PIMS online graduate course on Optimal Transport + Gradient Flows

Overview

In the fall term of 2023, Soumik Pal (UW) and Young-Heon Kim (UBC), will offer a graduate course on Optimal Transport + Gradient Flows. This course is part of the PIMS Network Wide Graduate Courses program and will be accessible remotely. Students in the PIMS network of Universities will be able to register for credit through the Western Deans Agreement.

This course is on the theory of Wasserstein gradient flows based on the formal Riemannian calculus due to Otto. Apart from the classical examples, we will also discuss many modern variations such as Wasserstein mirror gradient flows that come up in applications. A fruitful interaction between probability, geometry, and PDE theory will be developed simultaneously.

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.

Professors

• Young-Heon Kim (UBC and KAIST) yhkim@math.ubc.ca
• Soumik Pal (UW) soumikpal@gmail.com

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

Canadian PIMS students can register through the Western Deans Agreement as course Math 606D:101 at UBC. Please see the PIMS Network wide courses webpage](https://courses.pims.math.ca) for more information about the WDA and registration deadlines at various universities. UW students can register for MATH 581 F. Otherwise please write to 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. For those of you who are already familiar with basics of OT, this part will be mostly reviews. The lecture videos will be recorded and provided for your review.. This will be covered between AUG 29 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

This event is part of the Pacific Interdisciplinary Hub on Optimal Transport (PIHOT) which is a collaborative research group (CRG) of the Pacific Institute for the Mathematical Sciences (PIMS).