Atiyah–Singer index theorem

E84379

index theorem mathematical theorem

The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.

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All labels observed (4)

Label	Occurrences
Atiyah–Singer index theorem canonical	17
Atiyah–Singer index theorem for a single operator	1
Dirac operator	1
index theorems	1

Statements (47)

Predicate	Object
instanceOf	index theorem ⓘ mathematical theorem ⓘ
appliesTo	compact manifolds ⓘ elliptic pseudodifferential operators ⓘ
concerns	Dirac operator ⓘ index of elliptic operators ⓘ
connects	analysis ⓘ geometry ⓘ topology ⓘ
coreStatement	analytic index equals topological index ⓘ
field	K-theory ⓘ algebraic topology ⓘ differential geometry ⓘ global analysis ⓘ mathematical physics ⓘ operator theory ⓘ topology ⓘ
generalizes	Gauss–Bonnet theorem (early form) ⓘ surface form: Gauss–Bonnet theorem Hirzebruch–Riemann–Roch theorem ⓘ Poincaré–Hopf theorem ⓘ surface form: Hopf index theorem Riemann–Roch theorem ⓘ
hasApplicationIn	gauge theory ⓘ noncommutative geometry ⓘ quantum field theory ⓘ representation theory ⓘ spectral geometry ⓘ string theory ⓘ
hasVariant	equivariant index theorem ⓘ families index theorem ⓘ index theorem for manifolds with boundary ⓘ
implies	integrality of certain characteristic numbers ⓘ
influenced	development of K-theory ⓘ development of modern differential topology ⓘ
namedAfter	Isadore Singer ⓘ Michael Atiyah ⓘ
publishedIn	Annals of Mathematics ⓘ
recognizedAs	landmark result in 20th-century mathematics ⓘ
relates	analytic index ⓘ elliptic differential operators ⓘ topological index ⓘ topological invariants ⓘ
usesConcept	Chern character ⓘ Fredholm operator ⓘ K-theory of vector bundles ⓘ Todd class ⓘ Â-genus ⓘ
yearProved	1963 ⓘ

Referenced by (20)

Full triples — surface form annotated when it differs from this entity's canonical label.

Isadore Singer → notableWork → Atiyah–Singer index theorem ⓘ

Dirac equation → relatedTo → Atiyah–Singer index theorem ⓘ

this entity surface form: Dirac operator

Michael Atiyah → knownFor → Atiyah–Singer index theorem ⓘ

Riemann–Roch theorem → relatedTo → Atiyah–Singer index theorem ⓘ

Poincaré–Hopf theorem → specialCaseOf → Atiyah–Singer index theorem ⓘ

Chern–Weil theory → relatedTo → Atiyah–Singer index theorem ⓘ

Witten index → mathematicallyRelatedTo → Atiyah–Singer index theorem ⓘ

Grothendieck–Riemann–Roch theorem → relatedTo → Atiyah–Singer index theorem ⓘ

K-theory → relatedTo → Atiyah–Singer index theorem ⓘ

“Elliptic Operators and Compact Groups” (with Raoul Bott) → topic → Atiyah–Singer index theorem ⓘ

subject surface form: Elliptic Operators and Compact Groups

Atiyah–Bott fixed-point theorem → relatedTo → Atiyah–Singer index theorem ⓘ

Hirzebruch–Riemann–Roch theorem → relatedTo → Atiyah–Singer index theorem ⓘ

Hirzebruch–Riemann–Roch theorem → category → Atiyah–Singer index theorem ⓘ

this entity surface form: index theorems

Connes–Moscovici index theorem → generalizes → Atiyah–Singer index theorem ⓘ

Chern character → usedIn → Atiyah–Singer index theorem ⓘ

Todd class → roleIn → Atiyah–Singer index theorem ⓘ

Dirac operator → centralTo → Atiyah–Singer index theorem ⓘ

families index theorem → generalizationOf → Atiyah–Singer index theorem ⓘ

families index theorem → relatedTo → Atiyah–Singer index theorem ⓘ

this entity surface form: Atiyah–Singer index theorem for a single operator

equivariant index theorem → generalizes → Atiyah–Singer index theorem ⓘ