Connections
Every cross-unit synthesis claim is registered here with a stable ID. Connections type a relationship (dual / equivalence / foundation-of / bridging-theorem / generalisation / specialisation / analogy / limit / recurrence) and rate its strength (load-bearing or synthesis).
452
total
115
load-bearing
337
synthesis
9
types
452 shown
- canonical sheaf dual to Serre duality$\omega_X$ is the dualising sheaf appearing in the duality.
- Riemann-Roch (curves) equivalent to Riemann-Roch (RS)Riemann-Roch for compact Riemann surfaces — the analytic version, equivalent via Serre's GAGA.
- theorem bridging scheme and Riemann-Roch (curves)The algebraic-geometric setting; smooth projective curves are 1-dimensional smooth proper schemes.
- theorem bridging Serre duality and Hodge decompositionSerre duality is the algebraic Hodge symmetry $H^{p, q} \cong H^{n - p, n - q \vee}$.
- theorem bridging Serre duality and Kodaira vanishingSerre duality dualises Kodaira vanishing into the Akizuki-Nakano statement.
- theorem bridging sheaf cohomology and Riemann-Roch (curves)Direct application of sheaf cohomology of line bundles on a curve, computing $\dim H^0(C; \mathcal{L}) - \dim H^1(C; \mathcal{L})$.
- sheaf cohomology equivalent to Riemann-Roch (RS)Riemann-Roch for compact Riemann surfaces — analytic version, equivalent to the algebraic statement via GAGA.
- theorem bridging Chern-Weil and Atiyah-Singer indexThe topological side of the index formula is written in characteristic classes obtained through Chern-Weil theory.
- theorem bridging Pontryagin/Chern and Atiyah-Singer indexCharacteristic classes are the topological side of the formula.
- theorem bridging Pontryagin/Chern and Bott periodicityChern classes and Chern characters mediate between vector bundles and cohomology in complex $K$-theory.
- theorem bridging holomorphic function and Riemann-Roch (RS)Riemann-Roch for compact Riemann surfaces — the dimension formula for spaces of holomorphic sections of line bundles, the analytic version of algebraic Riemann-Roch.
- Riemann surface equivalent to Riemann-Roch (RS)Riemann-Roch for compact Riemann surfaces — the dimension formula on compact Riemann surfaces, equivalent to algebraic Riemann-Roch.
- theorem bridging exterior derivative and StokesRelates $d$ to integration via $\int_M d\omega = \int_{\partial M} \omega$.
- theorem bridging Yang-Mills and Atiyah-Singer indexYang-Mills action — elliptic deformation complexes in gauge theory use index formulas to compute expected moduli dimensions.
- theorem bridging Stable homotopy and Homotopy and homotopy groupStable homotopy depends on homotopy theory, suspension, and spheres. It connects to classifying spaces through stable classical groups and to topological K-theory through spectra and Bott periodicity.
- theorem bridging Stable homotopy and SpectrumStable homotopy depends on homotopy theory, suspension, and spheres. It connects to classifying spaces through stable classical groups and to topological K-theory through spectra and Bott periodicity.
- theorem bridging Stable homotopy and SuspensionStable homotopy depends on homotopy theory, suspension, and spheres. It connects to classifying spaces through stable classical groups and to topological K-theory through spectra and Bott periodicity.
- theorem bridging Bott periodicity and Atiyah-Singer indexBott periodicity is part of the topological index construction.
- theorem bridging Classifying space and Homotopy and homotopy groupClassifying spaces depend on principal bundles and homotopy theory. They feed topological K-theory by organizing stable vector bundles through $BU$ and $BO$.
- theorem bridging K-theory and Classifying spaceClassifying spaces depend on principal bundles and homotopy theory. They feed topological K-theory by organizing stable vector bundles through $BU$ and $BO$.
- theorem bridging CFT basics and Atiyah-Singer indexAnomalies in two-dimensional field theories are organized by index-theoretic constructions.
- theorem bridging character and Frobenius reciprocityThe character form $\langle \chi_{\mathrm{Ind}}, \chi_V \rangle_G = \langle \chi_W, \chi_V|_H \rangle_H$ reduces multiplicity computation to subgroup arithmetic.
- theorem bridging character and MaschkeCharacters are well-defined invariants because of complete reducibility — every representation has a character determined by its irreducible decomposition.
- theorem bridging Character orthogonality and MaschkeComplete reducibility ensures every representation has a unique decomposition into orthogonal isotypic components.
- theorem bridging group rep and SchurSchur's lemma is the foundational result organising the structure of representations.
- theorem bridging regular rep and Frobenius reciprocityThe regular-representation decomposition is a direct application of Frobenius reciprocity to $\mathrm{Ind}_{\{e\}}^G \mathbb{C}$.
- theorem bridging regular rep and MaschkeThe regular representation $\mathbb{C}G$ decomposes via Maschke into irreducible isotypic components, giving the dimension formula $\sum (\dim V)^2 = |G|$.
- theorem bridging Specht module and Young diagram(symmetric group representation) is the parent unit; Young diagrams are its combinatorial backbone, and each irreducible $V^\lambda$ is built from a tableau of shape $\lambda$. (Specht module) gives the modular construction of $V^\lambda$ via polytabloids — explicit basis vectors built from tableaux.
- theorem bridging symmetric group rep and Specht module(symmetric group representation) is the parent unit; Specht modules are the *characteristic-flexible* realisation of the irreducibles classified by Frobenius-Young. (Young diagrams) provides the combinatorial scaffolding — every Specht module has an explicit basis indexed by standard Young tableaux.
- theorem bridging symmetric group rep and Young diagram(symmetric group representation) is the parent unit; Young diagrams are its combinatorial backbone, and each irreducible $V^\lambda$ is built from a tableau of shape $\lambda$. (Specht module) gives the modular construction of $V^\lambda$ via polytabloids — explicit basis vectors built from tableaux.
- theorem bridging Clifford algebra and Bott periodicityThe eight-fold real / two-fold complex Clifford periodicities ($\mathrm{Cl}_{p, q+8} \cong \mathrm{Cl}_{p,q} \otimes M_{16}(\mathbb{R})$ and $\mathrm{Cl}_{2k} \cong M_{2^k}(\mathbb{C})$) are not coincidences but the algebraic shadow of Bott periodicity in $KO$- and $K$-theory.
