: Optimal Transport: Theory and Selected Applications 1000-1M26OTZ

: Most of the lecture follows the lecture notes by Peyré and Cuturi, Computational Optimal Transport, and Ambrosio and Gigli, A User’s Guide to Optimal Transport, restricted to the case of Euclidean spaces.

Lecture

1. Formulation of the optimal transport problem according to Monge and Kantorovich.
2. Fundamental Theorem of Optimal Transport.
3. Existence of optimal maps.
4. Introduction to Wasserstein metrics. The W2 metric.
5. Basic properties of spaces of measures equipped with the W2​ metric.
6. Measures equipped with the W2​ metric as a geodesic space.
7. Absolutely continuous curves and the continuity equation.
8. The Benamou–Brenier formula.

Exercises

Will be conducted partly in a workshop format, and their direction will depend on the number of interested students.

Topic 1: Stochastic Interpolants as Curves in Wasserstein Space. Stochastic interpolants are a tool used in generative artificial intelligence. The aim is to explain what this application consists of and what natural mathematical questions arise in this context. In particular, we will analyze the connections with the continuity equation and see how the argument from the work of Albergo et al. can be simplified using modern optimal transport theory. This topic is based on source [4].

Topic 2: Gradient Flow in a Metric Space. Gradient flows are simple dynamics that we observe every day. The aim is to take an abstract view of the notion of gradient flow in situations where the very notion of a gradient loses its classical meaning. We will show how optimal transport can be used to describe various phenomena as gradient flows. This topic is based on source [2].

Topic 3: Computational Optimal Transport. A practical component of the lecture related to concrete computational methods with direct applications. This is probably the most interesting workshop direction, but also the most demanding for interested students. This topic is based on the computational part of source [1].