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Title: Laplacian quadratic forms, function regularity, graphs, and optimal transport
Abstract: In this talk, I will discuss two different applications of harmonic analysis to problems motivated by data science. Both problems involve using Laplacian quadratic forms to measure the regularity of functions. In both cases the key idea is to understand how to modify these quadratic forms to achieve a specific goal. First, in the graph setting, we suppose that a collection of m graphs G_1 = (V, E_1),..., G_ m=(V, E_ m) on a common set of vertices V is given, and consider the problem of finding the 'smoothest' function f : V -> R with respect to all graphs simultaneously, where the notion of smoothness is defined using graph Laplacian quadratic forms. Second, on the unit square [0, 1] ^2, we consider the problem of efficiently computing linearization of 2-Wasserstein distance; here, the solution involves quadratic forms of a Witten Laplacian.
Nicholas Marshall is an NSF postdoctoral research fellow at Princeton University. He received his Ph.D. from Yale University in 2019 under the supervision of Ronald Coifman and Stefan Steinerberger.