School of Mathematics

Resonances are complex analogs of eigenvalues for Laplacians on noncompact manifolds, arising in long time resonance expansions of linear waves. We prove a Weyl type asymptotic formula for the number of resonances in a strip, provided that the set of trapped geodesics is r-normally hyperbolic for large r and satisfies a pinching condition. Our dynamical assumptions are stable under small smooth perturbations and motivated by applications to black holes. We also establish a high frequency analog of resonance expansions.

In its simplest form, Hirzebruch's 1953 Riemann-Roch theorem is an identity between certain combinations of Hodge numbers on the one hand and certain combinations of Chern numbers on the other. I will show that there are no other such identities, beyond HRR. I will also discuss the topological non-invariance of Hodge and Chern numbers, exhibiting systematic examples of diffeomorphic varieties with distinct Hodge and Chern numbers. This leads to the complete solution of a problem posed by Hirzebruch in connection with Riemann-Roch. (This talk is based in part on joint work with S.