Customer Reviews

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

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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.