schwartz initial data with the heat kernel gives a smooth solution attaining its initial value.
Solvability compatibility condition A necessary condition on equation data imposed by the requested solution class and boundary conditions.
Supremum growth obstructs continuous Sobolev continuation An embedding into bounded functions turns unbounded supremum norm into failure of continuous Hs extension.
Transport equation An equation prescribing the material derivative of a scalar field along a given velocity.
Weak formulation of a differential equation An equation expressed by integral or distributional identities against specified test functions.

Probability50 knowls

Bernoulli distribution The distribution on zero and one with a specified probability of one.
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 An integrable random variable characterized by its averages over events in a given 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
Covariance matrix The matrix of centered pairwise products of a random vector with finite second moment.
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.
Cumulative distribution function The function assigning to each real threshold the probability of a value at or below it.
Derivatives of the log moment generating function The first two derivatives of a scalar log moment generating function are the tilted mean and tilted variance.
Differential entropy The negative integral of a probability density times its logarithm, relative to Lebesgue measure.
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.
Exponential tilt A probability measure reweighted by an exponential and divided by its total mass.
Gaussian Measure on a Hilbert Space (Segal) An isonormal Gaussian process indexed by a real Hilbert space, realized on an auxiliary probability space.
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
Normal distribution The probability distribution on the real line with a Gaussian density determined by a mean and positive variance.
Pinsker's inequality An inequality bounding total variation distance by the square root of Kullback–Leibler divergence.
Probability density function A nonnegative measurable function whose integrals give probabilities relative to Lebesgue measure.
Probability mass function The point probabilities of a probability measure on a countable state space.
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.
Tame Function (Segal) A function on an infinite-dimensional Hilbert space depending on finitely many coordinates
Total variation distance A distance between two probability distributions defined by the largest possible difference they assign to the same event.
Uniform distribution The probability distribution with constant density on a specified finite interval.
Variance A measure of how spread out a random variable is around its mean.
Variance of a normalized exponential weight The variance of an exponential likelihood ratio is a ratio of moment generating functions and increases with a positive tilt parameter.
Weak law of large numbers Sample averages of iid variables converge in probability to the mean.

Quantum foundations21 knowls

Antiunitary operator A conjugate-linear surjective isometry of a complex Hilbert space.
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 normalized, weakly countably additive measure taking measurable sets to positive operators.
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.
Qubit A two-level quantum system with state Hilbert space C^2.
Qutrit A three-level quantum system with state Hilbert space C^3.
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.

Quantum statistical mechanics8 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 and an observable, the expectation value is the trace of their product.
Quantum Gibbs state For a finite quantum system, the Gibbs state is the normalized exponential of minus inverse temperature times the Hamiltonian.
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

