Topics in Mathematical Analysis

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Proposal DMS-0401260PI: F. Michael Christ- University of California, BerkeleyTitle: Topics in Mathematical AnalysisABSTRACTResearch will be conducted on a broad array of problemsin harmonic analysis, linear and nonlinear partial differential equations,several complex variables, Schroedinger operators,and ordinary differential equations.In harmonic analysis proper, the decay properties of multilinear oscillatoryintegral operators will be investigated, and Lebesgue space mapping propertiesof generalized Radon transforms will be analyzed in terms of underlying geometry. Techniques from harmonic analysiswill be used to develop a theory of almost everywhere WKB asymptotics for solutionsof ordinary differential equations depending on a parameter. In complex analysis, compactness of the d-bar Neumann problem, theanalytic hypoellipticity of related operators, and Toeplitz operators will be investigated. The Cauchy problem will be studied for nonlinear evolution equations, with the aim ofclarifying phenomena involving instablility and growth of norms of solutions. The spectra and generalized eigenfunctions of time-independent Schroedinger operators,in one and higher dimensions, will be investigated.The most fundamental laws of physical science are formulated as partial differentialequations. These equations are sometimes linear, and sometimes nonlinear, modelingvarious types of self-interaction. The nonlinear Schrodinger equation arises inconnection with Bose-Einstein condensates, fiber optics, and assorted other physicallydifferent phenomena. Time-independent Schrodinger operators with long-range potentialsmodel the quantum properties of disordered electrical media. Other sources of problems concerning differential equations are internal to mathematics,such as complex analysis in several variables. One of the most basic tools for the analysis of differential equations is harmonicanalysis. This research will focus both on the use of harmonic analysis methods tofurther the understanding of the nature of solutions of differential equations,and on the development of new tools within harmonic analysis which may ultimately providefurther insight into differential equations, as well as solving problems within harmonicanalysis itself.