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

University Of Kentucky Research Foundation, Lexington KY

Investigators

Abstract

Global Harmonic Analysis NSF/CBMS Regional Conference in the Mathematical Sciences University of Kentucky, June 20-24, 2011 Global harmonic analysis may be viewed as an extension of classical Fourier theory on the line and on the circle to the geometric setting of Riemannian manifolds. Riemannian manifolds provide models of both com- pletely integrable and chaotic dynamical systems, and also provide a setting in which the connection between "classical" and "quantum" behavior can be carefully studied. Moreover, just as harmonic analysis on the line and the circle are a fundamental tool in the study of both linear and nonlinear differ- ential equations in these settings, so global harmonic analysis is fundamental to the study of di¤erential equations, such as the wave and Schrödinger equa- tion, on manifolds. On a general Riemannian manifold, harmonic analysis is the 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 the eigenfunctions and eigenvalues of the Laplacian on a manifold. Since global eigenfunctions of the Laplacian on a manifold are eigenfunctions of the wave group, 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 quantum complete 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 lectures on global harmonic analysis. He will begin with basic examples (constant curvature 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 integrable and ergodic systems will be considered in depth.