- theorem bridging ample line bundle and Riemann-Roch (curves)Riemann-Roch and Castelnuovo's bound classify very ample bundles on curves by degree relative to genus.appears in:
04.05.05 - theorem bridging Blowup and Riemann-Roch (curves)Strict transforms of curves under blowup change their genus and degree predictably; key in classification of surfaces.appears in:
04.07.02 - theorem bridging canonical sheaf and Riemann-Roch (curves)Uses $\omega_C$ via $\dim H^0(\omega \otimes \mathcal{L}^{-1})$ as the correction term.appears in:
04.08.02 - theorem bridging Cartier divisor and Riemann-Roch (curves)Riemann-Roch is naturally stated in terms of (effective) Cartier divisors.appears in:
04.05.04 - theorem bridging coherent sheaf and Riemann-Roch (curves)Riemann-Roch is a calculation of coherent Euler characteristics.appears in:
04.06.02 - theorem bridging Kodaira vanishing and Riemann-Roch (curves)Kodaira vanishing on curves recovers $H^1(\mathcal{O}(D)) = 0$ for $\deg D > 2g - 2$.appears in:
04.09.02 - theorem bridging Kodaira vanishing and sheaf cohomologyKodaira vanishing is a fundamental theorem about sheaf cohomology vanishing.appears in:
04.09.02 - theorem bridging Moduli of curves and Riemann-Roch (curves)The dimension count $3g - 3$ comes via deformation theory and Riemann-Roch.appears in:
04.10.01 - morphism (schemes) dual to affine schemeAffine morphisms correspond to ring maps via $\mathrm{Spec}/\Gamma$ anti-equivalence.appears in:
04.02.04 - theorem bridging morphism (schemes) and Riemann-Roch (curves)Riemann-Roch is a statement about morphisms to a point and properties of the corresponding pushforward.appears in:
04.02.04 - theorem bridging Picard group and Riemann-Roch (curves)Riemann-Roch is naturally a statement about $\mathrm{Pic}(C)$ for a curve.appears in:
04.05.02 - theorem bridging projective scheme and Riemann-Roch (curves)Most natural application: smooth projective curves are projective schemes of dimension 1.appears in:
04.02.03 - theorem bridging projective space and Riemann-Roch (curves)Projective curves embed in $\mathbb{P}^n$ via line bundles; Riemann-Roch is computed by reduction to projective embedding.appears in:
04.07.01 - theorem bridging Serre duality and Riemann-Roch (curves)Serre duality bridges the Euler-characteristic and classical $\ell - \ell$ forms of Riemann-Roch.appears in:
04.08.03 - theorem bridging Serre duality and Riemann-Roch (RS)Riemann-Roch for compact Riemann surfaces — analytic Serre duality, via Hodge theory and the Dolbeault resolution.appears in:
04.08.03 - theorem bridging sheaf of differentials and Riemann-Roch (curves)The canonical divisor $K_C$ is the divisor class of $\omega_C = \Omega^1_C$.appears in:
04.08.01 - sheaf of differentials dual to Serre dualityUses the canonical sheaf $\omega_X$ as the dualising sheaf.appears in:
04.08.01 - theorem bridging Weil divisor and Riemann-Roch (curves)The classical statement is in terms of Weil divisors on curves.appears in:
04.05.01 - theorem bridging complex vector bundle and Bott periodicityComplex K-theory has period 2; the algebraic shadow is the $\mathrm{Cl}_n$ classification's complex 2-periodicity.appears in:
03.05.08 - Riemann surface equivalent to schemeSmooth projective complex algebraic curves are compact Riemann surfaces (GAGA).appears in:
06.03.01 - theorem bridging elliptic operators and Atiyah-Singer indexThe index theorem applies to elliptic operators through their symbol class.appears in:
03.09.09 - theorem bridging symbol and Atiyah-Singer indexThe symbol defines the topological K-theory class whose index is computed.appears in:
03.09.07 - theorem bridging compact operators and Hilbert spaceWhere the spectral theorem and Hilbert-Schmidt / trace-class refinements are sharpest.appears in:
02.11.05 - theorem bridging compact operators and Atiyah-Singer indexIndex theory of elliptic operators uses compact resolvents on Sobolev pairs.appears in:
02.11.05 - theorem bridging Stable homotopy and Atiyah-Singer indexThe Atiyah-Singer index theorem uses K-theoretic stable information in the construction of the topological index.appears in:
03.08.06 - theorem bridging Stable homotopy and Bott periodicityStable homotopy depends on homotopy theory, suspension, and spheres. It connects to classifying spaces through stable classical groups and to topological K-theory through spectra and Bott periodicity.appears in:
03.08.06 - theorem bridging Stable homotopy and Classifying spaceStable homotopy depends on homotopy theory, suspension, and spheres. It connects to classifying spaces through stable classical groups and to topological K-theory through spectra and Bott periodicity.appears in:
03.08.06 - theorem bridging Stable homotopy and K-theoryStable homotopy depends on homotopy theory, suspension, and spheres. It connects to classifying spaces through stable classical groups and to topological K-theory through spectra and Bott periodicity.appears in:
03.08.06 - theorem bridging Classifying space and principal bundleClassifying spaces depend on principal bundles and homotopy theory. They feed topological K-theory by organizing stable vector bundles through $BU$ and $BO$.appears in:
03.08.04 - theorem bridging Classifying space and Bott periodicityClassifying spaces depend on principal bundles and homotopy theory. They feed topological K-theory by organizing stable vector bundles through $BU$ and $BO$.appears in:
03.08.04 - theorem bridging K-theory and vector bundleTopological K-theory depends on vector bundles and classifying spaces. It is the input for Bott periodicity, which turns the suspension-defined groups into a periodic theory.appears in:
03.08.01 - theorem bridging K-theory and Atiyah-Singer indexTopological K-theory depends on vector bundles and classifying spaces. It is the input for Bott periodicity, which turns the suspension-defined groups into a periodic theory.appears in:
03.08.01 - theorem bridging K-theory and Bott periodicityTopological K-theory depends on vector bundles and classifying spaces. It is the input for Bott periodicity, which turns the suspension-defined groups into a periodic theory.appears in:
03.08.01 - Lie algebra equivalent to spin groupThe Lie algebra $\mathfrak{spin}(n)$ is isomorphic to $\mathfrak{so}(n)$, and rotations live as exponentials of skew-symmetric matrices.appears in:
03.04.01 - theorem bridging Virasoro and Atiyah-Singer indexThe central-charge parameter also connects to anomalies in quantum field theory and to index-theoretic anomaly calculations, which later meet the Atiyah-Singer theorem.appears in:
03.11.03 - character equivalent to group repCharacters are an invariant of the underlying representation; the structural results reduce representation isomorphism to character equality.appears in:
07.01.03 - theorem bridging Maschke and group repMaschke is the structural theorem organising all of finite-group representation theory.appears in:
07.02.01 - theorem bridging Specht module and group rep(symmetric group representation) is the parent unit; Specht modules are the *characteristic-flexible* realisation of the irreducibles classified by Frobenius-Young. (Young diagrams) provides the combinatorial scaffolding — every Specht module has an explicit basis indexed by standard Young tableaux.appears in:
07.05.03 - theorem bridging Specht module and Schur(symmetric group representation) is the parent unit; Specht modules are the *characteristic-flexible* realisation of the irreducibles classified by Frobenius-Young. (Young diagrams) provides the combinatorial scaffolding — every Specht module has an explicit basis indexed by standard Young tableaux.appears in:
07.05.03 - theorem bridging Specht module and Verma module(symmetric group representation) is the parent unit; Specht modules are the *characteristic-flexible* realisation of the irreducibles classified by Frobenius-Young. (Young diagrams) provides the combinatorial scaffolding — every Specht module has an explicit basis indexed by standard Young tableaux.appears in:
07.05.03 - theorem bridging symmetric group rep and Hodge decompositionGives the underlying notion of group representation; the symmetric-group case is the quintessential finite example. (Schur's lemma) is invoked at the irreducibility step in the Young symmetriser proof.appears in:
07.05.01 - theorem bridging symmetric group rep and Cartan-Weyl classificationGives the underlying notion of group representation; the symmetric-group case is the quintessential finite example. (Schur's lemma) is invoked at the irreducibility step in the Young symmetriser proof.appears in:
07.05.01 - theorem bridging symmetric group rep and group repGives the underlying notion of group representation; the symmetric-group case is the quintessential finite example. (Schur's lemma) is invoked at the irreducibility step in the Young symmetriser proof.appears in:
07.05.01 - theorem bridging symmetric group rep and highest weight repGives the underlying notion of group representation; the symmetric-group case is the quintessential finite example. (Schur's lemma) is invoked at the irreducibility step in the Young symmetriser proof.appears in:
07.05.01 - theorem bridging symmetric group rep and SchurGives the underlying notion of group representation; the symmetric-group case is the quintessential finite example. (Schur's lemma) is invoked at the irreducibility step in the Young symmetriser proof.appears in:
07.05.01 - theorem bridging Clifford algebra and Atiyah-Singer indexThe Â-genus on the topological side of the index formula for $\not{\!\partial}$ is built from Pontryagin classes via the splitting principle; the analytic side is a symbol calculation in which the principal symbol is Clifford multiplication. The index theorem itself routes through the Clifford-algebraic universal property of this unit.appears in:
03.09.02 - theorem bridging spinor bundle and Atiyah-Singer indexComputes the index of the chiral Dirac operator using spinor-bundle data.appears in:
03.09.05 - theorem bridging spin structure and Atiyah-Singer indexThe Â-genus side of the index formula for the Dirac operator integrates Pontryagin classes against the spin manifold's tangent bundle; spin is exactly what makes the analytic side definable.appears in:
03.09.04 - theorem bridging de Rham cohomology and Atiyah-Singer indexCohomological index formulas pair characteristic classes with fundamental classes.appears in:
03.04.06 - coherent sheaf built on affine schemeCoherent sheaves on Noetherian $\mathrm{Spec}(R)$ correspond to finitely-presented $R$-modules.
- projective scheme built on affine schemeProjective schemes are covered by affine schemes; both are core scheme constructions.
- quasi-coherent sheaf built on affine schemeQuasi-coherent sheaves on $\mathrm{Spec}(R)$ correspond bijectively to $R$-modules.
- Cartier divisor generalises ample line bundleAmpleness extends to Cartier divisors and $\mathbb{Q}$-Cartier divisors in birational geometry.
- coherent sheaf built on ample line bundleThe Cartan-Serre-Grothendieck criterion is a coherent-sheaf property.
- projective space built on ample line bundle$\mathcal{O}(d)$ on $\mathbb{P}^n$ is the canonical example of an ample (very ample for $d \geq 1$) bundle.
- Hodge decomposition built on canonical sheaf$\omega_X = \Omega^n$ is the top component, contributing $H^{n, q}$ to the Hodge decomposition.