Real analysis335 knowls

A sufficient criterion for smoothness at a cylindrical axis Smooth dependence on r squared with the correct component factors yields smooth Cartesian fields.
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 A map of the linearity and interval-additivity rules for Riemann and Riemann–Stieltjes integrals
Additivity of the Riemann integral over subintervals Splitting a Riemann integral at an interior point
Additivity of the Riemann–Stieltjes integral over subintervals Splitting a Riemann–Stieltjes integral at an interior point
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.
Anisotropic dilation A coordinate dilation with a separate scaling exponent in each direction.
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.
Asymptotic summation preserving divergence freedom Summing localized potentials before taking curl retains incompressibility and the asymptotic tail estimates.
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.
Borel extension of a prescribed smooth jet Arbitrary smooth coefficients can be realized as all normal derivatives of one smooth function.
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 range is a bounded subset of its metric space.
C^2 implies equal mixed partials A twice continuously differentiable real function has equal mixed second partial derivatives.
Cartesian coordinates Coordinates relative to a fixed orthonormal basis and an origin in Euclidean space.
Cartesian jet of a function The finite array of all Cartesian partial derivatives up to a specified order at a point.
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 principal value A symmetric limiting prescription for certain singular or non-absolutely convergent integrals.
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
Chain rule for iterated torus variables Integer matrix iterations amplify auxiliary directional derivatives by their action on the direction vector.
Chain rule in implicit anisotropic coordinates Explicit time and axial derivatives of a rescaled profile in an implicitly defined chart.
Change of variables formula A multivariable substitution rule involving the Jacobian determinant.
Circle reparametrization by a positive density Integrating a positive normalized periodic density gives an orientation-preserving circle diffeomorphism.
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.
Compact derivative seminorm A seminorm measuring all derivatives through a fixed order on a compact subset of an open domain.
Compactly supported function A function whose nonzero set has compact closure in its specified domain.
Compactly supported weighted radial primitive A weighted first-order radial equation has a compactly supported solution exactly when its weighted source integral vanishes.
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 The least-upper-bound, Cauchy, monotone-convergence, and nested-interval properties are equivalent completeness principles for the real numbers.
Completing the square Writing a quadratic polynomial as a shifted square plus a constant.
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.
Cosine function The even trigonometric function defined by its entire power series.
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
Cumulative weighted integral The integral of a weighted source up to a variable upper endpoint.
Curl of a three-dimensional vector field The oriented antisymmetric first derivatives of a vector field on a subset of R^3.
Cylindrical coordinates and their orthonormal frame The coordinates x=r cos(theta), y=r sin(theta), z=z and their moving unit vectors.
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 a function's difference quotient at an interior point.
Derivative estimates for scaled cutoffs Rescaling a smooth cutoff by a length δ costs one factor δ⁻¹ for each derivative.
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.
Derivatives of a cutoff composed with a controlled scale Scale derivatives compensate for large cutoff parameters on the transition region.
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 A map between Euclidean spaces is differentiable at a point when it has a first-order linear approximation there.
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 The derivative of a function along a line through a point 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
Divergence of a Euclidean vector field The trace of the Cartesian derivative of a vector field.
Divergence-free field A vector field whose divergence vanishes.
Divergent series A series whose partial sums do not converge to a finite limit.
Envelope-controlled amplitude A smooth coefficient whose derivatives obey prescribed pointwise spatial and temporal envelopes.
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 metric space, an equicontinuous, pointwise bounded family of real-valued functions is uniformly bounded.
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
Euclidean Laplacian The sum of the unmixed second Cartesian partial derivatives.
Evaluation along an auxiliary map Composition of a function of independent physical and periodic auxiliary variables with a prescribed auxiliary map.
Even and odd parity of a function The transformation rules f(-x)=f(x) and f(-x)=-f(x) on a reflection-invariant domain.
Every bounded sequence in R^k has a convergent subsequence A direct corollary form of the Bolzano–Weierstrass theorem
