Degenerate Microlocal Methods and Geometric Analysis

Stanford University, Stanford CA

Investigators

Abstract

NSF Proposal DMS - 0204730: Rafe Mazzeo The proposed work in this project involves the continuing development of analytic tools to study a variety of problems in geometric analysis. These include the global theory of the moduli space of constant mean curvature surfaces in Euclidean space, the application of new gluing techniques to construct new types of Einstein metrics, a more detailed study of the analytic and geometric behaviour of conformally compact Einstein metrics and the deformation theory of such metrics, with special attention to self-dual conformally compact Einstein metrics in four dimensions. The proposed techniques here include further extensions of Cauchy data matching, as developed by the PI and Pacard, as well as refinements of the PI's `edge calculus'.Abstract for Other parts of the proposal involve use of the pseudodifferential calculus of fibred boundary operators, as developed by the PI and Melrose, to the study of gravitational instantons, particularly their L2 cohomology. Finally, the PI and Vasy propose to investigate the connections between geometric scattering theory on symmetric spaces of rank greater than one, as well as a class of spaces asymptotically modelled on these, and the microlocal theory of quantum N-body scattering. For the human resources component, the PI proposes to continue his directorship of the Stanford University Math Camp, a residential summer program for talented high school students, and also to continue his other outreach efforts to disseminate mathematics appreciation to the general public. From a more general point of view, the PI's research concerns problems arising in geometry and analysis involving what are known as curvature equations (the theory of Einstein metrics in general relativity being the best-known case) as well as scattering theory on spaces which possess high degrees of symmetry `at infinity'. A central concern throughout is the application of somewhat novel techniques from harmonic and microlocal analysis to these problems. The theme is that one should develop analytic techniques which are specifically adapted to each geometric problem, and these geometric settings in turn should suggest new developments in the analytic technology. This approach has proved very successful in the PI's previous research. The problems considered here are inspired by main trends in various aspects of mathematical physics, most specifically the two somewhat separate fields of quantum scattering and some parts of string theory. Some of the current and proposed work has already stimulated interest on the part of some communities of physicists, and their intuitions provide an interesting guide for further mathematical directions in this work. Beyond these motivations, the PI regards this particular interplay between geometry and analysis as an important one, particularly because the types of geometric objects studied here are becoming increasingly important in many other places in mathematics. The PI has also undertaken extensive human resources development, including the above-mentioned summer program, and is active in mentoring a number of young researchers.