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:** otwiki.xyz

** Recommended References:**

- Functional Analysis:

Brezis,*Functional Analysis, Sobolev Spaces and Partial Differential Equations*, - Convex Analysis:

Bauschke and Combettes,*Convex Analysis and Monotone Operator Theory in Hilbert Spaces*, - Fluid Mechanics and PDEs:

Chorin and Marsden,*A Mathematical Introduction to Fluid Mechanics*, - Optimal Transport:

Villani,*Topics in Optimal Transportation*,

Santambrogio,*Optimal Transport for Applied Mathematicians*,

Figalli, Glaudo,*An Invitation to Optimal Transport*, (I have a copy you can borrow)

Ambrosio, Brué,Semola*Lectures on Optimal Transport*,

- Gradient Flows:

Ambrosio, Gigli, Savaré,*Gradient Flows in Metric Spaces and in the Space of Probability Measures*,

Ambrosio and Savaré,*Gradient Flows of Probability Measures*,

Santambrogio,*{Euclidean, metric, and Wasserstein} gradient flows*,

- Computational Optimal Transport:

Peyré and Cuturi,*Computational Optimal Transport*,

** 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.**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 |

**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) | 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 |