- Kodaira vanishing built on canonical sheafVanishing $H^i(X; \omega_X \otimes L) = 0$ for $L$ ample, $i > 0$.
- Moduli of curves built on canonical sheafThe canonical line bundle on $\overline{\mathcal{M}_g}$ relates to tautological classes $\kappa, \lambda$.
- morphism (schemes) built on Direct and inverse image of sheavesDirect/inverse image is foundational data of a morphism.
- Kodaira vanishing built on Hodge decompositionThe proof uses harmonic-form representatives and the Hodge-theoretic framework.
- ample line bundle built on line bundle (scheme)Ampleness is a property of line bundles; the ample cone sits inside $\mathrm{Pic}(X) \otimes \mathbb{R}$.
- Cartier divisor built on line bundle (scheme)Cartier divisors and line bundles correspond bijectively.
- coherent sheaf built on line bundle (scheme)Line bundles are coherent sheaves with locally-free-rank-1 structure.
- quasi-coherent sheaf built on line bundle (scheme)Line bundles are rank-1 locally free quasi-coherent sheaves.
- Geometric invariant theory built on Moduli of curvesMumford's original GIT application; constructs $\mathcal{M}_g$ as a quasi-projective scheme.
- ample line bundle built on Picard groupThe ample cone is the natural positivity structure inside $\mathrm{Pic}(X) \otimes \mathbb{R}$.
- Cartier divisor built on Picard groupDefined as Cartier divisors modulo principal divisors (or equivalently as line bundle isomorphism classes).
- line bundle (scheme) built on Picard group$\mathrm{Pic}(X)$ is by definition the abelian group of line bundles up to isomorphism.
- projective space built on Picard group$\mathrm{Pic}(\mathbb{P}^n_k) = \mathbb{Z}$ is the canonical foundational example.
- projective space built on projective scheme$\mathbb{P}^n_k = \mathrm{Proj}\,k[x_0, \ldots, x_n]$ is the universal example.
- quasi-coherent sheaf built on projective schemeQuasi-coherent sheaf and Coherent sheaf — coherent sheaves on $\mathrm{Proj}(S)$ correspond to graded $S$-modules modulo torsion.
- coherent sheaf built on quasi-coherent sheafCoherent sheaves are quasi-coherent + finiteness conditions.
- scheme built on SheafSchemes are locally ringed spaces; their structure sheaf is a sheaf of rings.
- Riemann surface built on SheafSheaves of holomorphic functions, meromorphic functions, and differentials on a Riemann surface.
- canonical sheaf built on sheaf of differentials$\omega_X = \det \Omega^1_X$, the top exterior power.
- Hodge decomposition built on sheaf of differentialsRelies on the spectral sequence $H^q(\Omega^p) \Rightarrow H^{p+q}_{\mathrm{dR}}$.
- Direct and inverse image of sheaves built on sheafificationThe inverse-image presheaf is generally not a sheaf; $f^{-1}\mathcal{G}$ is its sheafification.
- direct/inverse image built on stalkThe inverse image $f^{-1}\mathcal{G}$ at a point $x$ has stalk $\mathcal{G}_{f(x)}$; this is the key stalk-functoriality property.
- sheafification built on Stalk of a sheafSheafification preserves stalks: $\mathcal{F}_x = (\mathcal{F}^+)_x$ for any presheaf $\mathcal{F}$.
- Cartier divisor built on Weil divisorWeil and Cartier divisors agree on locally factorial schemes; differ in general.
- line bundle (scheme) built on Weil divisorWeil divisors and line bundles agree via $\mathrm{Pic} = \mathrm{Cl}$ on locally factorial schemes.
- Picard group built on Weil divisorDivisor class group of locally factorial schemes equals the Picard group; in general there is an injection $\mathrm{Pic}(X) \hookrightarrow \mathrm{Cl}(X)$ for normal $X$.
- Ideal in an algebra built on associative algebraAssociative algebra supplies the multiplication that ideals must absorb.
- group action built on groupGroup actions build on groups and feed principal bundles, where each fiber is a right torsor for the structure group. Lie groups act smoothly on manifolds, producing homogeneous spaces and associated bundles.
- orthogonal group built on groupOrthogonal groups depend on groups, bilinear forms, and Lie groups. They act on orthonormal frames, giving the structure group of the orthogonal frame bundle.
- Quotient algebra built on tensor algebraQuotient algebra follows tensor algebra in the Clifford chain. Clifford algebra is the quotient of a tensor algebra by the quadratic-form ideal from.
- tensor algebra built on tensor productTensor algebra depends on vector spaces and the tensor product of vector spaces. It feeds directly into quotient algebra, where relations are imposed by two-sided ideals.
- Chern-Weil built on complex vector bundleChern classes have de Rham representatives via curvature of a Hermitian connection.
- vector bundle built on principal bundleFrame bundles are principal bundles, and vector bundles are associated bundles.
- Chern-Weil built on principal bundleChern-Weil theory is a construction on principal bundles before it becomes a theory of vector bundles.
- Principal bundle with connection built on principal bundleA connection is additional differential data on a principal bundle.
- Yang-Mills built on Chern-WeilGauge theory uses the same curvature forms, though with metric-dependent action functionals rather than topological cohomology classes.
- Chern-Weil built on Pontryagin/ChernSupplies curvature representatives for the real de Rham images of Chern and Pontryagin classes.
- Dirac operator built on Pontryagin/ChernThe Â-class and Chern character in the index formula are built from these.
- Pontryagin/Chern built on Stiefel-WhitneyStiefel-Whitney classes — mod-2 companions for real bundles; $w_2$ obstructs spin structures.
- spin structure built on Stiefel-WhitneyStiefel-Whitney classes — the precise cohomological obstruction $w_2$ to spin existence; relatedly $w_1$ for orientability and $w_1^2 + w_2$ for Pin⁻.
- holomorphic line bundle (RS) built on divisor (RS)Supplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.
- Jacobian variety built on divisor (RS)Supplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.
- Riemann surface generalises holomorphic functionThe structural data of a Riemann surface is exactly the system of holomorphic functions on each chart with holomorphic transition maps.
- Jacobian variety built on holomorphic line bundle (RS)Supplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.
- Chern-Weil built on Principal bundle with connectionInvariant polynomials evaluated on curvature produce de Rham cohomology classes.
- Principal bundle with connection built on connection (vector bundle)Vector-bundle connections are induced from principal connections on frame or associated bundles.
- Variational calculus on manifolds built on integration on manifoldsAction functionals integrate Lagrangian densities over manifolds.
- elliptic operators built on symbolEllipticity is invertibility of the principal symbol away from the zero section.
- compact operators built on Banach spaceCompactness in infinite-dimensional Banach spaces controls Fredholm perturbations.
- Fredholm operators built on Banach spaceFredholm theory is a Banach-space theory before it becomes an elliptic-operator theory.
- Hilbert space built on Banach spaceHilbert spaces are Banach spaces whose norm comes from an inner product.
- Banach space built on bounded operatorsOperator norms and continuity are formulated between Banach spaces.
- compact operators built on bounded operators$\mathcal{K}(X) \subseteq \mathcal{B}(X)$ as a closed two-sided ideal.
- Fredholm operators built on bounded operatorsBounded operators invertible modulo compacts; the Atkinson characterisation.
- Fredholm operators built on compact operatorsFredholm $\iff$ invertible modulo $\mathcal{K}$ in $\mathcal{B}(X) / \mathcal{K}(X)$ (Atkinson).
- elliptic operators built on Fredholm operatorsCompact-manifold elliptic operators are Fredholm after Sobolev completion.
- theorem bridging Fredholm operators and Atiyah-Singer indexThe topological formula for the Fredholm index.
- theorem bridging Fredholm operators and Bott periodicityManifests through the path-component structure of $\mathcal{F}$ classifying $K$-theory.
- Dirac operator built on Fredholm operatorsClosed-manifold elliptic theory turns $D$ into a Fredholm operator.
- inner product space built on normed spaceInner product spaces depend on normed vector spaces and bilinear forms. Hilbert spaces are complete inner product spaces.
- CFT basics built on Yang-MillsTwo-dimensional gauge and conformal theories share stress-energy and symmetry methods, though Yang-Mills itself is not conformal in every dimension.
- infinite-dim reps built on central extensionThis unit depends on Lie algebras, Hilbert-space language, and central extensions. It feeds Virasoro algebra, where highest-weight and positive-energy representations are primary examples.
