Introduction to optimal transportation (course 287)

Department of Mathematics

Stanford University

Winter 2012

Time: M,W, F at 11am (twice a week among these).

Location: 381-U.

Teacher: Y. A. Rubinstein

This will be an introductory graduate level course on Optimal Transportation theory. We will study Monge's problem, Kantorovich's problem, c-concave functions (also in the Riemannian setting), Wasserstein distance and geodesics (including a PDE formulation), applications to inequalities in convex analysis, as well as other topics, time permitting. Participating graduate students will be required to present some of the topics.

Main references:

L. Ambrosio, N. Gigli, A user's guide to optimal transport, 2011. (Older version)
C. Villani, Topics in Optimal Transportation, AMS, 2003.
C. Villani, Optimal Transport: Old and New, Springer, 2008.


  • December 5, 3-4pm
    Introduction (Alessandro Carlotto)
  • January 9
    c-concave functions, Rockafellar's theorem.
  • January 18
    Brenier's theorem and optimal maps.
  • January 23
    Examples. Polar factorization.
  • January 27
    McCann's theorem.
  • January 30
    Wasserstein space, I (Otis Chodosh).
  • February 3
    Wasserstein space, II (Otis Chodosh).
  • February 6
    Wasserstein space, III.
  • February 10
    Wasserstein space, IV.
  • February 13
    Wasserstein space, V.
  • February 15
    Gradient flows (basic existence and uniqueness) (Ehsan Kamalinejad).
  • February 17
    Gradient flows, I (Alessandro Carlotto).
  • February 22
    Gradient flows, II (Alessandro Carlotto).
  • February 24
    Gradient flows and Wasserstein space, I.
  • February 27
    The Riemannian structure on Wasserstein space, I.
  • March 2
    Gradient flows and Wasserstein space, II.
  • March 5
    The Riemannian structure on Wasserstein space, II.
  • March 9
    Analytic applications: Isoperimetric, Sobolev, and Brunn-Minkowski inequalities.