Topics in Mathematical Analysis

提案DMS-0401260PI: F. Michael Christ-加州大学伯克利分校题目：数学分析主题摘要：研究将在谐波分析、线性和非线性偏微分方程、几种复变量、薛定谔算子和常微分方程等广泛的问题上进行。在谐波分析中，将研究多线性振荡积分算子的衰减性质，并从基础几何的角度分析广义Radon变换的勒贝格空间映射性质。谐波分析的技术将用于发展一个理论，几乎处处WKB渐近解的常微分方程依赖于一个参数。在复分析中，研究了d-bar Neumann问题的紧性、相关算子的解析亚椭圆性和Toeplitz算子。本文将研究非线性演化方程的柯西问题，目的是澄清涉及解的不稳定性和范数增长的现象。在一维和高维中，将研究与时间无关的薛定谔算子的谱和广义特征函数。物理科学最基本的定律是用偏微分方程表述的。这些方程有时是线性的，有时是非线性的，模拟各种类型的自相互作用。非线性薛定谔方程与玻色-爱因斯坦凝聚、光纤和其他各种不同的物理现象有关。具有远程势的时间无关薛定谔算符模拟无序电介质的量子特性。关于微分方程的问题的其他来源是数学内部的，例如多个变量的复分析。微分方程分析最基本的工具之一是谐波分析。这项研究将集中在谐波分析方法的使用上，以进一步理解微分方程解的本质，并在谐波分析中开发新的工具，这可能最终提供对微分方程的进一步了解，以及解决谐波分析本身的问题。

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.