Exponential function The real or complex function exp(z) defined by the everywhere convergent series sum z^n/n!.
Field axioms Axioms for addition and multiplication in a field, as used for the real numbers.
Finite subcover lemma Every open cover of a compact subset of a metric space has a finite subcover.
Fixed point A point x satisfying T(x)=x for a self-map T
Flat exponential An exponential cutoff whose extension by zero has vanishing derivatives of every order.
Flat factors absorb polynomial derivative growth A flat factor makes a product smoothly zero-extendible when every derivative of the other factor grows at most polynomially.
Flat function along a set A smooth function whose derivatives of every order vanish on a specified set.
Flat logarithmic interval weight A smooth weight on a positive interval that decays faster than every power of the logarithmic boundary margin.
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.
Gamma average of an algebraic factor A normalized Gamma integral averaging a negative power of one plus a nonnegative parameter.
Gamma function on positive real arguments Euler’s convergent integral interpolates factorials on the positive real axis.
Gaussian integral The integral of exp minus a positive multiple of the square is the square root of pi divided by that multiple.
Geometric series A series with constant ratio, with sum a/(1-r) for |r|<1.
Global extrema A continuous real function on a compact set attains its maximum and minimum.
Global maximizer and global minimizer A point where a function attains its maximum or minimum value on its entire domain.
Gradient Vector of first partial derivatives of a scalar function
Greatest Lower Bound Theorem Every nonempty subset of the real numbers that is bounded below has a real infimum.
Hadamard division lemma in one normal variable Finite-order vanishing along a hyperplane factors as a power of its normal coordinate times a smooth function.
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 anisotropic similarity coordinates A smooth positive scale defined by an implicit equation balancing time distance and an axial coordinate.
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.
Inverse loss when two moment exponents coalesce A two-row moment matrix from translated positive bumps has inverse size of reciprocal order in the exponent separation.
Invertibility of moments on ordered bump supports Distinct power weights paired with positive bumps on ordered disjoint intervals give an invertible moment matrix.
Irrational number A real number that cannot be written as a ratio of two integers.
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
Japanese bracket The smooth positive weight (1 + norm-squared)^(1/2) used to combine local and asymptotic estimates.
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 supremum of the infima of the tails of an extended-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.
Limit superior of a function at an endpoint The infimum of the suprema over shrinking one-sided neighborhoods of the endpoint.
Linearity of the Riemann–Stieltjes integral The Riemann–Stieltjes integral is linear in both the integrand and the integrator when the relevant integrals exist.
Linearization of a nonlinear map The derivative at a reference point and the resulting first-order approximation.
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.
Localization through a vector potential Taking the curl after inserting a cutoff preserves divergence freedom and produces an explicit cutoff correction.
Logarithmic radial derivative The dilation derivative r partial_r, equal to differentiation with respect to log r.
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
Matching prescribed integral moments Choosing a function so a specified finite family of integral functionals takes prescribed values.
Matrix-valued tensor field in Euclidean coordinates A field of matrices representing rank-two tensors in a chosen Euclidean basis.
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 The Cauchy product of two convergent series converges to the product of their sums if one series converges absolutely.
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.
Mollifier A smooth nonnegative unit-mass kernel rescaled to approximate the identity.
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.
Multi-index Leibniz rule The formula for all partial derivatives of a product, with multi-index binomial coefficients.
Multi-index notation Compact notation for products, factorials, and iterated partial derivatives in several variables.
Multiple Riemann integral Riemann integration of a bounded function over a rectangular region in Euclidean space.
Natural logarithm The inverse of the real exponential on the positive real axis.
Newton–Leibniz formula For continuous f and an antiderivative F, the integral of f over [a,b] equals F(b)-F(a).
Nonnegative square root The unique nonnegative real number whose square is a prescribed nonnegative real number.
Normalized dyadic chart Coordinates divided by a fixed band scale so that a scale-localized region has order-one size.
Normalized primitive of an exponential flat factor Dividing an integral containing exp(-c/t²) by the same exponential leaves a smooth coefficient times a cubic power.
Norms of an anisotropically concentrating profile The amplitude and volume factors in the Lp norm of a rescaled fixed profile.
