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Johns Hopkins University
Global Harmonic Analysis and the Concentration of Eigenfunctions.
We shall go over several problems related to the concentration of eigenfunctions. The study of eigenfunction concentration naturally arises in analysis, number theory and mathematical physics. The tools that we use to analyze these problems are based on microlocal and harmonic analysis, as well as Riemannian geometry. More specifically, we shall show how classical results from the propagation of singularities, bilinear oscillatory integral estimates and the Cartan-Hadamard theorem as well as comparison theorems from geometry play a natural role in the analysis of this phenomena.
We shall show that, even though extreme concentration occurs on manifolds
of positive curvature, it cannot occur on manifolds of negative curvature.