Index Theorem. 1
About this Title
Mikio Furuta, University of Tokyo, Tokyo, Japan. Translated by Kauro Ono
Publication: Translations of Mathematical Monographs
Publication Year: 2007; Volume 235
ISBNs: 978-0-8218-2097-1 (print); 1-4704-4789-2 (online)
MathSciNet review: MR2361481
MSC: Primary 58J20
The Atiyah-Singer index theorem is a remarkable result that allows one to compute the space of solutions of a linear elliptic partial differential operator on a manifold in terms of purely topological data related to the manifold and the symbol of the operator. First proved by Atiyah and Singer in 1963, it marked the beginning of a completely new direction of research in mathematics with relations to differential geometry, partial differential equations, differential topology, K-theory, physics, and other areas.
The author's main goal in this volume is to give a complete proof of the index theorem. The version of the proof he chooses to present is the one based on the localization theorem. The prerequisites include a first course in differential geometry, some linear algebra, and some facts about partial differential equations in Euclidean spaces.
Graduate students interested in index theory.
Table of Contents
- Manifolds, vector bundles and elliptic complexes
- Index and its localization
- Examples of the localization of the index
- Localization of eigenfunctions of the operator of Laplace type
- Formulation and proof of the index theorem
- Characteristic classes