One-sided jet limits from uniform derivative bounds Bounds on one more time derivative give compatible endpoint limits for every spatial and temporal derivative.
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.
Periodic function A function unchanged by a specified nonzero translation.
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.
Positive powers of flat weighted profiles Two-sided flat-weight control and weighted derivative bounds give smooth zero extension of every fixed positive power.
Power series A real or complex series in powers of a variable about a chosen center.
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.
Preservation of exterior primitives by moment matching Equal exterior sources and equal accumulated integrals give equal exterior primitives.
Product bounds for weighted coefficients and amplitudes Leibniz estimates that add parameter orders and combine spatial envelope factors.
Product rule for derivatives The derivative of a product differentiates each factor once while retaining the other.
Projection removing one weighted moment Subtracting a fixed normalized bump removes an integral obstruction linearly.
Quadratic algebraic conjugate The other root of the irreducible rational quadratic polynomial satisfied by a quadratic irrational.
Quadratic irrational An irrational real root of a quadratic polynomial with integer coefficients.
Quadratic irrational lower bound Integer linear forms in a fixed quadratic irrational have a reciprocal polynomial lower bound.
Quotient rule Differentiation of f/g at a point where g is nonzero.
Radial averaging operator from the origin The regularized average Y inverse times the integral from zero to Y.
Radial Laplacian The scalar Euclidean Laplacian of a radial function has a dimension-dependent first-derivative term.
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
Real power of a positive number The definition x^a = exp(a log x) for x > 0 and real a.
Real-analytic function A function locally represented by a convergent real power series.
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.
Row divergence of a matrix field The vector obtained by differentiating a matrix field in its second index.
Scaling of weighted radial moments The exact amplitude and length factors in a weighted one-dimensional radial integral.
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.
Sine function The odd trigonometric function defined by its entire power series.
Smooth cutoff function A smooth function equal to one near a chosen set and supported in a specified open region.
Smooth dyadic partition of a positive scale A locally finite partition of unity with uniform derivative bounds after dilation.
Smooth extension from bounded endpoint jets Compatible derivative limits can be continued through an endpoint by realizing their normal jet on the other side.
Smooth extension of a function A smooth function on a larger domain that agrees with the given function on its original domain.
Smooth nonanalytic endpoint of a Gamma average All one-sided derivatives of the Gamma average exist at zero, although its Taylor series has zero radius of convergence.
Smooth zero extension across a flat boundary Locally uniform vanishing of every mixed derivative permits smooth extension by zero.
Space of continuous functions The set of all real-valued continuous functions on a given topological space.
Squared partition of unity A locally finite smooth family whose squares sum to one.
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.
Stokes streamfunction for an axisymmetric meridional field A scalar S representing u_r=-S_z/r and u_z=S_r/r in cylindrical coordinates.
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.
Sum of nonnegative extended real terms The supremum of finite partial sums, with infinity allowed as a term or total.
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.
The constant pi The least positive zero of the series-defined sine function fixes the standard angular scale.
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.
Tricomi U for positive real arguments A convergent Laplace integral defining a confluent hypergeometric function on a positive real domain.
Unbounded values approaching a point prevent continuous extension Growth along a converging space-time sequence contradicts continuity at its limit point.
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 between metric spaces 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.
Vector field on a Euclidean open set A vector-valued function assigning an ambient vector to each point of a Euclidean open set.
Vector potential for a divergence-free field A field A whose curl equals the given three-dimensional vector field.
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.
Weighted coefficient class A parameter-dependent smooth family controlled derivative by derivative by a spatial weight and specified scale powers.
Weighted radial moment An integral of a radial function against a specified power of radius.
Weighted shifted radial primitive Changing the radial coordinate and conjugating by a power gives a compact inverse modulo a transported cutoff defect.
Zero count for distinct real powers A nonzero combination of m distinct powers has at most m minus one distinct positive zeros.
Zero derivative implies constant If f' vanishes everywhere on an interval, the function is constant

