- 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 ),
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 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!

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].

- ``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 (
*P*_{2}(*M*),*W*_{2})'' 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)