Title: Global Harmonic Analysis

Abstract:

Harmonic analysis began with Fourier series and the Fourier transform on Euclidean space and its quotients. Quantum mechanics led to its generalization to eigenfunctions of Laplacians on Riemannian manifolds. The Bohr correspondence principle suggests that in the high frequency limit, the behavior of eigenfunctions should reflect global dynamics of the geodesic flow.

Global harmonic analysis is about exploiting this global dynamics to study eigenfunctions and the heat and wave equation for long times. It rapidly yields results that local harmonic analysis cannot achieve.

My talk will introduce global harmonic analysis through its origins in quantum mechanics. By the end, I will discuss some new results on norms of eigenfunctions and their relation to geodesics. For instance, on a real analytic surface, if there are eigenfunctions which have saturate the Sogge Lp bounds, then there must exist a point through which all geodesics are closed
(joint work with Chris Sogge). I will also discuss the effect of ergodicity of the geodesic flow on nodal (zero sets).

Wednesday, April 1, 4:30 – 5:30 p.m.
Mathematics 520
Tea will be served at 4:00 p.m.

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