Optimal Transport as a Tool in Analytic Number Theory and PDEs
Optimal Transport is concerned with the question of how to best move one measure to another (this could be sand on a beach or products from a warehouse to consumers). I will explain the basic definition of Wasserstein distance and then describe how it can be used as a tool to say interesting things in other fields. (1) How to get new regularity statements for classical objects in number theory almost for free (irrational rotations on the torus, quadratic residues in finite fields). (2) How to best distribute coffee shops over downtown Minneapolis. (3) Finally, how to obtain higher dimensional analogues of classical Sturm-Liouville theory: simply put, Sturm-Liouville theory says that eigenfunctions of the operator Ly = -y''(x) +p(x)y(x) (think of sin(kx) and cos(kx)) cannot have an arbitrary number of roots; we present a generalization to higher dimensions that is based on a simple (geometric) inequality.