- Virasoro generalises central extensionVirasoro algebra depends on central extensions and infinite-dimensional Lie algebra representations. It refines the CFT symmetry discussion in, where the stress tensor has modes $L_n$ satisfying the Virasoro relations.
- Virasoro built on infinite-dim repsThis unit depends on Lie algebras, Hilbert-space language, and central extensions. It feeds Virasoro algebra, where highest-weight and positive-energy representations are primary examples.
- Invariant polynomial on a Lie algebra built on Lie algebraAdjoint-invariant polynomials on $\mathfrak{g}$ are the input to Chern-Weil theory.
- central extension generalises Lie algebraModifies the bracket by a 2-cocycle, producing the Heisenberg, Virasoro, and affine Kac-Moody algebras.
- spin group built on bilinear formThe quadratic form gives the unit-vector relation inside the Clifford algebra.
- tensor algebra built on vector spaceTensor algebra depends on vector spaces and the tensor product of vector spaces. It feeds directly into quotient algebra, where relations are imposed by two-sided ideals.
- vector bundle built on vector spaceEvery fiber is a vector space and all local charts are fiberwise linear.
- bilinear form built on vector spaceBilinear and quadratic forms add measurement data to vector spaces.
- Character orthogonality built on characterThe row and column orthogonality relations form the central computational engine of character theory.
- regular rep built on characterThe character $\chi_{\mathrm{reg}}(e) = |G|$, $\chi_{\mathrm{reg}}(g) = 0$ for $g \neq e$ is the defining feature.
- regular rep built on Character orthogonalityThe multiplicity formula $n_V = \langle \chi, \chi_V \rangle$ identifies irreducibles inside the regular representation, giving the dimension formula.
- Cartan-Weyl classification built on group repThe Cartan-Weyl classification organises representations of all simple Lie algebras and Lie groups.
- highest weight rep built on group repHighest-weight reps are the cornerstone of the rep theory of semisimple Lie groups / Lie algebras.
- Cartan-Weyl classification built on highest weight repThe classification of compact semisimple Lie groups via root systems is the structural backbone of highest-weight theory.
- Frobenius reciprocity built on induced repFrobenius reciprocity is the structural result organising the relationship between induction and restriction.
- Haar measure built on Peter-WeylGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.
- Cartan-Weyl classification built on SchurSchur is a building block in the classification of irreducible representations of semisimple Lie groups.
- highest weight rep built on SchurSchur's lemma underwrites the uniqueness statement: a highest-weight vector determines an irreducible representation up to scalar.
- induced rep built on tensor product (reps)The projection formula connects induction with tensor products via Mackey decomposition.
- Dirac operator built on Clifford algebraVia a chosen connection on the spinor bundle — to a first-order elliptic differential operator $\not{\!\partial}$ whose square recovers the Laplace-Beltrami operator (modulo curvature corrections). Atkinson's theorem makes $\not{\!\partial}$ a Fredholm operator on the appropriate Sobolev spaces (unit).
- spin group built on Clifford algebraSpin group and spin structure sit inside the even part $\mathrm{Cl}^0(V, q)$ as the closure of products of unit-norm vectors.
- spin structure built on Clifford algebraSpin group and spin structure sit inside the even part $\mathrm{Cl}^0(V, q)$ as the closure of products of unit-norm vectors.
- elliptic operators built on Dirac operatorDirac operators are the first-order model examples.
- theorem bridging Dirac operator and Atiyah-Singer indexComputes $\mathrm{ind}(D^+)$ topologically.
- spin structure built on spin groupThe double cover $\mathrm{Spin}(n)\to\mathrm{SO}(n)$ is the structural map used in the definition.
- Dirac operator built on spinor bundleThe natural first-order elliptic operator on the spinor bundle.
- Dirac operator built on spin structureThe elliptic operator on the spinor bundle whose existence presupposes a spin structure.
- spinor bundle built on spin structureThe associated bundle that requires a spin structure to define globally.
- free energy built on Boltzmann distribution and canonical ensembleThis unit links directly to,, and inside Strand E.
- Gaussian field theory and free boson built on Correlation functionsThis unit links directly to,, and inside Strand E.
- CFT criticality built on Gaussian fieldThis unit links directly to,, and inside Strand E.
- free energy built on Mean-field theory and Curie-Weiss modelThis unit links directly to,, and inside Strand E.
- Ising model built on Mean-field theory and Curie-Weiss modelThis unit links directly to,, and inside Strand E.
- SSB built on Mean-field theory and Curie-Weiss modelThis unit links directly to,, and inside Strand E.
- Ising model built on Onsager solutionThis unit links directly to,, and inside Strand E.
- transfer matrix built on Onsager solutionThis unit links directly to,, and inside Strand E.
- Boltzmann distribution and canonical ensemble built on Partition functionThis unit links directly to,, and inside Strand E.
- Ising model built on Partition functionThis unit links directly to,, and inside Strand E.
- Wilson-Fisher fixed point and universality built on Renormalisation groupThis unit links directly to,, and inside Strand E.
- Beta function built on Wilson-Fisher fixed point and universalityThis unit links directly to,, and inside Strand E.
- Wilson action built on Wilson's lattice gauge theoryThis unit links directly to,, and inside Strand E.
- pseudoholomorphic curve built on Almost-complex structure on a symplectic manifoldThis unit connects directly to,, and inside the symplectic strand.
- Lagrangian submanifold built on Arnold conjecture and Floer homology setupThis unit connects directly to,, and inside the symplectic strand.
- Poisson bracket built on coadjoint orbitThis unit connects directly to,, and inside the symplectic strand.
- Poisson bracket built on Hamiltonian vector fieldThis unit connects directly to,, and inside the symplectic strand.
- action-angle coords built on integrable systemThis unit connects directly to,, and inside the symplectic strand.
- Conley-Zehnder built on Maslov indexThis unit connects directly to,, and inside the symplectic strand.
- symplectic capacity built on Gromov non-squeezing theoremThis unit connects directly to,, and inside the symplectic strand.
- Gromov non-squeezing theorem built on pseudoholomorphic curveThis unit connects directly to,, and inside the symplectic strand.
- double cover is a special case of covering spaceDouble covers connect covering-space theory to principal bundles. The map $\mathrm{Spin}(n)\to\mathrm{SO}(n)$ is a double cover used by spin group theory.
- Chern-Weil built on de Rham cohomologyCurvature forms produce de Rham cohomology classes.
- covering space built on Homotopy and homotopy groupCovering spaces realise subgroups of $\pi_1$ geometrically.
- Eilenberg-MacLane space built on SpectrumIngredients of the Eilenberg-MacLane spectrum $HA$.
- Eilenberg-MacLane space built on Suspension$K(A, n)$ relates to $K(A, n+1)$ by loop space; suspension is "almost" the inverse.
- Spectrum built on SuspensionSequences with structure maps using suspension; the natural objects of stable homotopy theory.
