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

全球调和分析2011年6月20-24日在肯塔基大学举行的NSF/CBMS区域会议全球调和分析可以被视为经典傅立叶理论在直线和圆周上的延伸到黎曼流形的几何设置。黎曼流形提供了完全可积和混沌动力系统的模型，也提供了一个可以充分研究“经典”和“量子”行为之间联系的环境。此外，正如直线和圆周上的调和分析是研究这些背景下的线性和非线性微分方程组的基本工具一样，整体调和分析也是研究流形上的微分方程组，如波动方程和薛定谔方程的基础。在一般的黎曼流形上，调和分析是在给定的黎曼度量下研究拉普拉斯函数的本征函数，全局调和分析是指利用流形上的地线的整体动力学来研究流形上拉普拉斯的本征函数和本征值的大特征值渐近。由于流形上拉普拉斯函数的整体本征函数是波群的本征函数，而波群沿测地线传播奇点，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。两个特别有趣的例子是量子完全可积性，其中潜在的测地线流是完全可积的，以及量子混沌，其中潜在的测地线流是遍历的。西北大学的Steve Zelditch教授将就全球调和分析做10次演讲。他将从基本例子(常曲率空间)开始，发展关于伪随机算子和傅里叶积分算子的材料，推导包括著名的Duistermaat-Guillminn迹公式在内的迹公式，并讨论特征函数和特征值的性质以及逆特征值问题的应用。我们将深入研究完全可积的遍历系统。

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.