Editing Atiyah–Singer index theorem (section)

== History ==
The index problem for elliptic differential operators was posed  by [[Israel Gel'fand]].{{sfn|Gel'fand|1960}} He noticed the homotopy invariance of the index, and asked for a formula for it by means of [[topological invariant]]s. Some of the motivating examples included the [[Riemann–Roch theorem]] and its generalization the [[Hirzebruch–Riemann–Roch theorem]], and the [[Hirzebruch signature theorem]]. [[Friedrich Hirzebruch]] and [[Armand Borel]] had proved the integrality of the [[Â genus]] of a spin manifold, and Atiyah suggested that this integrality could be explained if it were the index of the [[Dirac operator]] (which was rediscovered by Atiyah and Singer in 1961).

The Atiyah–Singer theorem was announced in 1963.{{sfn|Atiyah|Singer|1963}} The proof sketched in this announcement was never published by them, though it appears in Palais's book.{{sfn|Palais|1965}} It appears also in the "Séminaire Cartan-Schwartz 1963/64"{{sfn|Cartan-Schwartz|1965}} that was held in Paris simultaneously with the seminar led by [[Richard Palais]] at [[Princeton University]]. The last talk in Paris was by Atiyah on manifolds with boundary. Their first published proof{{sfn|Atiyah|Singer|1968a}} replaced the [[cobordism]] theory of the first proof with [[K-theory]], and they used this to give proofs of various generalizations in another sequence of papers.{{sfnmp|1a1=Atiyah|1a2=Singer|2a1=Atiyah|2a2=Singer|3a1=Atiyah|3a2=Singer|4a1=Atiyah|4a2=Singer|1y=1968a|2y=1968b|3y=1971a|4y=1971b}}

*'''1965:''' [[Sergei Novikov (mathematician)|Sergey P. Novikov]] published his results on the topological invariance of the rational [[Pontryagin class]]es on smooth manifolds.{{sfn|Novikov|1965}}
* [[Robion Kirby]] and [[Laurent C. Siebenmann]]'s results,{{sfn|Kirby|Siebenmann|1969}} combined with [[René Thom]]'s paper{{sfn|Thom|1956}} proved the existence of rational Pontryagin classes on topological manifolds. The rational Pontryagin classes are essential ingredients of the index theorem on smooth and topological manifolds.
*'''1969:''' Michael Atiyah defines abstract elliptic operators on  arbitrary metric spaces. Abstract elliptic operators became protagonists in Kasparov's theory and Connes's noncommutative differential geometry.{{sfn|Atiyah|1970}}
*'''1971:''' Isadore Singer proposes a comprehensive program for future extensions of index theory.{{sfn|Singer|1971}}
*'''1972:''' Gennadi G. Kasparov publishes his work on the realization of K-homology by abstract elliptic operators.{{sfn|Kasparov|1972}}
*'''1973:''' Atiyah, [[Raoul Bott]], and [[Vijay Patodi]] gave a new proof of the index theorem{{sfn|Atiyah|Bott|Patodi|1973}} using the [[heat equation]], described in a paper by Melrose.{{sfn|Melrose|1993}}
*'''1977:''' [[Dennis Sullivan]] establishes his theorem on the existence and uniqueness of Lipschitz and [[quasiconformal mapping|quasiconformal]] structures on topological manifolds of dimension different from 4.{{sfn|Sullivan|1979}}
*'''1983:''' [[Ezra Getzler]]{{sfn|Getzler|1983}} motivated by ideas of Edward Witten{{sfn|Witten|1982}} and [[Luis Alvarez-Gaume]], gave a short proof  of the local index theorem for operators that are locally  [[Dirac operator]]s; this covers many of the useful cases.
*'''1983:''' Nicolae Teleman proves that the analytical indices of signature operators with values in vector bundles are topological invariants.{{sfn|Teleman|1983}}
*'''1984:''' Teleman establishes the index theorem on topological manifolds.{{sfn|Teleman|1984}}
*'''1986:''' [[Alain Connes]] publishes his fundamental paper on [[noncommutative geometry]].{{sfn|Connes|1986}}
*'''1989:''' [[Simon Donaldson|Simon K. Donaldson]] and Sullivan study Yang–Mills theory  on quasiconformal manifolds of dimension 4. They introduce the signature operator ''S'' defined on differential forms of degree two.{{sfn|Donaldson|Sullivan|1989}}
*'''1990:''' Connes and Henri Moscovici prove the local index formula in the context of non-commutative geometry.{{sfn|Connes|Moscovici|1990}}
*'''1994:''' Connes, Sullivan, and Teleman prove the index theorem for signature operators on quasiconformal manifolds.{{sfn|Connes|Sullivan|Teleman|1994}}