- Cartier divisor built on BlowupThe exceptional divisor is the canonical effective Cartier divisor on the blowup; the universal property says blowups make pulled-back ideal sheaves Cartier.appears in:
04.07.02 - Picard group built on BlowupBlowing up a smooth point on a smooth surface adds a $\mathbb{Z}$-summand to the Picard group, generated by the exceptional curve class.appears in:
04.07.02 - theorem bridging line bundle (scheme) and Riemann-Roch (curves)Riemann-Roch is a calculation involving line bundles.appears in:
04.05.03 - Atiyah-Singer index generalises Riemann-Roch (curves)The analytic generalisation of Riemann-Roch to elliptic operators on manifolds.appears in:
04.04.01 - coherent sheaf generalises Serre dualityThe duality holds for all coherent sheaves (with dualising-complex generalisation).appears in:
04.08.03 - sheaf of differentials analogous to de Rham cohomologyAlgebraic de Rham cohomology is $\mathbb{H}^*(X; \Omega^\bullet_X)$, mirrored by smooth de Rham theory.appears in:
04.08.01 - principal bundle built on groupGroup actions add a space on which a group acts. Orthogonal groups are groups preserving a bilinear form.appears in:
01.02.01 - Lie group built on groupGroup actions add a space on which a group acts. Orthogonal groups are groups preserving a bilinear form.appears in:
01.02.01 - principal bundle built on group actionGroup actions build on groups and feed principal bundles, where each fiber is a right torsor for the structure group. Lie groups act smoothly on manifolds, producing homogeneous spaces and associated bundles.appears in:
03.03.02 - Lie group built on group actionGroup actions build on groups and feed principal bundles, where each fiber is a right torsor for the structure group. Lie groups act smoothly on manifolds, producing homogeneous spaces and associated bundles.appears in:
03.03.02 - orthogonal group built on group actionGroup actions build on groups and feed principal bundles, where each fiber is a right torsor for the structure group. Lie groups act smoothly on manifolds, producing homogeneous spaces and associated bundles.appears in:
03.03.02 - Lie algebra built on Quotient algebraThe same factorization theorem underlies universal enveloping algebras for Lie algebras and coordinate rings in algebraic geometry. In each case, relations are encoded by a kernel, and maps out of the quotient are exactly maps that respect those relations.appears in:
03.01.05 - bilinear form built on Quotient algebraQuotient algebra follows tensor algebra in the Clifford chain. Clifford algebra is the quotient of a tensor algebra by the quadratic-form ideal from.appears in:
03.01.05 - Clifford algebra built on Quotient algebraQuotient algebra follows tensor algebra in the Clifford chain. Clifford algebra is the quotient of a tensor algebra by the quadratic-form ideal from.appears in:
03.01.05 - de Rham cohomology built on Quotient algebraQuotient algebra follows tensor algebra in the Clifford chain. Clifford algebra is the quotient of a tensor algebra by the quadratic-form ideal from.appears in:
03.01.05 - bilinear form built on tensor algebraTensor algebra depends on vector spaces and the tensor product of vector spaces. It feeds directly into quotient algebra, where relations are imposed by two-sided ideals.appears in:
03.01.04 - Clifford algebra built on tensor algebraTensor algebra depends on vector spaces and the tensor product of vector spaces. It feeds directly into quotient algebra, where relations are imposed by two-sided ideals.appears in:
03.01.04 - Pontryagin/Chern built on complex vector bundleThe natural characteristic classes; Chern classes are integral cohomology classes of complex bundles, Pontryagin classes are derived from complexified real bundles.appears in:
03.05.08 - principal bundle built on Orthogonal frame bundleOrthogonal frame bundles depend on vector bundles, principal bundles, and Lie groups. Spin structures lift the oriented orthonormal frame bundle through the double cover supplied by the spin group.appears in:
03.05.03 - vector bundle built on Orthogonal frame bundleOrthogonal frame bundles depend on vector bundles, principal bundles, and Lie groups. Spin structures lift the oriented orthonormal frame bundle through the double cover supplied by the spin group.appears in:
03.05.03 - differential forms built on Orthogonal frame bundleOrthogonal frame bundles depend on vector bundles, principal bundles, and Lie groups. Spin structures lift the oriented orthonormal frame bundle through the double cover supplied by the spin group.appears in:
03.05.03 - Dirac operator built on Orthogonal frame bundleOrthogonal frame bundles depend on vector bundles, principal bundles, and Lie groups. Spin structures lift the oriented orthonormal frame bundle through the double cover supplied by the spin group.appears in:
03.05.03 - spin group built on Orthogonal frame bundleOrthogonal frame bundles depend on vector bundles, principal bundles, and Lie groups. Spin structures lift the oriented orthonormal frame bundle through the double cover supplied by the spin group.appears in:
03.05.03 - spin structure built on Orthogonal frame bundleOrthogonal frame bundles depend on vector bundles, principal bundles, and Lie groups. Spin structures lift the oriented orthonormal frame bundle through the double cover supplied by the spin group.appears in:
03.05.03 - double cover built on Orthogonal frame bundleOrthogonal frame bundles depend on vector bundles, principal bundles, and Lie groups. Spin structures lift the oriented orthonormal frame bundle through the double cover supplied by the spin group.appears in:
03.05.03 - theorem bridging vector bundle and Bott periodicityTopological K-theory is built from vector bundles.appears in:
03.05.02 - Chern-Weil built on Invariant polynomial on a Lie algebraApplies invariant polynomials to curvature 2-forms to produce closed differential forms on the base manifold; this is the construction's downstream payoff.appears in:
03.06.05 - Yang-Mills built on Invariant polynomial on a Lie algebraThe action functional uses the *invariant inner product* on $\mathfrak{g}$, which is a degree-2 element of $S^\bullet(\mathfrak{g}^\ast)^G$ when $G$ is compact.appears in:
03.06.05 - divisor (RS) built on Abel-JacobiSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.04 - holomorphic line bundle (RS) built on Abel-JacobiSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.04 - Jacobian variety built on Abel-JacobiSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.04 - Riemann-Roch (RS) built on Abel-JacobiSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.04 - divisor (RS) built on analytic continuationSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.04 - holomorphic function built on analytic continuationSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.04 - holomorphic line bundle (RS) built on analytic continuationSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.04 - Riemann-Roch (RS) built on analytic continuationSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.04 - Riemann surface built on analytic continuationSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.04 - divisor (RS) built on branch pointSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.02.01 - holomorphic function built on branch pointSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.02.01 - holomorphic line bundle (RS) built on branch pointSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.02.01 - Riemann-Roch (RS) built on branch pointSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.02.01 - Riemann surface built on branch pointSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.02.01 - divisor (RS) built on Cauchy's formulaSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.02 - holomorphic function built on Cauchy's formulaSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.02 - holomorphic line bundle (RS) built on Cauchy's formulaSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.02 - Riemann-Roch (RS) built on Cauchy's formulaSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.02 - Riemann surface built on Cauchy's formulaSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.02 - Riemann-Roch (RS) built on divisor (RS)Supplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.05.01 - divisor (RS) built on genus of a RSSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.03.02 - holomorphic function built on genus of a RSSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.03.02 - holomorphic line bundle (RS) built on genus of a RSSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.03.02 - Riemann-Roch (RS) built on genus of a RSSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.03.02 - Riemann surface built on genus of a RSSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.03.02 - Hodge decomposition built on HartogsSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.07.02 - divisor (RS) built on HartogsSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.07.02 - holomorphic function built on HartogsSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.07.02 - holomorphic line bundle (RS) built on HartogsSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.07.02 - Riemann surface built on HartogsSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.07.02 - Riemann-Roch (RS) built on holomorphic line bundle (RS)Supplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.05.02 - divisor (RS) built on abelian differentialSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.01 - holomorphic line bundle (RS) built on abelian differentialSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.01 - Jacobian variety built on abelian differentialSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.01 - Riemann-Roch (RS) built on abelian differentialSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.01 - Hodge decomposition built on holomorphic (several variables)Supplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.07.01 - divisor (RS) built on holomorphic (several variables)Supplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.07.01 - holomorphic function built on holomorphic (several variables)Supplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.07.01 - holomorphic line bundle (RS) built on holomorphic (several variables)Supplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.07.01 - Riemann surface built on holomorphic (several variables)Supplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.07.01 - Riemann-Roch (RS) built on Jacobian varietySupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.03 - divisor (RS) built on meromorphic functionSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.05 - holomorphic function built on meromorphic functionSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.05 - holomorphic line bundle (RS) built on meromorphic functionSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.05 - Riemann-Roch (RS) built on meromorphic functionSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.05 - Riemann surface built on meromorphic functionSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.05 - divisor (RS) built on Period matrixSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.02 - holomorphic line bundle (RS) built on Period matrixSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.02 - Jacobian variety built on Period matrixSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.02 - Riemann-Roch (RS) built on Period matrixSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.02 - divisor (RS) built on residue theoremSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.03 - holomorphic function built on residue theoremSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.03 - holomorphic line bundle (RS) built on residue theoremSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.03 - Riemann-Roch (RS) built on residue theoremSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.03 - Riemann surface built on residue theoremSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.03 - divisor (RS) built on Riemann-HurwitzSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.05.03 - holomorphic line bundle (RS) built on Riemann-HurwitzSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.05.03 - Jacobian variety built on Riemann-HurwitzSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.05.03 - Riemann-Roch (RS) built on Riemann-HurwitzSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.05.03 - divisor (RS) built on Riemann mappingSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.06 - holomorphic function built on Riemann mappingSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.06 - holomorphic line bundle (RS) built on Riemann mappingSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.06 - Riemann-Roch (RS) built on Riemann mappingSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.06 - Riemann surface built on Riemann mappingSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.01.06 - Atiyah-Singer index generalises Riemann-Roch (RS)Analytic generalisation to elliptic operators.appears in:
