Degenerate Microlocal Methods in Geometric Analysis

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AbstractAward: DMS-0505709Principal Investigator: Rafe MazzzeoThe PI proposes several activities, all fitting in the frameworkof linear and nonlinear elliptic problems arising in geometricanalysis on noncompact and singular manifolds. He proposescontinuing his collaboration with Vasy on new microlocaltechniques for constructing the resolvent on locally symmetricspaces of arbitrary rank. This approach has already led to a newproof of a meromorphic continuation of this resolvent, which inturn will have applications in scattering theory and to nonlinearelliptic equations on asymptotically symmetric Einstein spaces. Agoal of this portion of the proposal is to create versatilemethods, adapted from N-body Schroedinger theory, to open theanalysis on higher rank spaces to a purely microlocal approach.This should lead to advances in geometric scattering theory onthese spaces. The other part of the proposal discussesdeformation theory of noncompact Ricci flat Einstein spaces andof Einstein spaces with conic singularities. Even for surfaces,the PI expects these methods to give new results about theexistence of constant curvature and Ricci solitonmetrics. Finally, the PI proposes to continue his role asdirector and teacher in SUMaC, the residential summer program formathematically talented high school students, and also tocontinue his other mentoring and outreach activities.The overall area of research in this proposal concerns a study ofthe relationship between the geometry of higher dimensionalcurved spaces and the behaviour of solutions of certain naturalequations on these spaces. One example is a subject calledgeometric scattering theory, which seeks to describe thebehaviour of waves, measured at large timescales, and to see howthese are influenced by the curvature of the underlyingspace. The PI has developed methods to study such problems on anatural class of spaces which appear in several differentbranches of mathematics. Other examples include the study ofsolutions to the Einstein equations, which are supposed todescribe possible configurations for the large-scale shape ofspace. There are many such solutions; one way to organize them isto describe their deformation theory, i.e. to see how they may bealtered continuously. The PI proposes to study these questionsfor certain of these spaces. A key novelty of his methods is thedevelopment of techniques from a field called microlocal analysisto these geometric problems. The nature of these topics will leadto interactions with mathematicians in diverse areas as well asphysicists, and it is hoped that the results of this proposal mayhave some impact in these other areas. The PI is also activelyinvolved in training and mentoring at various levels. His SUMaCprogram, now entering its eleventh year, is a proven success inmotivating gifted high school students to continue theirmathematical studies. He has also been active in graduatetraining and in collaborating with young postdoctoralresearchers.