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In differential geometry, the Atiyah–Singer index theorem, proved by , 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 Riemann–Roch theorem, as special cases, and has applications in theoretical physics. == History == The index problem for elliptic differential operators was posed by . 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. Hirzebruch and 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 by . The proof sketched in this announcement was never published by them, though it appears in the book . 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 the papers . *1965: S.P. Novikov published his results on the topological invariance of the rational Pontrjagin classes on smooth manifolds. *Kirby and Siebenmann's results , combined with René Thom's paper proved the existence of rational Pontryagin classes on topological manifolds. The rational Pontrjagin classes are essential ingredients of the index theorem on smooth and topological manifolds. *1969: M.F. Atiyah defines abstract elliptic operators on arbitrary metric spaces. Abstract elliptic operators became protagonists in Kasparov's theory and Connes's noncommutative differential geometry. *1971: I.M. Singer proposes a comprehensive program for future extensions of index theory. *1972: G.G. Kasparov publishes his work on the realization of the K-homology by abstract elliptic operators. * gave a new proof of the index theorem using the heat equation, described in . *1977: D. Sullivan establishes his theorem on the existence and uniqueness of Lipschitz and quasiconformal structures on topological manifolds of dimension different from 4. * motivated by ideas of and Alvarez-Gaume, gave a short proof of the local index theorem for operators that are locally Dirac operators; this covers many of the useful cases. *1983: N. Teleman proves that the analytical indices of signature operators with values in vector bundles are topological invariants. *1984: N. Teleman establishes the index theorem on topological manifolds. *1986: A. Connes publishes his fundamental paper on non-commutative geometry. *1989: S.K. Donaldson and D. Sullivan study Yang–Mills theory on quasiconformal manifolds of dimension 4. They introduce the signature operator ''S'' defined on differential forms of degree two. *1990: A. Connes and H. Moscovici prove the local index formula in the context of non-commutative geometry. *1994: A. Connes, D. Sullivan and N. Teleman prove the index theorem for signature operators on quasiconformal manifolds. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Atiyah–Singer index theorem」の詳細全文を読む スポンサード リンク
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