06.04.01 - theorem bridging Riemann surface and Riemann-Roch (curves)Algebraic version of the analytic statement.appears in:
06.03.01 - divisor (RS) built on Theta functionSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.05 - holomorphic line bundle (RS) built on Theta functionSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.05 - Jacobian variety built on Theta functionSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.05 - Riemann-Roch (RS) built on Theta functionSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.06.05 - divisor (RS) built on uniformizationSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.03.03 - holomorphic function built on uniformizationSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.03.03 - holomorphic line bundle (RS) built on uniformizationSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.03.03 - Riemann-Roch (RS) built on uniformizationSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.03.03 - Riemann surface built on uniformizationSupplies the local analytic language, supplies the Riemann-surface setting, and uses this unit as part of the global theory of curves, periods, or sheaf cohomology. The same ideas also interact with divisor and line-bundle constructions in and.appears in:
06.03.03 - unbounded self-adjoint generalises bounded operatorsWhat bounded operators are *not*; spectral theory extends to densely defined unbounded operators.appears in:
02.11.01 - Orthogonal frame bundle built on inner product spaceInner product spaces depend on normed vector spaces and bilinear forms. Hilbert spaces are complete inner product spaces.appears in:
02.11.07 - Hilbert space built on inner product spaceInner product spaces depend on normed vector spaces and bilinear forms. Hilbert spaces are complete inner product spaces.appears in:
02.11.07 - orthogonal group built on inner product spaceInner product spaces depend on normed vector spaces and bilinear forms. Hilbert spaces are complete inner product spaces.appears in:
02.11.07 - bilinear form built on inner product spaceInner product spaces depend on normed vector spaces and bilinear forms. Hilbert spaces are complete inner product spaces.appears in:
02.11.07 - Banach space built on normed spaceNormed vector spaces depend on vector spaces and metric spaces. Banach spaces are complete normed vector spaces.appears in:
02.11.06 - bounded operators built on normed spaceNormed vector spaces depend on vector spaces and metric spaces. Banach spaces are complete normed vector spaces.appears in:
02.11.06 - vector space built on normed spaceNormed vector spaces depend on vector spaces and metric spaces. Banach spaces are complete normed vector spaces.appears in:
02.11.06 - metric space built on normed spaceNormed vector spaces depend on vector spaces and metric spaces. Banach spaces are complete normed vector spaces.appears in:
02.11.06 - Chern-Weil built on Classifying spaceCharacteristic classes also factor through classifying spaces. A universal class on $BG$ pulls back along a classifying map to the characteristic class of the bundle over $X$, which connects this unit to Chern-Weil theory and Pontryagin/Chern classes.appears in:
03.08.04 - Pontryagin/Chern built on Classifying spaceCharacteristic classes also factor through classifying spaces. A universal class on $BG$ pulls back along a classifying map to the characteristic class of the bundle over $X$, which connects this unit to Chern-Weil theory and Pontryagin/Chern classes.appears in:
03.08.04 - Chern-Weil built on K-theoryCharacteristic classes give natural transformations out of K-theory after rationalization through the Chern character. This ties the vector-bundle bookkeeping of K-theory to the differential-form representatives built by Chern-Weil theory.appears in:
03.08.01 - Pontryagin/Chern built on K-theoryCharacteristic classes give natural transformations out of K-theory after rationalization through the Chern character. This ties the vector-bundle bookkeeping of K-theory to the differential-form representatives built by Chern-Weil theory.appears in:
03.08.01 - Hilbert space built on infinite-dim repsThis unit depends on Lie algebras, Hilbert-space language, and central extensions. It feeds Virasoro algebra, where highest-weight and positive-energy representations are primary examples.appears in:
03.11.02 - Lie algebra built on infinite-dim repsThis unit depends on Lie algebras, Hilbert-space language, and central extensions. It feeds Virasoro algebra, where highest-weight and positive-energy representations are primary examples.appears in:
03.11.02 - CFT basics built on infinite-dim repsThis unit depends on Lie algebras, Hilbert-space language, and central extensions. It feeds Virasoro algebra, where highest-weight and positive-energy representations are primary examples.appears in:
03.11.02 - Virasoro generalises Lie algebraThe central extension of the Witt algebra; the symmetry algebra of two-dimensional CFT.appears in:
03.04.01 - CFT basics built on VirasoroVirasoro algebra depends on central extensions and infinite-dimensional Lie algebra representations. It refines the CFT symmetry discussion in, where the stress tensor has modes $L_n$ satisfying the Virasoro relations.appears in:
03.11.03 - Orthogonal frame bundle built on orthogonal groupOrthogonal groups depend on groups, bilinear forms, and Lie groups. They act on orthonormal frames, giving the structure group of the orthogonal frame bundle.appears in:
03.03.03 - Lie group built on orthogonal groupOrthogonal groups depend on groups, bilinear forms, and Lie groups. They act on orthonormal frames, giving the structure group of the orthogonal frame bundle.appears in:
03.03.03 - bilinear form built on orthogonal groupOrthogonal groups depend on groups, bilinear forms, and Lie groups. They act on orthonormal frames, giving the structure group of the orthogonal frame bundle.appears in:
03.03.03 - Clifford algebra built on orthogonal groupOrthogonal groups depend on groups, bilinear forms, and Lie groups. They act on orthonormal frames, giving the structure group of the orthogonal frame bundle.appears in:
03.03.03 - spin group built on orthogonal groupOrthogonal groups depend on groups, bilinear forms, and Lie groups. They act on orthonormal frames, giving the structure group of the orthogonal frame bundle.appears in:
03.03.03 - CFT basics pattern recurs in Bott periodicityK-theoretic periodicity reappears in the classification of fermionic phases and anomaly data, outside the basic CFT unit.appears in:
03.10.02 - Lie algebra built on Borel-WeilGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.09 - Lie group built on Borel-WeilGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.09 - Cartan-Weyl classification built on Borel-WeilGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.09 - group rep built on Borel-WeilGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.09 - highest weight rep built on Borel-WeilGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.09 - Cartan-Weyl classification analogous to K-theoryClassifying spaces $BG$ for simple compact Lie groups have rational cohomology generated by Pontryagin / Chern classes whose structure mirrors the root system.appears in:
07.04.01 - Cartan-Weyl classification generalises characterWeyl's character formula generalises ordinary characters to compact semisimple Lie groups.appears in:
07.01.03 - Peter-Weyl generalises Character orthogonalityThe $L^2$-orthogonality of characters on compact groups generalises the finite-group statement.appears in:
07.01.04 - Lie algebra built on compact Lie repGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.07.01 - Lie group built on compact Lie repGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.07.01 - group rep built on compact Lie repGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.07.01 - Haar measure built on compact Lie repGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.07.01 - Peter-Weyl built on compact Lie repGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.07.01 - Lie algebra built on Dynkin diagramGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.05 - Lie group built on Dynkin diagramGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.05 - Cartan-Weyl classification built on Dynkin diagramGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.05 - group rep built on Dynkin diagramGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.05 - highest weight rep built on Dynkin diagramGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.05 - Frobenius reciprocity analogous to highest weight repParabolic induction (Lie-group analogue) is a primary source of $G$-representations from Levi subgroup data.appears in:
07.01.08 - Lie algebra built on Haar measureGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.07.03 - Lie group built on Haar measureGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.07.03 - group rep built on Haar measureGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.07.03 - induced rep analogous to highest weight repParabolic induction (the Lie-group analogue) is fundamental to the construction of admissible representations.appears in:
07.01.07 - Lie algebra built on Lie algebra repGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.01 - Lie group built on Lie algebra repGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.01 - Cartan-Weyl classification built on Lie algebra repGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.01 - group rep built on Lie algebra repGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.01 - highest weight rep built on Lie algebra repGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.01 - Maschke analogous to Cartan-Weyl classificationThe analogous compact-Lie-group statement is foundational to the classification of representations of compact semisimple Lie groups.appears in:
07.02.01 - Maschke analogous to highest weight repWeyl's complete-reducibility theorem for semisimple Lie algebras is the Lie-theoretic analogue of Maschke.appears in:
