Using heat equation methods, the index of an elliptic operator can be computed by a local formula. In this series of lectures, we will review the necessary analysis for defining the index of an elliptic operator, and derive a local formula for the index.

Using heat kernel methods developed by Vaillant, we will show how to obtain a local index formula for families of d-bar operators parametrized by the Teichmuller space of Riemann surfaces of genus g with n punctures. The formula also holds on the corresponding moduli space in the sense of orbifolds where it can be written in terms of Mumford-Morita-Miller classes. As we will indicate, the degree two part of the formula gives the curvature of the corresponding determinant line bundle equipped with the Quillen connection, a result originally obtained by Takhtajan and Zograf.

The Poincaré polynomial of a Weyl group calculates the Betti numbers of the projective homogeneous space G/B, while the h-vector of a simple polytope calculates the Betti numbers of the corresponding rationally smooth toric variety. There is a common generalization of these two extremes called the H-polynomial. It applies to projective, homogeneous spaces, toric varieties and, much more generally, to any algebraic variety X where there is a connected, solvable, algebraic group acting with a finite number of orbits. We illustrate this situation by describing the H-polynomials of certain projective (G x G)-varieties X, where G is a semisimple group and B is a Borel subgroup of G. This description is made possible by finding an appropriate cellular decomposition for X and then describing the cells combinatorially in terms of the underlying monoid of (B x B)-orbits. The most familiar example here is the wonderful compactification of a semisimple group of adjoint type.