Topology and Analysis: The Atiyah-Singer Index Formula and Gauge-Theoretic Physics

By B. Booss

The Motivation. With intensified use of mathematical ideas, the methods and techniques of the various sciences and those for the solution of practical problems demand of the mathematician not only greater readi- ness for extra-mathematical applications but also more comprehensive orientations within mathematics. In applications, it is frequently less important to draw the most far-reaching conclusions from a single mathe- matical idea than to cover a subject or problem area tentatively by a proper variety of mathematical theories. To do this the mathematician must be familiar with the shared as weIl as specific features of differ- ent mathematical approaches, and must have experience with their inter- connections. The Atiyah-Singer Index Formula, one of the deepest and hardest results in mathematics, probably has wider ramifications in topology and analysis than any other single result (F. Hirzebruch) and offers perhaps a particularly fitting example for such an introduction to Mathematics: In spi te of i ts difficulty and immensely rich interrela- tions, the realm of the Index Formula can be delimited, and thus its ideas and methods can be made accessible to students in their middle * semesters. In fact, the Atiyah-Singer Index Formula has become progressively easier and more transparent over the years. The discovery of deeper and more comprehensive applications (see Chapter 111. 4) brought with it, not only a vigorous exploration of its methods particularly in the many- facetted and always new presentations of the material by M. F.

• Format: Paperback
• Language: English
• ISBN: 0387961127
• ISBN13: 9780387961125
• Release Date: December 1984
• Publisher: Springer
• Length: 451 Pages
• Weight: 1.50 lbs.
• Dimensions: 0.9" x 6.1" x 9.2"

Customer Reviews

The most readable intro to Atiyah-Singer and Donaldson

The first edition of Booss' work was intended as a leisurely introduction to the Atiyah-Singer Index Theorem with minimal required background knowledge. It covered the first three chapters of the English translation:Operators with Index;Analysis on Manifolds;The Atiyah-Singer Index Formula.These three chapters cover a huge territory in few, but evocative, words, from undergraduate topics like Sturm-Liouville theory through the current view of partial differential equations on manifolds. Of all the expositions of the Atiyah-Singer Index Theorem that I've seen, this supplies the most background and motivation. While Atiyah and Singer still appear to be brilliant, they're not quite as god-like - the inspiration for their results are exposed, so the reader may imagine doing similarly amazing work. The original intent of this book was complicated by the work of Mike Freedman and Simon Donaldson on the topology and geometry of four-dimensional manifolds. Extending the techniques introduced in the first three chapters and inventing new ones, Freedman and Donaldson pushed our understanding of four-manifolds, the substrate of spacetime physics, far beyond the bounds that were known before 1983. This was addressed in the English edition of Booss' book by the addition of a fourth chapter,"The Index Formula and Gauge-Theoretical Physics", written with his usual clarity by Dave Bleecker, who was also the primary translator of the first three chapters.Much of the material in this book is part of the background folklore for workers in theoretical physics (especially things related to string theory) and several areas in mathematics. Springer should make it easier to find.