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Index Theory with Applications to Mathematics and Physics
David D. Bleecker, University of Hawaii at Manoa, Honolulu, and Bernhelm Booß-Bavnbek, Roskilde Universitet, Denmark
A publication of International Press.
cover
International Press
2013; 792 pp; hardcover
ISBN-10: 1-57146-264-3
ISBN-13: 978-1-57146-264-0
List Price: US$95
Member Price: US$76
Order Code: INPR/96
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This book describes, explains, and explores the Index Theorem of Atiyah and Singer, one of the truly great accomplishments of twentieth-century mathematics whose influence continues to grow, fifty years after its discovery. The Index Theorem has given birth to many mathematical research areas and exposed profound connections between analysis, geometry, topology, algebra, and mathematical physics. Hardly any topic of modern mathematics stands independent of its influence.

In this ambitious new work, the authors give two proofs of the Atiyah-Singer Index Theorem in impressive detail: one based on \(K\)-theory and the other on the heat kernel approach. As a preparation for this, the authors explain all the background information on such diverse topics as Fredholm operators, pseudo-differential operators, analysis on manifolds, principal bundles and curvature, and \(K\)-theory carefully and with concern for the reader. Many applications of the theorem are given, as well as an account of some of the most important recent developments in the subject, with emphasis on gauge theoretic physical models and low-dimensional topology.

The 18 chapters and two appendices of the book introduce different topics and aspects, often beginning from scratch, without presuming full knowledge of all the preceding chapters. Learning paths, through a restricted selection of topics and sections, are suggested and facilitated. The chapters are written for students of mathematics and physics: some for the upper-undergraduate level, some for the graduate level, and some as an inspiration and support for researchers.

Index Theory with Applications to Mathematics and Physics is a textbook, a reference book, a survey, and much more. Written in a lively fashion, it contains a wealth of basic examples and exercises. The authors have included many discussion sections that are both entertaining and informative and which illuminate the thinking behind the more general theory. A detailed bibliography and index facilitate the orientation.

A publication of International Press. Distributed worldwide by the American Mathematical Society.

Readership

Undergraduate and graduate students as well as research mathematicians interested in the Index Theorem of Atiyah-Singer.

Table of Contents

Part I. Operators with Index and Homotopy Theory
  • Fredholm operators
  • Analytic methods. Compact operators
  • Fredholm operator topology
  • Wiener-Hopf operators
Part II. Analysis on Manifolds
  • Partial Differential equations in euclidean space
  • Differential operators over manifolds
  • Sobolev spaces (crash course)
  • Pseudo-differential operators
  • Elliptic operators over closed manifolds
Part III. The Atiyah-Singer Index Formula
  • Introduction to topological \(K\)-Theory
  • The Index formula in the Euclidean case
  • The Index theorem for closed manifolds
  • Classical applications (survey)
Part IV. Index Theory in Physics and the Local Index Theorem
  • Physical motivation and overview
  • Geometric preliminaries
  • Gauge theoretic instantons
  • The local index theorem for twisted Dirac operators
  • Seiberg-Witten theory
  • Appendix A. Fourier series and integrals-fundamental principles
  • Appendix B. Vector bundles
  • Bibliography
  • Index of notation
  • Index of names/authors
  • Subject index
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