Optimal Transport and Wasserstein Geometry

Optimal transport asks: given two probability measures $\mu$ and $\nu$ on a metric space $(X, d)$, what is the most efficient way to rearrange $\mu$ into $\nu$? The Wasserstein distance $$W_p(\mu, \nu) = \left(\inf_{\gamma \in \Pi(\mu,\nu)} \int d(x,y)^p,dgamma\right)^{1/p}$$ metrizes the space of probability measures and inherits geometric properties from the base space. We discuss Brenier's theorem on the existence and uniqueness of optimal maps, the Riemannian structure of Wasserstein space, and connections to Ricci curvature lower bounds via the Lott-Sturm-Villani theory. Applications to gradient flows (the JKO scheme for the Fokker-Planck equation) and to generative models \in machine learning are outlined.