Research will be conducted on a broad array of problems in 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 oscillatory integral operators will be investigated, and Lebesgue space mapping properties of generalized Radon transforms will be analyzed in terms of underlying geometry. Techniques from harmonic analysis will be used to develop a theory of almost everywhere WKB asymptotics for solutions of ordinary differential equations depending on a parameter. In complex analysis, compactness of the d-bar Neumann problem, the analytic hypoellipticity of related operators, and Toeplitz operators will be investigated. The Cauchy problem will be studied for nonlinear evolution equations, with the aim of clarifying 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 differential equations. These equations are sometimes linear, and sometimes nonlinear, modeling various types of self-interaction. The nonlinear Schrodinger equation arises in connection with Bose-Einstein condensates, fiber optics, and assorted other physically different phenomena. Time-independent Schrodinger operators with long-range potentials model 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 harmonic analysis. This research will focus both on the use of harmonic analysis methods to further the understanding of the nature of solutions of differential equations, and on the development of new tools within harmonic analysis which may ultimately provide further insight into differential equations, as well as solving problems within harmonic analysis itself.
