Seminar Activity Project: A Primer on Optimal Mass Transportation and the Supremal Monge Problem (English)

Featuring a high cortisol graph for solving Monge problem in the \(L^\infty\) case.

This seminar explores the foundational concepts and central existence results in the theory of optimal transport. Starting from the intuitive but mathematically restrictive Monge problem, we transition to the Kantorovich relaxation, leveraging weak topologies and Prokhorov's theorem to guarantee the existence of optimal transport plans. By systematically constructing the dual problem and applying tools from convex analysis—specifically \(c\)-transforms and \(c\)-cyclical monotonicity—we rigorously establish strong duality. Utilizing this primal-dual framework, we demonstrate the existence and uniqueness of optimal transport maps for strictly convex costs, culminating in Brenier's Theorem for the quadratic cost in \(\mathbb{R}^d\). Finally, we explore the measure-theoretic stability of optimal plans and resolve the supremal (\(L^\infty\)) optimal transport problem. To tackle the supremal case, we consider a secondary variational problem, rigorously proving that its optimal solution is uniquely induced by a transport map \(T\).