Optimal transport in evolution equations 1000-1M23OTE

Most of the lecture follows the script of L. Ambrosio, N. Gigli ''A user's guide to optimal transport'' limiting itself to the case of Euclidean space.

Part I: Optimal transportation problem.

1. Formulating the problem of optimal transport according to Monge and Kantorowicz.
2. Conditions equivalent to optimality of the transport plan.
3. The existence of optimal mappings.

Part II: Wassestein Metrics.

1. Introduction of Wasserstein metrics. W2 metric.
2. Basic properties of the space of measures with the W2 metric.
3. Measures with the W2 metric as geodetic space.
4. Absolutely continuous curves and the continuity equation.
5. Weakly-Riemannian structure of the measure space with the W2 metric.

Part III: Gradient flows on metric spaces

1. The notion of a gradient flow on Hilbert and metric spaces.
2. Three definitions of a gradient flow and the relationships between them.
3. Gradient flows of geodesically convex functionals.
4. Three classic examples of gradient flow.

Supplementary material:

1. The boundary of the mean field for the Vlasov equation. The limit in the deterministic variant.

Type of course

elective monographs

Requirements

Functional Analysis

Prerequisites (description)

The participant should know the basic theorems of functional analysis. Advanced knowledge of measure theory is not required, but not advisable. The basics of measure theory related to such lectures as Mathematical Analysis and Probability should be sufficient. The participant can expect short excursions into differential equations.

Course coordinators

Jan Peszek

Learning outcomes

The students know and understand the concept of gradient flow and its relationship with transport of mass and the continuity equation. The students know where to look for extensions of the material from the lecture and have a sufficient understanding of the subject to continue studying on their own.

Assessment criteria

Two to choose from: activity in exercises, one fairly extensive homework, oral exam.

Bibliography

Ambrosio, Gigli ''A user's guide to optimal transport'',
Ambrosio, Gigli, Savare ''Gradient flows: In metric spaces and in the space of probability measures'',
François Golse, ''Mean-Field Limits in Statistical Dynamics''.

Additional information

Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:

Master's degree, second cycle programme, Mathematics

Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system:

Description of 1000-1M23OTE in USOSweb