Optimal Transportation Theory


Registration to the class is mandatory

IMPORTANT: the next lectures will take place in room 2L8!


Optimal transport is a powerful mathematical theory at the interface between optimization and probability theory with far reaching applications. It defines a natural tool to study probability distributions in the many situations where they appear: mathematical physics, data science, partial differential equations, statistics or shape processing. In this course we will present the classical theory of optimal transport, efficient algorithms to compute it and applications.

Pre-requisites: Notions on measure theory, weak convergence, and convex analysis. Some basic knowledge of Python.

Language: The class will be taught in French or English, depending on attendance.

Organisation: Lectures will take place on Wednesday afternoon from 2pm to 5pm at the Institut de mathématique d’Orsay (IMO) building 307 on the campus of Orsay, Room 2L8.

Evaluation: written exam + homework

Homework: Here you can find the homework to get some extra points for the final evaluation. The short report must be submit to luca.nenna@universite-paris-saclay.fr by March 25 at 6 a.m. (New York time).


Date Lecturer Topic Lecture notes
16/02/2022 LN Primal and dual formulations Lecture 1
23/02/2022 LN Entropic Optimal Transport Lecture 2, Slide
02/03/2022 LN Functionals over $\mathcal P(\Omega)$ Lecture 3
09/03/2022 LN Applications (Wasserstein Geodesics and MFGs) Lecture 4 slide
16/03/2022 YB Eulerian Approach I Lecture 5 and 6
23/03/2022 YB Eulerian Approach II
30/03/2022 EXAM

where YB:=Yann Brenier and LN:=Luca Nenna.

Here you can find some lecture notes in French by Nathael Gozlan, Paul-Marie Samson and Pierre-André Zitt.

Annals: Exam 2014


  • Ambrosio, Luigi, Nicola Gigli, and Giuseppe Savaré, Gradient flows: in metric spaces and in the space of probability measures, Springer Science & Business Media, 2005.
  • Gabriel Peyré, Marco Cuturi, Computational optimal transport, Foundations and Trends® in Machine Learning 11 (2019), no. 5-6, 355–607.
  • Filippo Santambrogio, Optimal transport for applied mathematicians, Springer, 2015.
  • Cédric Villani, Topics in optimal transportation, no. 58, American Mathematical Soc., 2003.
  • Cédric Villani, Optimal transport: old and new, vol. 338, Springer Science & Business Media, 2008.