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Wasserstein Distance in Analysis, Number Theory and PDEs

Stefan Steinerberger, Yale
Tuesday, October 15, 2019 - 1:30pm
PDL C-401

I will start by introducing the Wasserstein Distance, a notion of distance between two probability measures (basically: how far do you have to carry mass from one around until it looks like the other).  This notion is central in Optimal Transport -- I will try to argue that it can actually be fruitfully employed in a variety of other settings.  (1)  It can help in Number Theory, we describe new results for irrational rotations on the Torus as well as the distribution of quadratic residues in Finite Fields. (2) It can help in answering the question of how to regularly distribute points on a manifold (and, along the way, improve a result about the numerical integration of Lipschitz functions). (3) It's useful in geometry: a basic principle is that if it's always easy to buy milk, then there must be many supermarkets in the vicinity. I will translate this basic notion into a statement for zero sets of continuous functions that seems to be new and pretty exciting. (4)  I will conclude with an application to elliptic PDEs. An old question is how large the zero sets of eigenfunctions of elliptic PDEs have to be (with recent progress by Logunov & Malinnikova). I will use the geometry ideas from (3) to give lower bounds on the zero set of (possibly infinite) linear combinations of eigenfunctions which is the first instance of a full Sturm-Liouville theory in higher dimensions.

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