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Elliptic Operators, Topology, and Asymptotic Methods, Second Edition : Research Notes in Mathematics Series - John Roe

Elliptic Operators, Topology, and Asymptotic Methods, Second Edition

Research Notes in Mathematics Series

By: John Roe


Published: 6th January 1999
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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.

ResumĀ© of Riemannian geometry
Connections, curvature, and characteristic class
Clifford algebras and Dirac operators
The Spin groups
Analytic properties of Dirac operators
Hodge theory
The heat and wave equations
Traces and eigenvalue asymptotics
Some non-compact manifolds
The Lefschetz formula
The index problem
The Getzler calculus and the local index theorem
Applications of the index theorem
Witten's approach to Morse theory
Atiyah's GAMMA-index theorem
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780582325029
ISBN-10: 0582325021
Series: Research Notes in Mathematics Series
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 209
Published: 6th January 1999
Publisher: Pearson Education Limited
Country of Publication: GB
Dimensions (cm): 23.5 x 16.51  x 1.27
Weight (kg): 0.34
Edition Number: 2
Edition Type: New edition