Math 260J: Optimal transport

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

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

Lecture Location: weeks 1-2: zoom (please turn camera on), weeks 3-10: PHELP 1445

Office Hours: Tuesday 11:30am-12:30pm, Friday 2-3pm, and by appointment

Optimal Transport Wiki: otwiki.xyz

Recommended References:

Exams: none.

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

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. Suggested topics: TBD
  • Feb 11th: Finish your expository article for the wiki.
  • Feb 18th: Select an existing wiki article, for which you will complete a major revision. Suggested topics: TBD
  • 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


Syllabus:

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) the narrow topology LEC5 VID5
6 Jan 20 (Th) convexity and the subdifferential
7 Apr 20 (M) primal and dual optimization problems
8 Apr 22 (W) the dual Kantorovich problem
8 Apr 27 (M) equiv of primal and dual Kantorovich problems I
10 Apr 29 (W) equiv of primal and dual Kantorovich problems II
11 May 1 (F) optimal plans: the Knott-Smith criterion
12 May 6 (W) optimal maps: Brenier's characterization
13 May 11 (M) definition of Wasserstein metric
14 May 13 (W) approximation of measures by convolution
15 May 18 (M) Wasserstein metric: triangle inequality and topology
16 May 20 (W) curves in the space of probability measures
17 May 27 (W) duality again: Benamou Brenier
18 June 1 (M) the continuity equation and Wasserstein geodesics
20 June 3 (M) dynamic formulation of Wasserstein metric