Degenerate Microlocal Methods in Geometric Analysis

Stanford University, Stanford CA

Investigators

Abstract

Abstract Award: DMS-0505709 Principal Investigator: Rafe Mazzzeo The PI proposes several activities, all fitting in the framework of linear and nonlinear elliptic problems arising in geometric analysis on noncompact and singular manifolds. He proposes continuing his collaboration with Vasy on new microlocal techniques for constructing the resolvent on locally symmetric spaces of arbitrary rank. This approach has already led to a new proof of a meromorphic continuation of this resolvent, which in turn will have applications in scattering theory and to nonlinear elliptic equations on asymptotically symmetric Einstein spaces. A goal of this portion of the proposal is to create versatile methods, adapted from N-body Schroedinger theory, to open the analysis on higher rank spaces to a purely microlocal approach. This should lead to advances in geometric scattering theory on these spaces. The other part of the proposal discusses deformation theory of noncompact Ricci flat Einstein spaces and of Einstein spaces with conic singularities. Even for surfaces, the PI expects these methods to give new results about the existence of constant curvature and Ricci soliton metrics. Finally, the PI proposes to continue his role as director and teacher in SUMaC, the residential summer program for mathematically talented high school students, and also to continue his other mentoring and outreach activities. The overall area of research in this proposal concerns a study of the relationship between the geometry of higher dimensional curved spaces and the behaviour of solutions of certain natural equations on these spaces. One example is a subject called geometric scattering theory, which seeks to describe the behaviour of waves, measured at large timescales, and to see how these are influenced by the curvature of the underlying space. The PI has developed methods to study such problems on a natural class of spaces which appear in several different branches of mathematics. Other examples include the study of solutions to the Einstein equations, which are supposed to describe possible configurations for the large-scale shape of space. There are many such solutions; one way to organize them is to describe their deformation theory, i.e. to see how they may be altered continuously. The PI proposes to study these questions for certain of these spaces. A key novelty of his methods is the development of techniques from a field called microlocal analysis to these geometric problems. The nature of these topics will lead to interactions with mathematicians in diverse areas as well as physicists, and it is hoped that the results of this proposal may have some impact in these other areas. The PI is also actively involved in training and mentoring at various levels. His SUMaC program, now entering its eleventh year, is a proven success in motivating gifted high school students to continue their mathematical studies. He has also been active in graduate training and in collaborating with young postdoctoral researchers.