Bangalore probability seminar: Some applications of optimal transportation problem in PDEs

In 1781, French mathematician Gaspard Monge asked the following question: given two probability densities \mu and \nu in R^n and a cost function c(x, y) : R^n x R^n \to [0,\infty), what is the optimal way of transporting \mu to \nu with respect to the cost functional c? In other words, find a map T that transports \mu to \nu and minimizes the total transportation cost. The enormous applications of this theory became apparent only in late 1980s. In this seminar, after a brief introduction to the problem, its developments and historical backgrounds, I will explain some of its applications in PDEs. In particular, I will explain how this theory can be used to obtain various sharp geometric inequalities. In the last part of this talk, I will focus on PDEs that can be thought of as gradient flows (in an appropriate sense) on the metric space of probability measures with respect to the 2-Wasserstein distance.