Optimal transport theory and applications

(MATH-GA.2610 Fall 2026)

Office hours: TBA

This is a graduate introduction to the modern field of optimal transport (OT) consisting roughly of two parts. The first part of the course will cover the theoretical foundations of optimal transport as developed at the turn of the century (Brenier's theorem, Caffarelli's regularity theory, Caffarelli's contraction theorem for log concave measures, Benamou and Brenier's dynamic formulation, the JKO scheme and gradient flows). The second part will deal with a selection of topics in statistical inference, PDE (i.e. Langevin dynamics), and geometric data processing (e.g. Gromov-Wasserstein averaging) where optimal transport research has been specially active in the last few years.

We will not be following one particular textbook given the selection of topics, but some books I recommend are "Optimal transport for Applied Mathematicians" by Philippo Santambrogio, "Optimal transport on Euclidean Spaces" by Francesco Maggi, "Statistical Optimal Transport" by Sinho Chewi, Jonathan Niles-Weed, and Philippe Rigollet, "Computational optimal transport" by Gabriel Peyré and Marco Cuturi. There is also a very nice and short set of lecture notes by Justin Solomon, "Optimal transport on discrete domains".