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

全球调和分析NSF/CBMS区域会议在数学科学肯塔基州大学，2011年6月20日至24日全球调和分析可以被看作是经典傅立叶理论的延伸线上和圆上的几何设置黎曼流形。黎曼流形提供了完全可积和混沌动力学系统的模型，也提供了一个可以仔细研究“经典”和“量子”行为之间联系的环境。此外，正如线和圆上的调和分析是研究这些情形下的线性和非线性微分方程的基本工具一样，整体调和分析也是研究流形上的波动方程和薛定谔方程等微分方程的基础。调和分析是研究一般黎曼流形上拉普拉斯算子在给定黎曼度量下的本征函数，整体调和分析是利用流形上测地流的整体动力学来研究流形上拉普拉斯算子的本征函数和本征值的大本征值渐近性。由于流形上拉普拉斯算子的整体本征函数是波群的本征函数，而波群沿着测地线传播奇点，因此本征函数的高频性质与测地线流的长时间动力学有关。两种特别有趣的情况是量子完全可积性，其中底层的测地线流是完全不可积的，以及量子混沌，其中底层的测地线流是遍历的。西北大学的Steve Zelditch教授将在全球谐波分析上发表十次演讲。他将开始与基本的例子（constantcurvature空间），发展材料pseudodiéequisite和傅立叶积分算子，推导出跟踪公式，包括著名的Duistermaat-Guillemin跟踪公式，并讨论应用程序的性质，本征函数和本征值和逆特征值问题。完全可积和遍历系统将被深入地考虑。

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.