Representation theory27 knowls

Central character of a representation The character by which the center of a group acts on a representation.
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.
Characters separate semisimple conjugacy classes Semisimple conjugacy classes in a connected complex reductive group are determined by all algebraic character values.
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.
Contragredient representation The inverse-transpose action on an algebraic dual, or on the smooth dual for a locally profinite group.
Group algebra The associative algebra k[G] whose basis is a group G and whose multiplication extends the group law bilinearly.
Group algebra of a lattice The Laurent monomial algebra with basis indexed by a lattice, serving as the coordinate ring of a complex torus.
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 ring The Grothendieck ring of finite-dimensional representations, with direct sum as addition and tensor product as 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 A nonzero intertwiner between irreducible representations is an isomorphism; equivariant endomorphisms form a division ring and, under standard hypotheses, are scalars.
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.

Rings130 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.
Boolean semifield The two-element idempotent semifield with OR as addition and AND as multiplication.
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.
Commutative semiring A semiring whose multiplication is 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 divisibility criterion implying that a polynomial is irreducible over the fraction field of a UFD.
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 nonzero commutative ring with identity is a field exactly when its only ideals are zero and the whole ring.
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 inverse function theorem A pointed tuple of formal series is compositionally invertible exactly when its linear coefficient matrix is invertible.
Formal power series ring The complete x-adic ring of one-variable formal coefficient sequences with Cauchy multiplication.
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 A left, right, or two-sided additive subgroup stable under the corresponding multiplication by ring elements.
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.
Idempotent semifield A semifield with idempotent addition.
Idempotent semifields and lattice-ordered groups Idempotent semifields are equivalent to lattice-ordered abelian groups after adjoining or removing the additive bottom element.
Idempotent semiring A semiring whose addition satisfies a+a=a.
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 square matrices over a ring, with entrywise addition and matrix 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.
Multivariable formal power series ring The complete augmentation-adic ring of formal series in finitely many variables.
Natural order of an idempotent semiring Idempotent addition defines a partial order in which addition is binary join.
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.
Order in a quaternion algebra A unital integral subring that is a full finitely generated module in a quaternion algebra over a number field.
Ordered semiring A semiring with a partial order compatible with addition and multiplication.
Parasemifield A semiring-like algebra whose multiplicative reduct is a group but which has no required additive zero.
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.
Quaternion algebra over a field A four-dimensional algebra generated by two anticommuting square roots.
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.
Ramification of a quaternion algebra at a real place The real scalar extension is Hamilton’s division algebra rather than a matrix algebra.
Reduced norm in a quaternion algebra The scalar-valued multiplicative quadratic form obtained by multiplying an element by its standard conjugate.
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 ring homomorphism that is right-cancellative among ring homomorphisms.
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.
Semifield A nontrivial commutative semiring in which every nonzero element is multiplicatively invertible.
Semiprime ideal An ideal containing no nonzero nilpotent ideal modulo it; in commutative rings, the same as a radical ideal.
Semiring An additive commutative monoid with a distributive, unital multiplication and absorbing zero.
Semiring homomorphism A map preserving zero, one, addition, and multiplication.
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.
Split quaternion algebra A quaternion algebra isomorphic to the two-by-two matrix algebra over its base field.
Standard conjugation of a quaternion algebra The canonical linear involution that reverses the order of multiplication.
Subring A subset of a ring that is itself a ring under the inherited operations.
Substitution of formal power series Composition f(g_1,...,g_n) defined adically when each substituted series has zero constant term.
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).
Tropical semifield The max-plus idempotent semifield on the extended real line.
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.

Supergeometry27 knowls

Batchelor theorem Every finite-dimensional smooth real supermanifold is noncanonically split.
Category of super vector spaces The symmetric monoidal category of Z/2-graded vector spaces and even linear maps.
Equivalence of Lie supergroups and super Harish–Chandra pairs Finite-dimensional smooth real Lie supergroups are equivalent to finite-dimensional smooth real super Harish–Chandra pairs.
Functor of points of a supermanifold The contravariant functor sending each test supermanifold to its family of points in a given supermanifold.
Koszul sign rule The sign convention in which exchanging homogeneous factors of parities p and q contributes (-1)^(pq).
Lie superalgebra A Z/2-graded vector space with a graded-skew bracket satisfying the super Jacobi identity.
Lie superalgebra of a Lie supergroup The tangent superspace at the identity of a Lie supergroup, with bracket induced by invariant vector fields.
Lie supergroup A group object in the category of finite-dimensional smooth real supermanifolds.
Parity shift The functor that interchanges the even and odd parts of a super vector space.
Representation of a Lie superalgebra An even Lie-superalgebra morphism from a Lie superalgebra to the super-endomorphisms of a super vector space.
Split supermanifold A supermanifold globally presented by the exterior algebra of the dual of a vector bundle.
Super Harish–Chandra pair A Lie group and Lie superalgebra with compatible adjoint data encoding a smooth Lie supergroup.
Super internal Hom The super vector space whose even and odd parts are the parity-preserving and parity-reversing linear maps.
Super PBW theorem The associated graded algebra of a Lie superalgebra's enveloping algebra is its symmetric superalgebra.
Super vector space A vector space decomposed into even and odd subspaces.
Super-Minkowski space The affine superspace of spacetime and spinor coordinates equipped with the Lie supergroup structure integrating supertranslations.
Superalgebra An associative algebra whose multiplication preserves Z/2-degree.
Supercommutative algebra A superalgebra whose homogeneous elements commute with the Koszul sign.
Supercommutator The graded commutator ab - (-1)^(|a||b|)ba in an associative superalgebra.
Superdomain The local smooth model of dimension p|q with ordinary base U and exterior-algebra-valued smooth functions.
Supergeometry Super linear algebra, smooth real supermanifolds, Lie supergroups, and their mathematical-physics enrichments.
Supermanifold A locally superringed space locally isomorphic to a finite-dimensional smooth real superdomain.
Supermodule A graded module over a superalgebra with degree-preserving action.
Superspace A locally superringed space, the ambient geometric object from which supermanifolds are selected by a local model condition.
Supertranslation distribution The odd left-invariant distribution on super-Minkowski space whose Levi bracket is the supertranslation bracket.
Symmetric algebra of a super vector space The free supercommutative algebra generated by a super vector space.
Universal enveloping algebra of a Lie superalgebra The associative superalgebra universally realizing a Lie superalgebra bracket as a supercommutator.

