Math 260R: Optimal transport

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

Lecture: Tuesday and Thursday, 9:30-10:45am

Lecture Location: HSSB 2251

Office Hours: Thursday 11am-12pm, Friday 1-2pm, and by appointment

Optimal Transport Wiki:



Recommended References:

Exams: none.

Homework:

  • The main assignment will be to work together to continue building the Optimal Transport Wiki: https://otwiki.github.io/otwiki-main/.
    • Jan 31st: Select a topic on which you will write a new article for the wiki. Email me to "claim" this topic.
    • Feb 14th: Finish your expository article for the wiki.
    • Feb 21st: Select an existing wiki article, for which you will complete a major revision. Email me to "claim" this topic.
    • March 7th: Complete a major revision of an existing article for the wiki.

    To prepare articles for the wiki, you will need to download the following software:
    You can find the current files that compose the wiki here: https://github.com/otwiki/otwiki-main. To practice with quarto (and as something to model your own article on), download the following files:
    • Kantorovich_Problem.qmd
    • Kantorovich_Problem_Image.png
    • bibliography.bibtex

    To preview what your article will look like as a website, use the terminal to navigate to the folder containing your file and then enter the following command into the terminal: ``quarto preview FILENAME.qmd''

    For each article you work on, please include citations to all references you consulted in preparation of the article in bibtex format. When you complete your article, please email me the main .qmd file, any image files, and any additional citations you have added to the .bibtex file. (Later, we will learn how to collaborate on github--commit/push/pull/etc.)

  • As a smaller assignment, each student will type up a detailed solution to two or three of the exercises I give in class. Solutions should be typed in the shared Overleaf document linked below. To indicate that you have selected a solution to work on, please type your name after the word ``Solution'' in the shared Overleaf file. Solutions should be submitted within one week of when the exercise was assigned.

The above deadlines are chosen 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. If you would like to submit revised work, keep it mind that it typically takes me a week to provide feedback on the original version.

Grading Scheme: First article: 30%, Second article: 30%, Exercise Solution(s) 10%, Class participation: 30%

Prerequisites: Measure theory, functional analysis

Exercises: List of homework exercises and solutions

Syllabus:

`
topic lecture notes
1 Jan 7 (T) the optimal transport problem and transport of measures LEC1
2 Jan 9 (Th) normalizing flows and the Monge problem LEC2
3 Jan 14 (T) from transport maps to transport plans LEC3
4 Jan 16 (Th) the Kantorovich problem LEC4
5 Jan 21 (T) topologies on the space of measures LEC5
6 Jan 23 (Th) crash course in convex optimization LEC6
7 Jan 28 (T) primal and dual optimization problems LEC7
8 Jan 30 (Th) equiv of primal and dual optimiazation problems LEC8
9 Feb 4 (T) the dual of Kantorovich's problem LEC9
10 Feb 6 (Th) equiv of primal and dual Kantorovich problems LEC10
11 Feb 11 (T) optimal plans: the Knott-Smith criterion LEC11
12 Feb 13 (Th) optimal maps: Brenier's characterization LEC12
13 Feb 18 (T) definition of the p-Wasserstein metric LEC13
14 Feb 20 (Th) topology of p-Wasserstein metric LEC14
15 Feb 25 (T) the continuity equation LEC15
16 Feb 26 (Th) Wasserstein geodesics LEC16
17 Mar 4 (T) absolutely continuous curves LEC17