Kai-Hsiang Wang 望開翔 (Northwestern University): Introduction to Optimal Transport with Application to Geometric Inequalities

In this talk, I will first give an introduction to optimal transport (OT) theory and recall some fundamental results. This will then be illustrated by the common scenario where the cost function is given by distance squared and the measures are absolutely continuous. Under these assumptions, I will show the classical theorems of Brenier and McCann about OT maps for Euclidean spaces and manifolds, respectively, as well as the connection to the Monge-Amp\'ere equation and some geometric inequalities. After that, I will mention my previous work generalizing McCann's theorem to a submanifold setting and its application to the Michael-Simon inequality.

TBA