We will talk about what a coupling of two probability measures is, give some examples from probability theory, and state some simple tools that will be useful throughout the semester (chapter 1). Then we will introduce the basic problem of optimal transport, and prove that there exists an optimal coupling (chapter 4).

Today, for the first half, we will set up notions of cyclical monotonicity, and a generalized notion of convexity (called c-convexity). We will then start on the big theorem of Chapter 5, which is a central duality principle in optimal transport. We will start proving it by relying on properties of cyclically monotone sets.

Today we will continue with the proof of the Kantorovich duality theorem, and will point out some interesting special cases.

We will finish the proof of Theorem 5.10, in particular we will show several equivalences of the optimality of a transference plans for real-valued cost functions.

We will focus on basic properties of Wasserstein distances and Wasserstein spaces, with an eye towards topology. In particular, we will characterize the relationship between convergence in the Wasserstein metric and weak convergence, and we will prove that \(P_p(X)\) is complete and separable whenever \(X\) is complete and separable.

We will now discuss a time-dependent version of optimal transport leading to a continuous displacement of measures, which gives a more complete description of the transport and a richer mathematical structure for future use. Several new concepts, including action and displacement interpolation, will be introduced along with examples and informal discussions before we move forward.

We will continue discussing displacement interpolation. Specifically, we will quickly review deterministic interpolation, prove some basic properties of Lagrangian actions, and finally introduce interpolation of random variables.

We will revise the properties of Lagrangian actions from the last time, and prove some of them. If time permits, we'll state the "main theorem" of displacement interpolation (Theorem 7.21) and discuss an outline of its proof.

We will discuss the main theorem on displacement interpolation (7.21), give an outline of the proof, and prove two corollaries. If time permits, we will discuss displacement interpolation between intermediate times and restriction.

We will discuss the main theorem on displacement interpolation (Theorem 7.21) and give its proof.

We will talk about some useful properties of optimal couplings; for instance, there is an optimal map between two sufficiently smooth densities. We will also discuss an example where the optimal transport map cannot be smooth. We will also note that between two log-concave distributions, the optimal transport map is Lipschitz, and, if we have time, we will explain how that fact allows transporting inequalities from one random variable to another.

Refreence: Optimal transport methods in economics by Alfred Galichon.