A brief introduction to the regularity theory of optimal transport

Optimal transport is a classic field of mathematics which studies the most cost-efficient allocation of resources. It has many applications, both in the real world as well as in theoretical mathematics. In this video, I discuss the regularity theory of optimal transport, which tries to understand when transport will be continuous. The goal is to give a history of this problem and to indicate some current areas of research.



00:00 Introduction
01:10 What is optimal transport?
04:41 When is optimal transport deterministic?
06:35 When is optimal transport continuous?
08:59 The work of Ma, Trudinger and Wang
09:46 The MTW condition
12:27 What is the MTW tensor?
15:01 An open question
15:51 Final thoughts

This video is adapted from a talk that I gave at GSI2021.

I apologize in advance for the uneven audio. In the future, I'll try to record in a single location to keep the levels consistent. Please let me know if you have any other suggests for how to improve this video, or any future videos.

If you are interested in the fugue excerpt at the end of the video, you can download a rough version of the sheet music here.
https://differentialgeometri.files.wo...
There are no dynamic markings or indications where each statement comes in.

Other than the videos by Flavien Léger and the diagram taken from Villani's book, the other animations and images were created using Procreate and Keynote.


Technical notes:

The animation first appearing at :39 in the video depicts the notion of displacement interpolation. Roughly speaking, this considers a continuous-time version of optimal transport which flows from the initial distribution to the final one. By doing so, you can study the distributions at intermediate times. This idea was invented by McCann and plays an important role in functional analysis and the notion of Ricci curvature for metric-measure spaces.

The animation first appearing at :57 into the video (and later at 13:43) is meant to show that the injectivity domains of the round unit sphere are convex (they are simply disks of radius pi). This fact plays a central role for optimal transport on a Riemannian manifold. A very deep theorem of Figalli, Rifford and Villani shows that this remains true for small deformations of the sphere.

At 4:35 in the video, I state that the solution to the Kantorovich problem exists. To be more accurate, you need to make some mild assumptions on the costs and measure to obtain a solution. Roughly speaking, you want the cost function to be continuous and for a coupling to exist whose total cost is finite. The precise statements of Kantorovich's duality theorem are covered in great detail in Chapter 5 of Villani's text "Optimal Transport: Old and New."

The dimension of the castle and sandpile is taken to be n. I realized that I didn't state what n is and only had it written on the slide with the existence theorem for the Monge problem.

At 8:30 in the video, I noted that the Jacobian equation simplifies considerably for the squared-distance cost in Euclidean space. It is worth remarking that the transport also simplifies: it is given by the usual sub-differential of the potential u.

When describing Loeper's result at 13:13 in the video, I state that if the MTW condition fails that we can find smooth measures with "convex supports" such that the solution to the Monge map has discontinuities. Here, the precise condition is actually that the supports are relatively c-convex; the phrase "convex supports" is used as a short-hand for this.