Elliptic Operators, Topology, and Asymptotic Methods / Edition 2

by John Roe

Overview

Ten years after publication of the popular first edition of this volume, the index theorem continues to stand as a central result of modern mathematics-one of the most important foci for the interaction of topology, geometry, and analysis. Retaining its concise presentation but offering streamlined analyses and expanded coverage of important examples and applications, Elliptic Operators, Topology, and Asymptotic Methods, Second Edition introduces the ideas surrounding the heat equation proof of the Atiyah-Singer index theorem.
The author builds towards proof of the Lefschetz formula and the full index theorem with four chapters of geometry, five chapters of analysis, and four chapters of topology. The topics addressed include Hodge theory, Weyl's theorem on the distribution of the eigenvalues of the Laplacian, the asymptotic expansion for the heat kernel, and the index theorem for Dirac-type operators using Getzler's direct method. As a "dessert," the final two chapters offer discussion of Witten's analytic approach to the Morse inequalities and the L2-index theorem of Atiyah for Galois coverings.
The text assumes some background in differential geometry and functional analysis. With the partial differential equation theory developed within the text and the exercises in each chapter, Elliptic Operators, Topology, and Asymptotic Methods becomes the ideal vehicle for self-study or coursework. Mathematicians, researchers, and physicists working with index theory or supersymmetry will find it a concise but wide-ranging introduction to this important and intriguing field.

Product Details

ISBN-13:	9780582325029
Publisher:	Taylor & Francis
Publication date:	01/06/1999
Series:	Chapman & Hall/CRC Research Notes in Mathematics Series , #395
Edition description:	Revised
Pages:	218
Product dimensions:	6.12(w) x 9.19(h) x (d)

Table of Contents

Chapter 1. Resume of Riemannian geometry, Chapter 2. Connections, curvature, and characteristic classes, 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 T-index theorem, References

Editorial Reviews

Based on a course of lectures delivered at Oxford. Introduces the circle of ideas surrounding the heat-equation proof of the Atiyeh- 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. Analysis is worked out in the context of Dirac operators on Clifford bundles. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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