International Press of Boston, Inc. 2013; 792 pp; hardcover ISBN10: 1571462643 ISBN13: 9781571462640 List Price: US$95 Member Price: US$76 Order Code: INPR/96
 This book describes, explains, and explores the Index Theorem of Atiyah and Singer, one of the truly great accomplishments of twentiethcentury 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 AtiyahSinger 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, pseudodifferential 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 lowdimensional 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 upperundergraduate 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 of Boston, Inc. Distributed worldwide by the American Mathematical Society. Readership Undergraduate and graduate students as well as research mathematicians interested in the Index Theorem of AtiyahSinger. Table of Contents Part I. Operators with Index and Homotopy Theory  Fredholm operators
 Analytic methods. Compact operators
 Fredholm operator topology
 WienerHopf operators
Part II. Analysis on Manifolds  Partial Differential equations in euclidean space
 Differential operators over manifolds
 Sobolev spaces (crash course)
 Pseudodifferential operators
 Elliptic operators over closed manifolds
Part III. The AtiyahSinger 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
 SeibergWitten theory
 Appendix A. Fourier series and integralsfundamental principles
 Appendix B. Vector bundles
 Bibliography
 Index of notation
 Index of names/authors
 Subject index
