Optimal transport
The topic of optimal transport goes back to XVIII century, but the last years have witnessed rapid progress in the field, and new emerging applications to the variety of different areas including mathematical finance, geometry, AI, statistics, and engineering.
We will cover basic notions and principles — the existence and uniqueness of optimal transport maps, the Kantorovich duality, cyclical monotonicity, the Brenier theorem, Wasserstein spaces. We will pay special attention to the Wasserstein 1-space and present the related needle decomposition and its applications in geometry. If time and the class interest permits, we will also cover Wasserstein gradient flows.
We will cover basic notions and principles — the existence and uniqueness of optimal transport maps, the Kantorovich duality, cyclical monotonicity, the Brenier theorem, Wasserstein spaces. We will pay special attention to the Wasserstein 1-space and present the related needle decomposition and its applications in geometry. If time and the class interest permits, we will also cover Wasserstein gradient flows.
Lecturer
Date
5th March ~ 28th May, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Thursday,Friday | 09:50 - 11:25 | Shuangqing | ZOOM 02 | 518 868 7656 | BIMSA |
Website
Prerequisite
An understanding of real analysis, with basic measure theory, and of probability theory are prerequisites for the course. Acquaintanceship with convex analysis concepts such as duality, will be an advantage. Some examples may involve PDEs, so knowledge of the PDE theory will be beneficial, but not necessary.
Syllabus
W1 March 5,6. Definition of the problem. Examples. Transport maps and transport plans. Kantorovich relaxation. One-dimensional problem.
W2: March 12, 13. Kantorovich duality. Existence of optimal plans.
W3: March 19, 20. Cyclical monotonicity as sufficient and necessary condition for optimality. c-cyclical monotone sets, c-subdifferentials, existence of optimal transport maps.
W4: March 26, 27. Brenier theorem for quadratic cost.
W5: April 2, 3. Optimal trasnport for the linear cost. Transport rays, Lipschitz potentials, needle
decomposition.
W6: April 9, 10. Optimal trasnport for the linear cost. Transport rays, Lipschitz potentials, needle
decomposition.
W7: April 16, 17. Wasserstein distances.
W8: April 23, 24. Displacement interpolation and displacement convexity.
W9: April 30. Applications: geometric inequalities.
W10: May 7, 8. Applications: geometric inequalities continued.
W11: May. 14, 15. The Monge-AmpÅLere equation.
W12: May. 21, 22. Gradient flows.
W13: May. 28. Gradient flows continued.
W2: March 12, 13. Kantorovich duality. Existence of optimal plans.
W3: March 19, 20. Cyclical monotonicity as sufficient and necessary condition for optimality. c-cyclical monotone sets, c-subdifferentials, existence of optimal transport maps.
W4: March 26, 27. Brenier theorem for quadratic cost.
W5: April 2, 3. Optimal trasnport for the linear cost. Transport rays, Lipschitz potentials, needle
decomposition.
W6: April 9, 10. Optimal trasnport for the linear cost. Transport rays, Lipschitz potentials, needle
decomposition.
W7: April 16, 17. Wasserstein distances.
W8: April 23, 24. Displacement interpolation and displacement convexity.
W9: April 30. Applications: geometric inequalities.
W10: May 7, 8. Applications: geometric inequalities continued.
W11: May. 14, 15. The Monge-AmpÅLere equation.
W12: May. 21, 22. Gradient flows.
W13: May. 28. Gradient flows continued.
Reference
References: Topics in Optimal Transportation by Villani
Optimal Transport for Applied Mathematicians by Santambrogio
Optimal Mass Transportation on Euclidean Spaces by Maggi
Supplementary references: Optimal Transport: Old and New by VillaniGradient Flows in Metric Spaces and in the Space
of Probability Measures by Ambrosio, Gigli, Savare
Mass Transportation Problems vol. 1 & 2 by Rachev, Ruschendorf
An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows by Figalli, Glaudo
Lectures on Optimal Transport by Ambrosio, Brue, Semola
Optimal Transport for Applied Mathematicians by Santambrogio
Optimal Mass Transportation on Euclidean Spaces by Maggi
Supplementary references: Optimal Transport: Old and New by VillaniGradient Flows in Metric Spaces and in the Space
of Probability Measures by Ambrosio, Gigli, Savare
Mass Transportation Problems vol. 1 & 2 by Rachev, Ruschendorf
An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows by Figalli, Glaudo
Lectures on Optimal Transport by Ambrosio, Brue, Semola
Audience
Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
No
Language
English
Lecturer Intro
After graduating from the University of Warsaw, Krzysztof Ciosmak joined the Mathematical Institute of the University of Oxford, where he defended my doctoral thesis on Optimal transport and 1-Lipschitz maps, and subsequently held a postdoctoral position. Before joining BIMSA, he held the Fields-Ontario Postdoctoral Fellowship at the University of Toronto and the Fields Institute.