Harmonic Analysis, Partial Differential Equations, and Complex Analysis

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Research will be conducted into a broad array of problems in Fourier analysis, partial differential equations, and complex analysis. The PI will investigate upper bounds for multilinear oscillatory integral operators, seeking to further develop the stationary phase method and to characterize what it means for a phase function to give rise to genuinely oscillatory behavior in the multilinear context. The analytic and the underlying geometric theory of Radon-like transforms will be developed, with emphasis on Lebesgue space inequalities. The PI will use Fourier analytic methods to investigate solutions of the nonlinear Schrodinger equation, seeking to rigorously establish strongly nonlinear behavior, to analyze the transfer of energy between Fourier modes and between scales, to understand instability and stability of solutions, and to shed light on uniqueness questions. He will work to develop geometric criteria, phrased in terms of the symplectic geometry of phase space, which characterize compactness and hypoellipticity for the Neumann problem for the Cauchy-Riemann complex in several complex variables, and for related linear partial differential equations.For centuries, the fundamental laws of physical science have been most precisely formulated as differential equations, which express relationships between physical quantities and the rates at which they change. Fourier analysis was first introduced as a tool for the solution of the specific differential equation which governs heat flow, and has subsequently become a ubiquitous tool in engineering, in applied physical science, in theoretical physics, and throughout mathematics itself. This project is concerned with several different types of differential equations, with potential applications of Fourier analysis to them, and with fundamental issues internal to Fourier theory. The most challenging mathematical issues around differential equations today concern nonlinear equations, which model self-interacting physical systems. Basic nonlinear interactions are multilinear, e.g. one physical quantity multiplied by the rate of change of another; these appear for instance in the Navier-Stokes equations describing viscous fluid flow, and in the nonlinear Schrodinger equation, describing quantum optics in certain situations. Multilinear Fourier analysis is potentially a valuable tool in this context, yet is only partially developed and presents challenges. In one part of the project, multilinear Fourier operator integrals themselves are the object of study. In another, the PI will focus more narrowly on the behavior of solutions of the nonlinear Schrodinger equation, and will employ multilinear Fourier integrals as tools with the hope of analyzing stable and unstable behavior. As an integral part of this project, the PI will mentor individual PhD students in both research and teaching, so that they can in turn become productive researchers and college/university level teachers.