CBMS Regional Conference in the Mathematical Sciences - Global Harmonic Analysis - June 2011

CBMS 数学科学区域会议 - 全球调和分析 - 2011 年 6 月

• 批准号: 1040927
• 负责人: Peter Perry
• 金额: $ 3.64万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-01-15 至 2011-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1040927&HistoricalAwards=false
• 关键词: CBMS; Regional; Conference; Mathematical; Sciences

Global Harmonic AnalysisNSF/CBMS Regional Conference in the Mathematical SciencesUniversity of Kentucky, June 20-24, 2011Global harmonic analysis may be viewed as an extension of classicalFourier theory on the line and on the circle to the geometric setting ofRiemannian manifolds. Riemannian manifolds provide models of both com-pletely integrable and chaotic dynamical systems, and also provide a settingin which the connection between "classical" and "quantum" behavior can becarefully studied. Moreover, just as harmonic analysis on the line and thecircle are a fundamental tool in the study of both linear and nonlinear differ-ential equations in these settings, so global harmonic analysis is fundamentalto the study of di¤erential equations, such as the wave and Schrödinger equa-tion, on manifolds. On a general Riemannian manifold, harmonic analysis isthe study of eigenfunctions of the Laplacian in the given Riemannian metric.Global harmonic analysis refers to the use of the global dynamics of geo-desic fow on the manifold to study the large-eigenvalue asymptotics of theeigenfunctions and eigenvalues of the Laplacian on a manifold. Since globaleigenfunctions of the Laplacian on a manifold are eigenfunctions of the wavegroup, and the wave group propagates singularities along geodesics, the high-frequency properties of eigenfunctions are connected to the long-time dy-namics of the geodesic flow. Two cases of particular interest are quantumcomplete integrability, where the underlying geodesic flow is completely in-tegrable, and quantum chaos, where the underlying geodesic flow is ergodic.Professor Steve Zelditch of Northwestern University will given ten lectureson global harmonic analysis. He will begin with basic examples (constantcurvature spaces), develop material on pseudodi¤erential and Fourier inte-gral operators, derive trace formulas including the celebrated Duistermaat-Guillemin trace formula, and discuss applications to properties of eigenfunc-tions and eigenvalues and inverse eigenvalue problems. Completely integrableand ergodic systems will be considered in depth.

全局调和分析美国肯塔基大学数学科学区域会议，2011 年 6 月 20-24 日全局调和分析可以被视为经典傅里叶理论在直线和圆上对黎曼流形几何设置的延伸，提供了黎曼流形的模型。既是完全可积的混沌动力系统，也提供了一种将“经典”和“经典”之间的联系联系起来的环境。此外，正如直线和圆的调和分析是研究线性和非线性微分方程的基本工具一样，全局调和分析也是微分研究的基础。流形上的方程，例如波动方程和薛定谔方程 在一般黎曼流形上，调和分析是研究流形中拉普拉斯算子的本征函数。全局调和分析是指利用流形上的测地线流的全局动力学来研究流形上拉普拉斯算子的特征函数和特征值的大特征值渐近性。流形是波群的本征函数，波群沿着测地线传播奇点，高频特性特征函数与测地线流的长期动力学相关的两个特别令人感兴趣的情况是量子完全可积性，其中基础测地线流是完全不可积的，以及量子混沌，其中基础测地线流是遍历的。教授西北大学的 Steve Zelditch 将发表十场关于全局调和分析的讲座，他将从基本示例（常曲率空间）开始，开发有关伪微分和傅里叶的材料。积分算子，推导出迹公式，包括著名的 Duistermaat-Guillemin 迹公式，并深入讨论特征函数和特征值的性质以及逆特征值问题的应用。