Introduction to Optimal Transport
Autumn 2018

Alessio Figalli
Giada Franz, Federico Glaudo
Mo 10-12
ETH Course Page

Given two distributions of mass, it is natural to ask ourselves what is the "best way" to transport one into the other. What are mathematically acceptable definitions of "distributions of mass" and "to transport one into the other"? Measures are perfectly suited to play the role of the distributions of mass, whereas a map that pushes-forward one measure into the other is the equivalent of transporting the distributions. By "best way" we mean that we want to minimize the map in some norm.

The original problem of Monge is to understand whether there is an optimal map and to study its properties. In order to attack the problem we will need to relax the formulation (Kantorovich's statement) and to apply a little bit of duality theory. The main theorem we will prove in this direction is Brenier's theorem that answers positively to the existence problem of optimal maps (under certain conditions). The Helmotz's decomposition and the isoperimetric inequality will then follow rather easily as applications of the theory.

Finally, we will see how the optimal transport problem gives a natural way to define a distance on the space of probabilities (Wasserstein distance) and we will study some of its properties.


The students are expected to have mastered the content of the first two years taught at ETH, especially Measure Theory. The seminar is mainly intended for Bachelor students.

The seminar consists of \(10\) talks and each of them will be split between two students (talking \(45\) minutes each). Thus every student is assumed to give two talks: one talk in the first half and one talk in the second half of the semester, each of \(45\) minutes length. The two speakers for week \(n\) will be determined at the end of the seminar in week \(n-2\).

As a preparation for your talk, we will meet with you before your talk once you have it ready. This meeting should take place around five days before the talk will take place. We are of course also willing to answer questions by email or in person in our offices (F 27.1 and F 28.3).

Week Speakers Topics References
1 Ataei Alireza, Cédéric Hiltbrunner Monge formulation and Kantorovich formulation, existence theorem for Kantorovich problem. [Sa] §1.1 and §1.5
2 Erik Bertolino, Matteo Giardi Introduction to duality, convex analysis tools. [Sa] §1.2, §1.6.1 and §1.6.2
until Theorem 1.37 included
3 Davide Apolloni, Lara Fratini Proof of duality. [Sa] §1.6.2 from Theorem 1.38
4 Giovanni Compagnoni, Marc Nübel Existence of an optimal map and conditions for optimality. [Sa] §1.3 without §1.3.2, Thm 1.47 and 1.48
5 Gian Luca Cola, Amr Umeri Remarks and applications. [Sa] §1.4, §1.7.2 and §2.5.3
6 Matteo Giardi, Cédéric Hiltbrunner Introduction to Wasserstein distance. [Sa] Introduction of §5, §5.1 and §5.2
7 Davide Apolloni, Erik Bertolino Preliminaries for the optimal transport with linear cost. [Sa] §3.1.1, §3.1.2 and §3.1.3
until Corollary 3.8 included
8 Lara Fratini, Marc Nübel Transport rays and existence of the map for the linear cost. [Sa] §3.1.3 from Lemma 3.9
and Theorem 3.18
9 Ataei Alireza, Gian Luca Cola L^\infty cost. [Sa] §3.2
10 Giovanni Compagnoni, Amr Umeri Recap of the whole course.
The first meeting will be on September 24th. The lecture on October 29th will not be held. During the seminar we will assign exercises mostly concerning theorical facts needed during the talks. Trying to solve them is an integral part of the seminar and we suggest to write them down once solved. The solutions of the exercises can be found in this Overleaf Document.