The Atiyah–Singer index theorem
نویسندگان
چکیده
The Atiyah–Singer index theorem, a landmark achievement of the early 1960s, brings together ideas in analysis, geometry, and topology. We recount some antecedents motivations, various forms its implications, which extend to present.
منابع مشابه
The Index of a Holomorphic Mapping and the Index Theorem
The index theorem for a harmonic mapping of riemannian manifolds is given. Let /: M -> N be a holomorphic mapping of Kaehler manifolds. Then it is shown that the index of / is zero and that a Jacobi field along/is a holomorphic section of the bundle f*T(N) induced by/.
متن کاملA Digital Index Theorem
This paper is devoted to prove a Digital Index Theorem for digital (n − 1)-manifolds in a digital space (Rn, f), where f belongs to a large family of lighting functions on the standard cubical decomposition Rn of the n-dimensional Euclidean space. As an immediate consequence we obtain the corresponding theorems for all (α, β)-surfaces of Kong-Roscoe, with α, β ∈ {6, 18, 26} and (α, β) 6= (6, 6)...
متن کامل- Patodi - Singer Index Theorem ∗
In [Wu], the noncommutative Atiyah-Patodi-Singer index theorem was proved. In this paper, we extend this theorem to the equivariant case.
متن کاملGerbes, Clifford Modules and the Index Theorem
The use of bundle gerbes and bundle gerbe modules is considered as a replacement for the usual theory of Clifford modules on manifolds that fail to be spin. It is shown that both sides of the Atiyah-Singer index formula for coupled Dirac operators can be given natural interpretations using this language and that the resulting formula is still an identity.
متن کاملNotes on the Atiyah-Singer Index Theorem
This is arguably one of the deepest and most beautiful results in modern geometry, and in my view is a must know for any geometer/topologist. It has to do with elliptic partial differential operators on a compact manifold, namely those operators P with the property that dim ker P, dim coker P < ∞. In general these integers are very difficult to compute without some very precise information abou...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 2021
ISSN: ['0002-9904', '1936-881X']
DOI: https://doi.org/10.1090/bull/1747