Topological algebra3 knowls

Adic ring A topological ring whose topology is defined by the powers of a finitely generated ideal.
Completed tensor product The completion of a tensor product for the topology induced by specified filtrations.
Ideal of definition An ideal whose powers form a neighborhood basis of zero in a linearly topologized ring.

Topology138 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.
Common torus for iterated integer coverings Several levels of an iterated torus map can be represented on the least active level.
Compact exhaustion of an open Euclidean set An increasing sequence of compact subsets whose interiors cover the open domain.
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 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 subsets of its domain to compact subsets of its codomain.
Continuous map A function whose preimage of every open set is open.
Contractible space A space whose identity map is homotopic to a constant map.
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.
Covering space A space locally homeomorphic to a fixed base through a continuous surjection whose fibers are evenly covered.
Cup product and cohomology ring The natural graded multiplication on singular cohomology induced by multiplying cochains.
Curve A continuous map from an interval of real numbers into a space.
CW complex A space built by attaching cells inductively and equipped with the closure-finite and weak topology conditions.
Deck transformation A homeomorphism of a covering space that preserves its projection.
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.
Euclidean annulus The region between two concentric Euclidean spheres.
Extreme value theorem A continuous real-valued function on a nonempty 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
Flat torus A Euclidean space modulo a full-rank lattice, with its quotient distance and local Euclidean coordinates.
Fundamental class The top-dimensional homology class determined by an orientation of a closed manifold.
Fundamental group The group of based loops in a space modulo endpoint-preserving homotopy.
Hausdorff distance The maximum discrepancy between two nonempty compact subsets of a metric space.
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 A continuous map on a product with the unit interval that interpolates between two maps.
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.
Integer-matrix torus covering A nonsingular integer matrix induces a finite covering of the unit torus.
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.
Local lifts of cells under a torus covering A sufficiently small cell has finitely many disjoint lifted copies, labeled by an integer-lattice quotient.
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.
Locally finite family A family of subsets for which every point has a neighborhood meeting only finitely many members.
Locally Lipschitz map A map satisfying a Lipschitz bound on some neighborhood of each point.
Locally profinite group A locally compact totally disconnected Hausdorff group, equivalently one with compact-open subgroups forming an identity basis.
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 bounded closed intervals in the real line has nonempty intersection.
Nowhere dense set A set whose closure has empty interior
One-point compactification A compact space obtained by adjoining one point whose neighborhoods have compact complements.
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 The open subsets of a metric space satisfy the axioms of a topology.
Parabolic metric on space-time A space-time metric in which a time separation has the size of its square root.
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.
Polish space A separable topological space whose topology is induced by a complete metric.
Positive margin around a compact subset of an open set A compact subset of a Euclidean open set has a uniform positive distance from the complement.
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.
Reduced cohomology The cohomology theory whose degree-zero constant contribution is removed from ordinary singular cohomology.
Refinement of an open cover A cover that is finer than another, with each set contained in a member of the original cover.
Relative singular cohomology group The cohomology of the cochain complex dual to the relative singular chain complex of a pair of spaces.
Relative singular homology Homology of the quotient singular chain complex of a pair of spaces.
Relatively compact set A subset whose closure is compact in the ambient space.
Residual set A set whose complement is meager
Second-countable space A topological space whose topology has a countable basis.
Separable space A topological space containing a countable dense subset.
Separated centers under finitely many torus covers Finitely many centers can be chosen to avoid a fixed finite collection of iterated covering coincidences.
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.
Simply connected space A path-connected topological space with trivial fundamental group.
Singular chain complex The chain complex generated by singular simplices in a topological space.
Singular cohomology group The cohomology of the cochain complex dual to the singular chain complex of a topological space.
Singular homology group The homology of the singular chain complex of a topological space.
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.
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.
Totally disconnected space A topological space whose connected subsets have at most one point.
T₀ space A topological space in which some open set distinguishes each pair of distinct points.
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.
Universal cover A connected simply connected covering space that covers a given base space.