07.02.01 - Lie algebra built on Peter-WeylGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.07.02 - Lie group built on Peter-WeylGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.07.02 - group rep built on Peter-WeylGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.07.02 - Cartan-Weyl classification built on regular repThe dimension formula $\sum (\dim V)^2 = |G|$ is the algebraic counterpart of the analytic statement for compact Lie groups.appears in:
07.01.05 - Peter-Weyl generalises regular repThe $L^2$-decomposition of compact groups generalises the regular-representation decomposition to compact Lie groups.appears in:
07.01.05 - Lie algebra built on root systemGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.03 - Lie group built on root systemGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.03 - Cartan-Weyl classification built on root systemGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.03 - group rep built on root systemGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.03 - highest weight rep built on root systemGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.03 - character built on tensor product (reps)The multiplicativity $\chi_{V \otimes W} = \chi_V \chi_W$ makes characters a ring homomorphism from the representation ring to class functions.appears in:
07.01.06 - Lie algebra built on universal enveloping algebraGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.02 - Lie group built on universal enveloping algebraGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.02 - Cartan-Weyl classification built on universal enveloping algebraGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.02 - group rep built on universal enveloping algebraGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.02 - highest weight rep built on universal enveloping algebraGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.02 - Lie algebra built on Verma moduleGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.06 - Lie group built on Verma moduleGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.06 - Cartan-Weyl classification built on Verma moduleGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.06 - group rep built on Verma moduleGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.06 - highest weight rep built on Verma moduleGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.06 - Lie algebra built on Weyl characterGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.07 - Lie group built on Weyl characterGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.07 - Cartan-Weyl classification built on Weyl characterGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.07 - group rep built on Weyl characterGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.07 - highest weight rep built on Weyl characterGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.07 - Lie algebra built on Weyl dimensionGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.08 - Lie group built on Weyl dimensionGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.08 - Cartan-Weyl classification built on Weyl dimensionGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.08 - group rep built on Weyl dimensionGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.08 - highest weight rep built on Weyl dimensionGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.08 - Lie algebra built on Weyl groupGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.04 - Lie group built on Weyl groupGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.04 - Cartan-Weyl classification built on Weyl groupGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.04 - group rep built on Weyl groupGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.04 - highest weight rep built on Weyl groupGives the group-representation starting point, supplies highest-weight or compact averaging methods, and uses this unit in classification and harmonic analysis. The Lie-algebraic chain also connects to through brackets and to through differentiation of Lie group actions.appears in:
07.06.04 - projective space built on Young diagram(symmetric group representation) is the parent unit; Young diagrams are its combinatorial backbone, and each irreducible $V^\lambda$ is built from a tableau of shape $\lambda$. (Specht module) gives the modular construction of $V^\lambda$ via polytabloids — explicit basis vectors built from tableaux.appears in:
07.05.02 - Cartan-Weyl classification built on Young diagram(symmetric group representation) is the parent unit; Young diagrams are its combinatorial backbone, and each irreducible $V^\lambda$ is built from a tableau of shape $\lambda$. (Specht module) gives the modular construction of $V^\lambda$ via polytabloids — explicit basis vectors built from tableaux.appears in:
07.05.02 - highest weight rep built on Young diagram(symmetric group representation) is the parent unit; Young diagrams are its combinatorial backbone, and each irreducible $V^\lambda$ is built from a tableau of shape $\lambda$. (Specht module) gives the modular construction of $V^\lambda$ via polytabloids — explicit basis vectors built from tableaux.appears in:
07.05.02 - Fredholm operators built on Clifford algebraDirac operator. The Clifford action of $V$ on the spinor module promotes — via a chosen connection on the spinor bundle — to a first-order elliptic differential operator $\not{\!\partial}$ whose square recovers the Laplace-Beltrami operator (modulo curvature corrections).appears in:
03.09.02 - principal bundle built on continuous mapContinuous maps depend on topological spaces and feed metric spaces, classifying spaces, and bundle pullback constructions. Smooth maps between manifolds are continuous maps with differentiability added.appears in:
02.01.02 - smooth manifold built on continuous mapContinuous maps depend on topological spaces and feed metric spaces, classifying spaces, and bundle pullback constructions. Smooth maps between manifolds are continuous maps with differentiability added.appears in:
02.01.02 - Classifying space built on continuous mapContinuous maps depend on topological spaces and feed metric spaces, classifying spaces, and bundle pullback constructions. Smooth maps between manifolds are continuous maps with differentiability added.appears in:
02.01.02 - metric space built on continuous mapContinuous maps depend on topological spaces and feed metric spaces, classifying spaces, and bundle pullback constructions. Smooth maps between manifolds are continuous maps with differentiability added.appears in:
02.01.02 - topological space built on continuous mapContinuous maps depend on topological spaces and feed metric spaces, classifying spaces, and bundle pullback constructions. Smooth maps between manifolds are continuous maps with differentiability added.appears in:
02.01.02 - principal bundle built on double coverDouble covers connect covering-space theory to principal bundles. The map $\mathrm{Spin}(n)\to\mathrm{SO}(n)$ is a double cover used by spin group theory.appears in:
03.05.05 - Stiefel-Whitney built on double coverThe same two-valued monodromy viewpoint appears in characteristic-class obstructions: the failure to lift structure groups is recorded by cohomological data, especially Stiefel-Whitney classes in the spin case.appears in:
03.05.05 - spin group built on double coverDouble covers connect covering-space theory to principal bundles. The map $\mathrm{Spin}(n)\to\mathrm{SO}(n)$ is a double cover used by spin group theory.appears in:
03.05.05 - spin structure built on double coverDouble covers connect covering-space theory to principal bundles. The map $\mathrm{Spin}(n)\to\mathrm{SO}(n)$ is a double cover used by spin group theory.appears in:
03.05.05 - theorem bridging Homotopy and homotopy group and Bott periodicityPeriodicity of stable homotopy of classical groups.appears in:
03.12.01 - theorem bridging Spectrum and Bott periodicityThe structure maps of $\mathbf{KU}$ via Bott periodicity.appears in:
03.12.04 - Clifford chessboard equivalent to real Bott periodicityAtiyah-Bott-Shapiro identify the module quotient \widehat{\mathfrak{M}}_n \cong KO^{-n}(\mathrm{pt}), exposing the eight-fold algebraic rhythm of \mathrm{Cl}_{p,q} as the algebraic shadow of real Bott periodicity in KO-theory.appears in:
03.09.11 - KR (1,1)-periodicity equivalent to Cl_{r+1,s+1} = Cl_{r,s} ⊗ Cl_{1,1}The K-theoretic statement and the algebraic statement are the same theorem at different abstraction levels; both rest on \mathrm{Cl}_{1,1} \cong M_2(\mathbb{R}) and a contractibility argument for \mathrm{Cl}_{1,1}-module structures.
- calibrations built on Spin(7)/G_2 spinor squaring via trialityThe Spin(7) and G_2 calibrating forms of Harvey-Lawson 1982 are constructed by spinor squaring under triality; calibrated submanifolds are exactly those preserved by this form's pointwise structure. Forward-promise: target unit ships in Batch 2.appears in:
03.09.13 - Hodge Laplacian as Dirac square of the de Rham Dirac bundleThe Hodge Laplacian on differential forms is the Dirac square of the de Rham Dirac bundle (\Lambda^* T^*M, c(\xi) = \xi\wedge - \iota_{\xi^\sharp}, \nabla^{LC}); the Bochner curvature operator is the obstruction tensor for harmonic forms.appears in:
03.09.14 - α-invariant built on Cl_k-linear Dirac index in KOHitchin's α-invariant on a closed spin manifold is the Clifford-index of the spin Dirac operator viewed as a Cl_n-linear Dirac bundle; the entire psc obstruction chain (Hitchin → Gromov-Lawson → Stolz) is built on this Cl_k-linear refinement. Forward-promise: target unit ships in Batch 2 as 03.09.16.appears in:
03.09.15 - theorem bridging McKean-Singer supertrace and analytic indexFor every t > 0, Str(e^{-tD^2}) = ind(D^+); the eigenspace pairing under D collapses every positive-eigenvalue contribution. The heat-kernel proof of Atiyah-Singer takes this as its starting point and evaluates the supertrace at small t.
- Fredholmness of elliptic operators built on parametrix constructionAn elliptic pseudodifferential operator A admits a parametrix Q in Ψ^{-m} with QA-I, AQ-I smoothing. Smoothing operators are compact between Sobolev spaces; Atkinson identifies operators invertible modulo compacts as Fredholm. The parametrix construction is therefore the analytic substrate of every elliptic-Fredholm theorem.
- family-index Chern character lives in cohomology of the baseFor a smooth family of elliptic operators D = {D_b} over B, ind(D) ∈ K^0(B) and ch(ind(D)) = π_*(ch(σ) Td(T_{M/B} ⊗ ℂ)) ∈ H^*(B; ℚ) by the Atiyah-Hirzebruch Riemann-Roch identity. The Bismut superconnection produces the cohomology representative as a closed differential form.appears in:
03.09.21 - equivariant index localises to the fixed-point setFor a G-equivariant elliptic operator D with isolated non-degenerate fixed points of g ∈ G, the equivariant character of ind_G(D) at g equals a sum of local contributions at M^g built from \hat{A}, ch, and det(I - dg^{-1}|_{T_pM}). Atiyah-Bott 1968 fixed-point formula is recovered by the equivariant heat-kernel localisation argument.appears in:
03.09.21 - psc obstruction chain built on Lichnerowicz formulaThe spinor-Bochner identity D^2 = nabla^* nabla + (1/4) Scal is the structural foundation of the entire psc obstruction chain. Without Lichnerowicz, no Hitchin-α; without Hitchin, no Gromov-Lawson; without Gromov-Lawson, no Stolz classification.appears in:
03.09.16 - α-invariant equivalent to Cl_n-linear Clifford-index of the spin Dirac operatorHitchin's α-invariant on a closed spin manifold is exactly the Cl_n-linear Clifford-index of the spin Dirac operator viewed as a Cl_n-linear Dirac bundle. Equivalence of two definitions: one analytical (kernel of D as graded Cl_n-module), one K-theoretic (image under Atiyah-Bott-Shapiro orientation).
