Math 260L: Optimal transport

Professor: Katy Craig, katy•craig at ucsb • edu

Lecture: Monday and Wednesday, 10:30-11:45am, zoom (recordings will be posted below)

Office Hours: Monday and Wednesday, 11:45am-12:45pm, and by appointment, zoom

Optimal Transport Wiki:

Recommended References:

Exams: none.

Homework: Each student will write 2-3 articles for the Optimal Transport Wiki that we will create together. Potential topics are listed on the wiki. At least one article must be submitted by Friday, May 8th. Students will have the opportunity to revise submitted articles and will only be graded on the final version.
Here is a short video explaining how to create a new article on the wiki.
Grading Scheme: Homework: 10%, Surviving Covid-19: 90%
Prerequisites: Measure theory, functional analysis

Outline of Course:

Part I: Optimal Transport Part II: The Wasserstein Metric
Monge and Kantorovich Problems approximation by convolution
convex analysis and duality in optimization topology of Wasserstein metric
dual Kantorovich problem Benamou-Brenier and dynamic characterization of Wasserstein metric
characterization of OT maps Wasserstein geodesics and displacement interpolation

A sample video on the topology of the Wasserstein metric and absolutely continuous curves (Lecture 16) is publicly available. The password to access to the remaining video is available by request.


topic lecture notes video typed notes
1 Mar 30 (M) general remarks LEC1
2 Apr 1 (W) the Monge problem LEC2 VID2 NOTE2
3 Apr 6 (M) from transport maps to transport plans LEC3 VID3 NOTE3
4 Apr 8 (W) the Kantorovich problem LEC4 VID4 NOTE4
5 Apr 13 (M) convergence of measures LEC5 VID5 NOTE5
6 Apr 15 (W) convexity and the subdifferential LEC6 VID6 NOTE6
7 Apr 20 (M) primal and dual optimization problems LEC7 VID7 NOTE7
8 Apr 22 (W) the dual Kantorovich problem LEC8 VID8 NOTE8
8 Apr 27 (M) equiv of primal and dual Kantorovich problems I LEC9 VID9 NOTE9
10 Apr 29 (W) equiv of primal and dual Kantorovich problems II LEC10 VID10 NOTE10
11 May 1 (F) optimal plans: the Knott-Smith criterion LEC11 VID11 NOTE11
12 May 6 (W) optimal maps: Brenier's characterization LEC12 VID12 NOTE12
13 May 11 (M) definition of Wasserstein metric LEC13 VID13 NOTE13
14 May 13 (W) approximation of measures by convolution LEC14 VID14 NOTE14
15 May 18 (M) Wasserstein metric: triangle inequality and topology LEC15 VID15 NOTE15
16 May 20 (W) curves in the space of probability measures LEC16 VID16 NOTE16
17 May 27 (W) duality again: Benamou Brenier LEC17 VID17 NOTE17
18 June 1 (M) the continuity equation and Wasserstein geodesics LEC18 VID18 NOTE18
20 June 3 (M) dynamic formulation of Wasserstein metric LEC19 VID19 NOTE19