Math 260J: Optimal transport

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

Lecture: Tuesday and Thursday, 12:30-1:45pm

Lecture Location: weeks 1-4: zoom (please turn camera on), weeks 5-10: PHELP 1445 (google map)

Office Hours: Tuesday 11:30am-12:30pm, Friday 2-3pm, and by appointment (either in SH 6507 or on zoom)

Optimal Transport Wiki:

Recommended References:

Exams: none.

Homework: Students will work together to build the Optimal Transport Wiki:

For each article you work on, please include citations to all references you used in preparation of the article.

All assignments are due on the assigned date at 11:59pm.

Notify me via email when you complete your assignment.

Deadlines are given to help prevent you from getting behind. All deadlines are flexible, though all work (including revised work, submitted for a regrade), must be submitted by March 14th.

Students will have the opportunity to revise their work and will only be graded on the final version. If you would like the opportunity to submit revised work, keep it mind that it typically takes me a week to provide feedback on the original version.

  • Jan 28th: Select a topic on which you will write a new article for the wiki.
  • Feb 11th: Finish your expository article for the wiki.
  • Feb 18th: Select an existing wiki article, for which you will complete a major revision.
  • March 4th: Complete a major revision of an existing article for the wiki.

Here is a short video explaining how to create a new article on the wiki.

Grading Scheme: First article: 35%, Second article: 35%, Class participation: 30%

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


topic lecture notes video
1 Jan 4 (T) general remarks LEC1 VID1
2 Jan 6 (Th) the Monge problem LEC2 VID2
3 Jan 11 (T) from transport maps to transport plans LEC3 VID3
4 Jan 13 (Th) the Kantorovich problem LEC4 VID4
5 Jan 18 (T) compactness in the narrow topology LEC5 VID5
6 Jan 20 (Th) lower semicontinuity in the narrow topology LEC6 VID6
7 Jan 25 (T) narrow vs wide topology and intro to convex analysis LEC7 VID7
8 Jan 27 (Th) the subdifferential and duality LEC8 VID8
9 Feb 1 (T) equiv of primal and dual optimiazation problems LEC9 VID9
10 Feb 3 (Th) equiv of primal and dual Kantorovich problems I LEC10 VID10
11 Feb 8 (T) optimal plans: the Knott-Smith criterion LEC11 VID11
12 Feb 10 (Th) optimal maps: Brenier's characterization LEC12 VID12
13 Feb 17 (Th) definition of Wasserstein metric LEC13 forgot :(
14 Feb 18 (F) approximation of measures by convolution LEC14 VID14
15 Feb 22 (T) Wasserstein metric: triangle inequality and topology LEC15 VID15
16 Feb 24 (Th) curves in the space of probability measures LEC16 VID16
17 May 27 (W) duality again: Benamou Brenier LEC17 VID17
18 March 3 (Th) the continuity equation and Wasserstein geodesics LEC18 VID18
19 March 8 (T) characterization of absolutely continuous curves (part 1) LEC19 VID19
10 March 10 (Th) characterization of absolutely continuous curves (part 1) LEC20 VID20