- theorem bridging enlargeable topology and Lichnerowicz vanishingGromov-Lawson 1983 §V: a closed enlargeable spin manifold admits no metric of positive scalar curvature. The bridge converts a topological condition (small-Lipschitz maps to S^n from finite covers) into an analytic Lichnerowicz vanishing via twisted Dirac operators.appears in:
03.09.16 - theorem bridging asymptotic flatness and harmonic spinor identityWitten 1981 bridges the ADM-mass formalism (general relativity, asymptotic geometry) and the Dirac-spinor formalism (analytic differential geometry) via a single integration-by-parts identity. The boundary term of Lichnerowicz on an AF spin 3-manifold equals -4π m_ADM.appears in:
03.09.17 - Berger holonomy bijection with parallel spinorsWang 1989: a simply-connected Riemannian spin manifold admits a non-zero parallel spinor iff its restricted holonomy is one of SU(n), Sp(n), G_2, Spin(7). Parallel-spinor counts: 2, n+1, 1, 1 respectively. The bijection is the structural foundation of calibrated geometry.appears in:
03.09.18 - calibrations require special holonomy structureEach Harvey-Lawson calibrated geometry lives on a manifold with one of Wang's special holonomy groups — SU(n) for SL, Sp(n) for hyperkähler, G_2 for associative/coassociative, Spin(7) for Cayley. The calibrating form is the spinor square of the parallel spinor.
- Mayer-Vietoris sequence is the foundation for good-cover inductionThe MV exact sequence is the engine that good-cover induction iterates to produce finite-dimensionality of de Rham cohomology on every compact manifold.
- de Rham theorem built on Mayer-Vietoris induction over a good coverThe de Rham theorem with R-coefficients is proved by comparing the de Rham and singular MV sequences on a good cover, with the integration pairing as the comparison map.
- Čech-de Rham double complex built on a good coverThe good-cover hypothesis makes column exactness in the double complex hold; without it, the column collapse fails and the double complex carries spectral-sequence content rather than collapse content.
- tic-tac-toe Künneth equivalent to MV-induction Künneth on finite-good-cover manifoldsKünneth admits two proofs: MV induction (Bott-Tu §5) and tic-tac-toe ascent on the product Čech-de Rham double complex (§9). Bott-Tu's dual-proof discipline.
- de Rham theorem built on Čech-de Rham double-complex collapseWeil's 1952 proof: row-collapse to H*_dR; column-collapse to Čech of the constant sheaf R = simplicial cohomology of the nerve = singular cohomology with R-coefficients.
- de Rham Thom class equivalent to Chern-Weil Euler formFor oriented Euclidean rank-r bundle E with metric connection, e(E) admits two de Rham realisations: i*Φ (zero-section pullback of Thom class) and (1/(2π)^{r/2})Pf(Ω) (Chern-Weil); the two routes give the same cohomology class.
- spin-geometry Â-genus machinery built on the global angular formOn a spin manifold, ψ on the spinor unit-sphere bundle with sign dψ = -π* e(E) is structural input for Â-genus density in local Atiyah-Singer; both Bismut superconnection and Getzler rescaling use this convention.
- Thom isomorphism equivalent to relative de Rham of the disc-sphere pairH*_{cv}(E) ≅ H*(D(E), S(E)); the Thom class corresponds to the relative fundamental class. Relative LES + Thom iso = Gysin sequence.
- Leray-Hirsch built on Künneth on each fiberWhen fibre cohomology is finite-dim and a global lift exists, the Leray-Hirsch theorem reduces to fibrewise Künneth times base cohomology.
- de Rham cohomology equivalent to singular cohomology with real coefficients (three routes)Three independent proofs: MV induction over a good cover, Čech-de Rham column collapse (Weil 1952), sheaf-cohomology fine resolution (Leray).
- local system on connected X equivalent to π_1(X)-representationThe category of local systems on a connected pointed space is equivalent to the category of π_1(X)-representations via monodromy; a structural insight from Poincaré 1883.appears in:
04.03.02 - twisted de Rham complex built on orientation local systemOn a non-orientable manifold, twisted Poincaré duality uses the orientation local system; the twisted de Rham complex computes cohomology with this twist.
- exact-couple spectral sequence equivalent to the Čech-de Rham double-complex spectral sequenceThe two filtrations on the Čech-de Rham double complex give two spectral sequences; both are concrete instances of the abstract exact-couple machinery (Massey 1952). The reader has already done several spectral sequences before meeting the abstract definition.
- Serre spectral sequence is the filtered-complex SS of a fibration's singular cochain filtrationSerre 1951 specialises the abstract spectral-sequence machinery to a Serre fibration; the convergence is to the cohomology of the total space, with E_2 = H^p(B; H^q(F)).
- Gysin sequence + Euler class derived from Serre spectral sequence of an oriented sphere bundleFor an oriented S^{r-1}-bundle, the Serre SS collapses at E_2 except for one differential of bidegree (r, 1-r); that differential is multiplication by the Euler class, recovering the Gysin LES.
- Serre SS of path-loop fibration computes loop-space cohomology and π_n(S^k)Serre 1951's path-loop fibration ΩX → PX → X with PX contractible gives an SS converging to a point and so allows ΩX cohomology to be read off; π_4(S^3) = Z/2 is computed via this device.appears in:
03.13.02 - splitting principle built on Leray-Hirsch theorem applied iteratively to flag-bundle projectionsIterating Leray-Hirsch on the flag-bundle Fl(E) → M splits E formally as a sum of line bundles after pulling back to Fl(E); characteristic classes are then computed as elementary symmetric functions of formal Chern roots.
- splitting principle equivalent to Borel presentation H*(BG) = H*(BT)^WBorel-Hirzebruch 1958/59: the universal splitting principle is the cohomology fact H*(BG; Q) = H*(BT; Q)^W; for G = U(n), W = S_n acts by permuting Chern roots.
- Hopf index theorem built on global angular form and integration of Euler classBott-Tu §11 derivation of Σ ind_p(V) = χ(M) rests on the global angular form ψ with fiber-integral 1 and dψ = -π* e(E), then applies Stokes in de Rham.
- Sullivan minimal model encodes rational homotopy type for simply-connected finite-type spacesSullivan's main theorem (1977 Publ. IHÉS 47): the indecomposable space V^n of the minimal model is dual to π_n(X) ⊗ ℚ; the differential encodes Whitehead products and higher Massey products. Rational Hurewicz is the bottom-degree statement.
- Sullivan model built on de Rham complex of polynomial formsThe piecewise-polynomial functor A_PL is the rational refinement of the smooth de Rham complex; on a smooth manifold A_PL(X) ⊗ ℝ is quasi-isomorphic to Ω*(X).
- Whitehead tower equivalent to dual Postnikov tower for connectivityPostnikov truncates from above; Whitehead tower from below. Each stage is a principal K(π, n)-fibration; the two towers bracket homotopy structure.
- Finiteness of π_k(S^n) for k > n built on Whitehead tower and Serre SSSerre 1953 Annals 58: π_k(S^n) finite for k > n except π_{4j-1}(S^{2j}) which has Z-summand from Hopf invariant. Proof via rational Hurewicz on Whitehead tower + finite-generation.
- Universal complex rank-k bundle γ_k = colim γ_k^n on infinite Grassmannian, equivalent to BU(k)Steenrod 1951; Milnor 1956: G_k(ℂ^∞) carries the universal complex rank-k bundle, equivalent up to homotopy to BU(k). Pullback implements [X, BU(k)] ↔ Vect^k_ℂ(X).
- Schubert calculus on Grassmannian built on Borel presentation of flag-manifold cohomologyThe Borel presentation H*(Fl_n; ℤ) = ℤ[x_1,…,x_n]/⟨e_1,…,e_n⟩ (S_n coinvariants) realises Schubert classes via BGG-Demazure 1973-74 divided-difference operators.appears in:
03.08.05