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Langlands's letter4 knowls

Historical L-group and Satake-parameter bridge The letter's bundled passage from the L-group to the semisimple conjugacy class encoding unramified local data.
Langlands Letter A historical annotated reading of Langlands' 1967 letter to Weil, cross-linked to modern Langlands vocabulary.
Langlands Letter Definitions Mathematical definitions for the Langlands letter
Langlands' Letter to Weil The 1967 letter that launched the Langlands program

Long-form documents2 knowls

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

Shale's paper11 knowls

C*-Algebra of Field Observables 𝔄 The observable algebra is the operator-norm closure of the algebra generated by the Weyl unitaries.
Decomposition K = M ⊕ M A choice of real subspace M giving coordinates (x,y) for phase space K
Field Automorphism θ(T) from a Symplectic Map The *-automorphism induced by sending Weyl operators V(z) to V(Tz)
Hilbert–Schmidt operator in the Shale paper 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
Projective unitary representation in the Shale paper A group action by unitaries defined only up to phase (unitary rays)
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 Invertible elementary, trace-class, and Hilbert–Schmidt perturbations of the identity.
State, Pure State, Regular State (CCR context) Positive normalized functionals, with purity and CCR-regularity conditions

Source expansion guides12 knowls

Baez E7 and exceptional Jordan-algebra paper expansion A permanent index of the Lie-theoretic, Standard Model, octonionic, and Jordan-algebra knowls added from two 2026 papers.
Bianchi groups: new knowl collection An index of 39 new knowls connecting imaginary quadratic arithmetic and hyperbolic three-orbifolds.
Charts, bundle cocycles, and metrics additions New knowls added while annotating a conversation about charts, principal-bundle cocycles, gauge transformations, and Riemannian metrics.
Fractal uncertainty and quantum chaos knowl batch Dependency-ordered index of atomic knowls supporting Alex Cohen's higher-dimensional fractal uncertainty paper.
Galois Extensions and Bundles — knowlified transcript A full conversation transcript with expandable definitions for its mathematical terminology.
July 2026 operator algebra and geometry additions Definitions added in the July 2026 expansion of operator algebras, noncompact representation theory, noncommutative geometry, and differential geometry.
July 30 mathematical knowl expansion Review index for 372 new and 64 substantively refined production knowls in the July 30 mathematical expansion, organized by durable subject and dependency order.
Navier–Stokes: new knowl collection The 433 knowls from batches B00–B14 plus one document-integration addition, grouped by batch and subject.
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.
Quaternionic plurisubharmonicity and convexity expansion A dependency-ordered index for harmonic, quaternionic, HKT, convex-valuation, and octonionic plurisubharmonic concepts.
Representation-theoretic quantization additions The new knowls added while annotating a discussion of symplectic symmetries, quantization maps, and infinitesimal unitary representations.

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