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 in nature, and the total curvature, which is geometric in nature. The Atiyah-Singer index formula, proved in 1963 for closed manifolds, is a higher dimensional analog of the Gauss-Bonnet formula that incorporates a functional analytic invariant into the formula called the index of an operator. This work helped Atiyah and Singer win the Abel prize in 2004. In 1975, the Atiyah-Singer formula was extended to manifolds with boundary and in the late 90's it was extended to manifolds that have corners, in some sense the easiest class of "singular spaces." In the lecture, I will discuss the Atiyah-Singer formula from the viewpoint of the Gauss-Bonnet formula, and then talk about some recent results in extending the Atiyah-Singer formula to manifolds that have corners.