Topics in Mathematical Analysis

提案DMS-0401260 PI：F. Michael Christ-加州大学伯克利分校数学分析专题摘要研究内容包括调和分析、线性和非线性偏微分方程、多复变量、薛定谔算子和常微分方程。在调和分析中，将研究多线性积分算子的衰减性质，和Lebesgue空间映射性质的广义Radon变换将分析的基础几何。从调和分析的技术将被用来发展一个理论的几乎处处WKB渐近解的常微分方程依赖于一个参数。在复分析中，d-杆Neumann问题的紧性，相关算子的解析亚椭圆性，以及Toeplitz算子将被研究。本文研究非线性发展方程的柯西问题，目的在于阐明解的不稳定性和范数增长性。本课程将研究一维和更高维的与时间无关的薛定谔算子的谱和广义本征函数，并将物理学最基本的定律表述为偏微分方程。这些方程有时是线性的，有时是非线性的，模拟了各种类型的自相互作用。非线性薛定谔方程的产生与玻色-爱因斯坦凝聚、纤维光学和其他各种不同的物理现象有关。含长程势的非时性薛定谔算符模拟了无序电介质的量子特性。微分方程问题的其他来源是数学内部的，例如多变量的复分析。分析微分方程最基本的工具之一是调和分析。这项研究将集中在调和分析方法的使用，以进一步了解微分方程的解的性质，并在调和分析中的新工具，最终可能提供进一步深入了解微分方程的发展，以及在harmonicanalysis本身解决问题。

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.