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

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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.