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.

Aliases (1)

Dirac operator ×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 → Hirzebruch–Riemann–Roch theorem → 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 (4)

Subject (surface form when different)	Predicate
Dirac equation ("Dirac operator") → Riemann–Roch theorem →	relatedTo
Michael Atiyah →	knownFor
Isadore Singer →	notableWork