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Concept
Atiyah–Singer index theorem
Summary
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.
The index problem for elliptic differential operators was posed by Israel Gel'fand. He noticed the homotopy invariance of the index, and asked for a formula for it by means of topological invariants. 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  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. The proof sketched in this announcement was never published by them, though it appears in Palais's book. It appears also in the "Séminaire Cartan-Schwartz 1963/64" 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 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.
1965: Sergey P. Novikov published his results on the topological invariance of the rational Pontryagin classes on smooth manifolds.
Robion Kirby and Laurent C. Siebenmann's results, combined with René Thom's paper 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.
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