Microlocal Methods in Geometric Analysis

Stanford University, Stanford CA

Investigators

Abstract

This project is broadly concerned with the development of new techniques to study geometric and analytic problems on an important class of singular spaces. Geometry and mathematical physics have traditionally focused on smooth geometric objects called manifolds, but a slightly more general class of geometric objects called stratified spaces is equally ubiquitous. Despite this, only recently have specific techniques been developed appropriate to study detailed analytic problems on these spaces. One goal of this project is a continuation of the PI's long-term goal of establishing certain strong analytic techniques to study a wide variety of problems in geometric analysis which are set on such stratified spaces. Other goals of this project include applications of these techniques to problems occurring in gauge theory and in mathematical relativity. This has significant cross-disciplinary appeal between partial differential equations, geometric analysis and mathematical physics, and modulates between the parallel goals of developing general techniques and solving specific problems of interest in various disciplines. Geometric microlocal analysis has emerged as a powerful tool to study a range of problems concerning differential and pseudodifferential operators with degeneracies. Such operators arise in many applications in geometric analysis and mathematical physics. The PI's current research interest centers on the development of a calculus of iterated edge pseudodifferential operators which can be used as an investigative tool to study elliptic, parabolic and hyperbolic operators on stratified pseudomanifolds. The ultimate goal in this is a theory paralleling and generalizing the Boutet de Monvel calculus, which provides the most general and natural setting to study elliptic boundary problems on manifolds with boundary. The PI intends to use this calculus to study specific problems, including Kaehler-Einstein metrics with edge singularities along normal crossing divisors and singular solutions of the Kapustin-Witten equations. Closely related research projects include a new approach to study the asymptotic geometry of certain gauge-theoretic moduli spaces, in particular the moduli space of solutions to the Hitchin equations on a Riemann surface, and the continuation of a program to understand the blowup profiles of families of solutions of the vacuum Einstein constraint equations in mathematical relativity.