Topics in analysis on metric spaces

EPFL March/April 2010

Marc Troyanov (EPFL) and Felix Stephane  (University of  Bonn)

The main goal of these lectures is to give a gentle introduction to the theory of optimal mass transport theory and its applications in Riemannian and metric geometry. We will try to be as elementary as possible.

We will discuss in particular the following topics:

1.   Introduction to Optimal Transport Theory (Monge-Kantorovitch problem)
2.   The Wasserstein distance between measures.
3.   The Riemannian case and relations to Ricci curvature.
4.   Generalization to metric measure spaces.

References

• Selected parts of the book Optimal Transport, old and new (998 pages) from C. Villani's Home page
• Ambrosio, Luigi; Gigli, Nicola; Savaré, Giuseppe Gradient flows in metric spaces and in the space of probability measures. Second edition. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008.
• 2001 Lecture notes from Ambrosio at ETHZ available here
• For chapter 4 the paper Optimal Transport and curvature  (35 pages) from C. Villani's Home page
  

Prerequisites  Some notions of measure theory and Riemannian geometry are (or should be) necessary and sufficient to follow the class.

Format  The course will be given in 5 blocks of 3 hours each, once a week.

Credits  Interested students can obtain one credit for the doctoral school

Schedule   Wednesday  9:15-12:00 (March 3, 10, 17, and April 14 and 21).

Where EPFL, Building BCH  (salle BCH 2101).

For any questions, please contact the organisers (Marc Troyanov or  Stéphane Félix)