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Explore the fascinating intersection of mathematical physics and convex geometry in this 45-minute lecture on variational functionals in the Gauss space. Delve into three classical functionals: electrostatic capacity, torsion, and the first Dirichlet eigenvalue of the Laplace operator. Examine their intriguing properties from a convex geometry perspective, including Brunn-Minkowski type inequalities, Hadamard formulas for first variation, and the well-posedness of the Minkowski problem. Discover how these functionals extend naturally to the Gauss space and investigate recent results and open problems, particularly focusing on the potential for proving Brunn-Minkowski type inequalities for these extended functionals. Learn about collaborative research findings with E. Francini, G. Livshyts, P. Salani, and L. Qin in this advanced mathematical exploration presented by Andrea Colesanti at the Hausdorff Center for Mathematics.
