Elliptic operators, topology and asymptotic methods
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Bibliographic Information
Elliptic operators, topology and asymptotic methods
（Research notes in mathematics, 395）
Chapman & Hall/CRC, 2001
2nd ed
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Note
First published 1988
First CRC reprint 2001
Bibliography: p. 203-206
Includes index
Description and Table of Contents
Description
This Research Note gives an introduction to the circle of ideas surrounding the `heat equation proof' of the Atiyah-Singer index theorem. Asymptotic expansions for the solutions of partial differential equations on compact manifolds are used to obtain topological information, by means of a `supersymmetric' cancellation of eigenspaces. The analysis is worked out in the context of Dirac operators on Clifford bundles.
The work includes proofs of the Hodge theorem; eigenvalue estimates; the Lefschetz theorem; the index theorem; and the Morse inequalities. Examples illustrate the general theory, and several recent results are included.
This new edition has been revised to streamline some of the analysis and to give better coverage of important examples and applications.
Readership: The book is aimed at researchers and graduate students with a background in differential geometry and functional analysis.
Table of Contents
Introduction
Chapter 1 Resume of Riemannian geometry
Chapter 2 Connections, curvature, and characteristic class
Chapter 3 Clifford algebras and Dirac operators
Chapter 4 The Spin groups
Chapter 5 Analytic properties of Dirac operators
Chapter 6 Hodge theory
Chapter 7 The heat and wave equations
Chapter 8 Traces and eigenvalue asymptotics
Chapter 9 Some non-compact manifolds
Chapter 10 The Lefschetz formula
Chapter 11 The index problem
Chapter 12 The Getzler calculus and the local index theorem
Chapter 13 Applications of the index theorem
Chapter 14 Witten's approach to Morse theory
Chapter 15 Atiyah's GAMMA-index theorem
References
by "Nielsen BookData"
