## Culture and History of Mathematics

Seminar on the Atiyah – Singer Index Theorem

Richard S. Palais

The Atiyah-Singer index theorem, proved in 1963, is one of the highlights of twentieth century mathematics. It subsumed many classical results that connected topological properties of manifolds with their differential-geometric properties, and led to a lot of new research in topology, geometry, analysis and physics.

This book published in 1965, was one of the first expositions of this theorem. It is a document of historical interest and much can be learned from it even now.

Contents

1: Statement of the Theorem, Outline of the Proof- A. Borel. 2: Review of K-Theory-R. Solovay.

3: The Topological Index of an Operator Associated To A G-Structure-R. Solovay.

4: Differential Operators on Vector Bundles-R.S. Palais.

5: Analytical Indices of Some Concrete Operators-M. Solovay.

6: Review of Functional Analysis-R.S. Palais.

7: Fredholm Operators- R. S. Palais.

8: Chains of Hilbertian Spaces-R. S. Palais.

9: The Discrete Sobolev Chain of a Vector Bundle- R. S. Palais.

10: The Continuous Sobolev Chain of a Vector Bundle-R. S. Palais.

11: The Seeley Algebra-R. S. Palais.

12: Homotopy Invariance of the Index- R. S. Palais.

13: Whitney Sums- R. S. Palais.

14: Tensor Products- R. S. Palais.

15: Definition of 1a and 1t on K(M)-R. M. Solovay.

16: Construction of Intk-R. S. Palais and R. T. Seeley.

17: Cobordism Invariance of the Analytical Index-R. S. Palais and R. T. Seeley.

18: Hordism Groups of Bundles-E. E. Floyd.

19: The Index Theorem: Applications-R.M. Solovay.

Appendix I: The Index Theorem for Manifolds with Boundary-M.F. Atiyah

Appendix II: Non Stable Characteristic Classes and the Topological Index of Classical Elliptic Operators-W. Shih

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Culture and History of Mathematics/ 2

2005, 9788185931531, 376 pages, paper cover, Out of Print

Culture and History of Mathematics/ 2

2005, 9788185931531, 376 pages, paper cover, Out of Print