# Optimal Transport + Gradient Flows

Soumik Pal (University of Washington) , Young-Heon Kim (University of British Columbia)Sep 1, 2023 — Dec 31, 2023

## About the course

The space of probability distributions with finite second moments can be made into a natural metric space, called the Wasserstein space, whose metric is defined by using the optimal transportation between probability distributions. On this metric space one can draw curves that represent motion along the steepest descent (AKA gradient flow) of functionals of probability measures. This is a very fruitful way to view many important families of probability measures that arise from PDEs and stochastic processes. For example, using this geometric framework, one may derive functional inequalities and infer rates of convergence of Markov processes. A striking example is that of the heat equation, whose solution can be interpreted as the family of marginal distributions of Brownian motion. In the Wasserstein space, this curve of probability laws is the gradient flow of the Shannon entropy.

We will discuss the theory of Wasserstein gradient flows, including the formal Riemannian calculus due to Otto, and the modern techniques of metric measures spaces. Apart from the classical examples, we will also discuss many modern variations such as Wasserstein mirror gradient flows that come up in statistical applications. A fruitful interaction between probability, geometry, and PDE theory will be developed simultaneously. This is a continuation of the sequence of OT+X courses under the Kantorovich Initiative.

# Registration

This course is available for registration under the Western Dean's Agreement. To register, you must obtain the approval of the course instructor and you must complete the Western Dean's agreement form , using the details below. The completed form should be signed by your home institution department and school of graduate studies, then returned to the host institution of the course.

## Enrollment Details

- Course Name
- Optimal Transport + Gradient Flows
- Date
- Sep 1, 2023 — Dec 31, 2023
- Course Number
- **University of Washington Students:** * University of Washington: Math 581 F **All Other WDA Students:** * University of British Columbia: Math 606D:101
- Section Number
- Section Code

## Instructor(s)

For help with completing the Western Dean’s agreement form, please contact the graduate student program coordinator at your institution. For more information about the agreement, please see the Western Dean's Agreement website

# Other Course Details

## Delivery Details

The course is being offered simultaneously at Korea Advanced Institute of
Science and Technology (KAIST) and the PIMS network, including the University of
Washington, Seattle. **Due to different time schedules for individual campuses
and
the time zones, the course has an unusual structure. Please read the details
below carefully.**

### Lectures

Lecture hours

**6:30pm - 8pm Pacific on Tuesdays and Thursdays**. Thus we will have two classes per week, each for 90 mins.Lectures will be taught over Zoom and videos and notes will be made available to everyone afterwards.

A Slack channel will be used to communicate with students and distribute teaching material.

There will be no exams in this course. Occasional homework problems will be provided.

### Registration

Students at Canadian PIMS Member Universities may register through the Western Deans Agreement. Students at UW may register directly for the UW course. Course codes and other registration details for students in either of these cases are listed in the registration section above. Students at other institutions should contact one of the instructors to attend the course as a non-registered student.

## Course Structure

### Part I

Part I is a recap of the basics of Monge-Kantorovich optimal transport theory.
*You do NOT need to take this part if you are already familiar with the basics*.
This will be covered between **AUG 28** and **SEP 26**. Topics covered during
this period are:

- linear programming
- Monge-Kantorovich problem
- Kantorovich duality
- Monge-Ampère PDE
- Brenier’s Theorem
- Wasserstein-2 metric

### Part II

This will start on **SEP 27** and continue through **DEC 7**. A rough syllabus
of topics covered are presented below in the order they will be covered. There
might be some changes depending on our progress.

#### core topics

- Wasserstein space
- metric property
- geodesics, displacement interpolation, generalized geodesic
- Geodesic convexity

- AC curves in the Waserstein space and the continuity equation
- Benamou-Brenier and dynamic OT
- Otto calculus
- tangent spaces to the Wasserstein space
- Riemannian gradient

- Diffusions as gradient flows via Otto calculus
- Brownian motion
- Langevin diffusions

#### Modern research topics that will be surveyed

- log-Sobolev and other functional inequalities
- Convergence of finite dimensional gradient flow of particles to the
**McKean-Vlasov diffusions**and gradient flow in the Wasserstein space. - The implicit Euler or JKO scheme
- Entropy regularization and gradient flows
- Schrödinger bridges
- Large deviation and gradient flows

- Mirror gradient flows, parabolic Monge-Ampere and the Sinkhorn algorithm