| DATE: | Wednesday, March 20, 2002 |
| TIME: | 4:40-5:40 PM |
| PLACE: | LN 2205 |
| SPEAKER: | Paul Loya, M.I.T. |
| TITLE: | Index formulas: a bridge between Topology, Geometry, Analysis, and Linear Algebra |
A fascinating feature of the Gauss-Bonnet formula is that it relates two invariants of a surface that are apparently distinct in nature, the Euler characteristic, which is topological, and the total curvature, which is geometric. The Atiyah-Singer index formula, proved in 1963 for closed manifolds, is a higher dimensional analog of the Gauss-Bonnet formula that incorporates an analytical invariant into the formula, the index of an operator. In 1975, the Atiyah-Singer formula was extended to manifolds with boundary, and only recently has it been extended to manifolds that have corners. In the lecture, I will discuss the Atiyah-Singer formula from the viewpoint of the Gauss-Bonnet formula, and then talk about recent joint work with Richard Melrose in extending the Atiyah-Singer formula to manifolds that have corners.