Seminar on Optimal Transport Theory

• Target audience: bacheler+master students in mathematics from 3rd year of studies
• required background: solid knowledge of calculus, including measure theory
• teaching: Thomas Schick, Bernadette Lessel
• contact: schick@uni-math.gwdg.de, blessel@uni-math.gwdg.de
• Organizational meeting: Friday, September 27, 14:15 in Sitzungssaal

Optimal Transport Theory is a modern and many-sided branch of mathematics. At the basis lies the 1781 question of Monge: how to redistribute a given mass distribution (e.g. in R^3) most cost-efficiently into another given mass distribution.

Formulated in the language of Measure theory, this means: Given two metric spacesX and Y (e.g. subsets of $\mathbb{R}^n$), probability measures m on X and m', on Y, and a cost function c from X times Y to R, find a transport-plan which optimizes the mass transportation between m and m'. A transport plan is a probability measure $\gamma$ on X x Y, with m,m' as marginals: That means it should hold g(A\times Y)=m(A) and g(X\times B)=m'(B) for all measurable A subset X and B subset Y. One can say that g(A\times B) is the mass transported from A to B. A plan g optimizes the mass transportation between m and m' if the following expression is minimal: W(m,m') the integral over X\times Y of c with respect to the measure g.

Beside questions of existence, uniqueness und regularity of optimal transport-plans, this theory also clarifies questions concerning the meaning of the numbers W(m,m'): For certain cost-functions, W is a metric on the space of all probability measures, the so called Wasserstein metric, which naturally extends the metric of the underlying metric space!

Topics

Our seminar will be based on ``A user's guide to optimal transport'' by L. Ambrosio and N. Gigli (see ``Literature''), supplemented by further literature. The first four talks treat the Optimal Transport problem, while the the following talks will be concerned with the Wasserstein metric and further structures, and will be chosen regarding the participants interests and background. Possible for further talks are topics regarding Displacement interpolation, formal Riemannian Structure on the Wasserstein-Space, theory of Gradient Flows on metric spaces and especially on the Wasserstein-Space. The talks can be held either in english or german.

More details: Thomas Schick's webpage and soon stud.ip.

Initial program:

• 1.) Chapter 1.1 of [1]: ``Monge and Kantorovich formulations of the optimal transport problem'', Introduction of [4], Example 4.9 of [3].
• 2.) Chapter 1.2 of [1]: ``Necessary and sufficient optimality conditions'', Section 2.1.3. of [4], ``Definitions and heuristics'' of chapter 5 of [3].
• 3.) Chapter 1.3 of [1]: ``The dual problem'', Sections 1.1-1.7, 1.2 of [4]. <'li>
• 4.) Chapter 1.4 of [1]: ``Existence of optimal maps'', 2.1.5-2.1.6 of [4].

Literature

• ``A user's guide to optimal transport'' by L. Ambrosio and N. Gigli (legally and for free available in the Internet)
• ``Second order analysis on (P₂(M),W₂)'' by N. Gigli (is also online)
• ``Optimal Transport. Old and New'' by C. Villani (Springer, 2009)
• ``Topics in Optimal Transportation'' by C. Villani (Springer, 2003)