Index Theory with Applications to Mathematics and Physics

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Description

Index Theory with Applications to Mathematics and Physics 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 work, authors David Bleecker and Bernhelm Booß-Bavnbek 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, which illuminate the thinking behind the more general theory. A detailed bibliography and index facilitate the orientation.

About the Authors

David D. Bleecker, Ph.D. (University of California at Berkeley under S.-S. Chern) taught on the faculty of the Department of Mathematics at the University of Hawaii, Manoa. Dr. Bleecker passed away in 2016. He was known for his work in differential geometry and gauge-theoretic physics. He was also known to a larger audience as a textbook author of unrivaled care for details. Among his student textbooks are the widely-read Basic Partial Differential Equations (International Press, 1996) with George Csordas; the more advanced Gauge Theory and Variational Principles; and this book’s predecessor with Bernhelm Booß-Bavnbek, Topology and Analysis: The Atiyah-Singer Index Formula and Gauge-Theoretic Physics.

Bernhelm Booß-Bavnbek, Ph.D. (Universität Bonn under F. Hirzebruch) is senior lecturer in mathematics and mathematical modeling with the Mathematics-Physics Group IMFUFA at Roskilde University (Denmark), where he has been for more than thirty years. He is known for his work in global analysis and spectral geometry. He is also known to a larger audience as a textbook author and editor with lively interest in a broad sweep of mathematics, applications of mathematics, and communication about mathematics. Among his research monographs are the now-classic treatise Elliptic boundary problems for Dirac operators, co-written with Krzysztof Wojciechowski; the volume Mathematics and War, with Jens Høyrup as co-editor; the monograph entitled BetaSys: Systems Biology of Regulated Exocytosis in Pancreatic ß-cells, as main editor; and this book’s first German-language predecessor, Topologie und Analysis – Eine Einführung in die Atiyah-Singer-Indexformel.

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