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:** 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*, - Optimal Transport:

Villani,*Topics in Optimal Transportation*,

Santambrogio,*Optimal Transport for Applied Mathematicians*,

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

- Computational Optimal Transport:

Peyré and Cuturi,*Computational Optimal Transport*,

** 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. **

**Syllabus:**

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 |