Harmonic Analysis, Partial Differential Equations, and Complex Analysis

该奖项由2009年美国复苏和再投资法案(公法111-5)资助。将对傅立叶分析、偏微分方程式和复数分析中的一系列广泛问题进行研究。PI将研究多线性振荡积分算子的上界，寻求进一步发展驻相方法，并刻画相函数在多线性背景下产生真正振荡行为的意义。本课程将发展类Radon变换的解析理论和基本几何理论，重点研究勒贝格空间不等式。PI将使用傅立叶分析方法来研究非线性薛定谔方程的解，寻求严格地建立强非线性行为，分析傅立叶模式之间和尺度之间的能量转移，了解解的不稳定性和稳定性，并阐明唯一性问题。他将致力于发展几何准则，用相空间的辛几何来描述几个复变量中柯西-黎曼复的诺依曼问题的紧致性和亚椭圆性，以及相关的线性偏微分方程组。几个世纪以来，物理科学的基本定律一直被最精确地表述为微分方程式，它表达了物理量和它们变化的速率之间的关系。傅立叶分析最初是作为解决控制热流的特定微分方程式的工具而引入的，随后在工程、应用物理科学、理论物理和整个数学本身中成为一种普遍的工具。这个项目涉及几种不同类型的微分方程，以及傅里叶分析对它们的潜在应用，以及傅里叶理论的基本问题。今天围绕微分方程最具挑战性的数学问题涉及到非线性方程，它模拟了自相互作用的物理系统。基本的非线性相互作用是多线性的，例如一个物理量乘以另一个物理量的变化率；例如，它们出现在描述粘性流体流动的Navier-Stokes方程中，以及在描述某些情况下的量子光学的非线性薛定谔方程中。多线性傅立叶分析在这种情况下可能是一个有价值的工具，但只有部分开发，并提出了挑战。在项目的一个部分，多线性傅立叶算子积分本身就是研究的对象。在另一种情况下，PI将更狭隘地关注非线性薛定谔方程的解的行为，并将使用多线性傅立叶积分作为工具，希望分析稳定和不稳定的行为。作为该项目不可或缺的一部分，PI将在研究和教学方面指导个别博士生，使他们能够反过来成为多产的研究人员和学院/大学水平的教师。

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.