Degenerate Microlocal Methods and Geometric Analysis

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NSF Proposal DMS - 0204730: Rafe MazzeoThe proposed work in this project involves the continuingdevelopment of analytic tools to study a variety of problemsin geometric analysis. These include the global theory ofthe moduli space of constant mean curvature surfaces in Euclideanspace, the application of new gluing techniques to constructnew types of Einstein metrics, a more detailed study of theanalytic and geometric behaviour of conformally compact Einsteinmetrics and the deformation theory of such metrics, with specialattention to self-dual conformally compact Einstein metrics infour dimensions. The proposed techniques here include furtherextensions of Cauchy data matching, as developed by the PI andPacard, as well as refinements of the PI's `edge calculus'.Abstract for Other parts of the proposal involve use of the pseudodifferentialcalculus of fibred boundary operators, as developed by the PI andMelrose, to the study of gravitational instantons, particularlytheir L2 cohomology. Finally, the PI and Vasy propose toinvestigate the connections between geometric scattering theoryon symmetric spaces of rank greater than one, as well as aclass of spaces asymptotically modelled on these, and themicrolocal theory of quantum N-body scattering. For the humanresources component, the PI proposes to continue his directorshipof the Stanford University Math Camp, a residential summerprogram for talented high school students, and also to continuehis other outreach efforts to disseminate mathematics appreciationto the general public.From a more general point of view, the PI's research concernsproblems arising in geometry and analysis involving what areknown as curvature equations (the theory of Einstein metrics ingeneral relativity being the best-known case) as well as scatteringtheory on spaces which possess high degrees of symmetry `at infinity'.A central concern throughout is the application of somewhat noveltechniques from harmonic and microlocal analysis to these problems. Thetheme is that one should develop analytic techniques which arespecifically adapted to each geometric problem, and these geometricsettings in turn should suggest new developments in the analytictechnology. This approach has proved very successful in the PI'sprevious research. The problems considered here are inspired by maintrends in various aspects of mathematical physics, most specificallythe two somewhat separate fields of quantum scattering and some partsof string theory. Some of the current and proposed work has alreadystimulated interest on the part of some communities of physicists,and their intuitions provide an interesting guide for further mathematicaldirections in this work. Beyond these motivations, the PI regards thisparticular interplay between geometry and analysis as an important one,particularly because the types of geometric objects studied here arebecoming 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 inmentoring a number of young researchers.