An Introduction to Optimal Transport and Wasserstein Space

Given a pile of sand, how should one move each grain into a nearby ditch so as to do the least amount of work? The theory of optimal transport provides an answer to such a question, but nowadays it has many applications both inside and outside of mathematics. In this expository talk, we will provide a survey of some standard results in optimal transport theory: the existence of optimal transport, duality, and Brenier's Theorem. We will then see how this theory elucidates the geometric structure of Wasserstein space by discussing from a high-level perspective absolutely continuous curves and gradient flows in this space.