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This lecture from the "Spectral Geometry in the clouds" seminar series features Rupert Frank from Ludwig-Maximilians-Universität München discussing the asymptotic optimization of Riesz means of Laplace eigenvalues on convex sets. Explore the mathematical problem of optimizing Riesz means Tr(−∆ − Λ γ −) of Laplace eigenvalues among convex sets in Rd with given measure, focusing on maximizing Riesz means of Dirichlet eigenvalues and minimizing Riesz means of Neumann eigenvalues. Learn about the behavior of optimizers in the asymptotic regime where Λ approaches infinity, with proof of convergence in Hausdorff distance to a disk for any Riesz exponent γ greater than 0 in 2D, and similar results in higher dimensions. The presentation combines uniform versions of Weyl asymptotics with partially semiclassical analysis of degenerating convex sets, based on joint work with Simon Larson.
