The moduli space of compact Riemann surfaces or compact Riemann surfaces with punctures is one of the most important spaces in mathematics and has been extensively studied in algebraic geometry, complex geometry, topology and mathematics physics. A closely related space is the Teichmueller space of Riemann surfaces where the mapping class group acts, and the quotient is equal to the moduli space of Riemann surfaces. The mapping class group is a basic group intensively studied in geometric group theory. The Teichmueller space and hence the moduli space of Riemann surfaces admit several natural Riemannian metrics such as the Weil-Petersson metric, the Ricci metric and Poincare type metrics. The main goal of this proposal is to study the spectral theory of the moduli space with respect to these Riemannian metrics and to understand its relations with the geometry and topology of the moduli space. Pursuing similarities between the moduli space and locally symmetric spaces, and between mapping class groups and arithmetic groups has been fruitful, the second goal of this proposal is to study some related problems for locally symmetric spaces. Specifically, the proposal consists of the following 4 projects: (1) Spectral theory of the incomplete Weil-Petersson metric on the moduli space. (2) Spectral theory for complete Riemannian metrics on the moduli space: geometric scattering theory. (3) Simplicial volume and spines of the moduli space. (4) Equivariant spines of symmetric spaces and L^p-spectral theory of locally symmetric spaces.
A drum corresponds to a domain in the plane, and its tones correspond to the eigenvalues of the Laplace operator of the domain with the Dirichlet boundary condition. A perhaps naive question is how these eigenvalues are related to and reflect the shape of the drum, i.e., the geometry of the domain. For example, a very large drum has a low pitch. A famous question raised by Marc Kac in 1966 is "Can one hear the shape of a drum?". This question has been one of the motivating forces for the subject of spectral geometry. Domains are special examples of Riemannian manifolds, and mathematicians have been trying to understand geometry and spectral theory of various Riemannian manifolds, for example, closed surfaces inside the three dimensional Euclidean space. An important class of spaces comes from collections of mathematical objects sharing similar properties, the so-called moduli spaces. The main purpose of this proposal is to understand the geometry and spectral of moduli spaces of Riemann surfaces.
