Index Theory with Applications to Mathematics and Physics (International Press) - Hardcover

David D. Bleecker (University Of Hawaii At Manoa); Bernhelm Booß-Bavnbek (Roskilde Universitet Denmark)

Hardcover

ISBN 10: 1571462643 ISBN 13: 9781571462640

Publisher: International Press of Boston, 2013

Synopsis

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 new 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 this title

About the Author

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, for more than thirty years, and is now retired. He is known for his work in differential geometry and gauge-theoretic physics. He is 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. (Universitaet 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 Hoyrup as co-editor; the monograph entitled BetaSys: Systems Biology of Regulated Exocytosis in Pancreatic beta-cells, as main editor; and this book's first German-language predecessor, Topologie und Analysis Eine Einführung in die Atiyah-Singer-Indexformel.

Review

This book ... describes in impressive detail one of the greatest achievements of 20th-century mathematics, the Atiyah-Singer index theorem. As a bonus, the author includes an extensive and detailed discussion of some of the applications of the theorem and more recent developments in the direction of gauge theoretic physical models. For the student, the gold lining in the book is that all the background is there -- starting at a level that is accessible to anyone with a reasonable mathematical education. There is also a bridge to the point of view from the physics side. The student can dip into the book at many points and use it to learn individual topics. The discussion covers a very broad sweep of mathematics which will be of interest to anyone aiming to understand geometric analysis, whether or not gauge theories are their primary focus. Two proofs of the Atiyah Singer theorem are given, one based on K-theory and the other on the heat kernel approach ... The authors have included many discussion sections that illuminate the thinking behind the more general theory ... There is much valuable historical context recorded here. Moreover, it captures the flavour of contemporary mathematics, with the interaction of geometric, topological, analytic, and physical perspectives and methods being in the forefront of the discussion. --Alan Carey, Mathematical Sciences Institute at Australian National University, Canberra

Professors Bleecker and Booss-Bavnbek have followed ... developments in index theory from the beginning, and made original contributions of their own ... Assuming only basic analysis and algebra, [this book] gives detailed constructions and proofs for all the necessary concepts, along with illuminating digressions on the various paths through the rich territory of index theory. --Robert Seeley, Professor Emeritus, University of Massachusetts, Boston

Two or three famous Index Formulas discovered and proved in the course of middle decades of the last century are some of the highest peaks in a mountain country rising from the vast plains of functional analysis, theory of smooth manifolds, and homotopical topology. This treatise, written with ambition, wit and (mathematical) eloquence, strives to combine the qualities of a guide-book, historical chronicles, and a hiking manual for enthusiastic travellers and budding future explorers of this vast territory. Any reader possessing will and enthusiasm can profit from studying (parts of) this book and enjoy finding his or her own path through this land. --Yuri Manin, Max Planck Institute for Mathematics, Bonn, Germany

Edition details

• Publisher: International Press of Boston
• Publication date: 2013
• ISBN 10: 1571462643
• ISBN 13: 9781571462640
• Binding: Hardcover
• Language: English
• Number of pages: 792