Spectral Geometry for the Riemann Moduli Space

Abstract

A report on joint work with Lizhen Ji, Werner Müller and Andras Vasy.

It is only within the last few decades that the study of analytic and geometric problems in the setting of spaces with structured singularities has become a real focus in geometric analysis and partial differential equations. Unlike for the many traditional problems concerning spaces and equations with overall low regularity, the focus here is on spaces which are mostly smooth, but which have stratified singular sets and metrics (or coefficients of the PDE) which are adapted to these singularities. Because of this, one can expect much sharper results. There are many reasons for studying such spaces, chief amongst which is that from many points of view they are just as natural as smooth manifolds. Examples of this type of singular space appear in the study of algebraic and analytic varieties, configuration and moduli spaces, and even just as level sets of smooth functions (Morse-Bott theory). Granting the interest of these spaces, one then wishes to extend the techniques, methods and goals of geometric analysis to these settings.