Mini-course "Metric Geometry and Optimal Transport"

On October 8-22 associate professor at the Ecole Centrale Marseille (France) Thibaut Le Gouic read the mini-course "Metric geometry and optimal transport"

Pfogram of mini-course:
  1 Length spaces

1. De nition of length spaces: length structure, length structure induced by metric, intrinsic metric

2. Characterization of intrinsic metric: existence shortest path, existence of mid-point, complete locally compact length spaces

3. Hopf-Rinow Theorem

4. Space of bounded curvature

      2 Optimal transport

1. Monge-Kantorovitch problem: statement of Monge problem, Kantorovitch relaxation, dualilty formula

2. Existence and tightness of transferance plans

3. Cyclical monotonicity of the support

4. Wasserstein space: de nition, triangular inequality, topology of the Wasserstein space

5. Interpolation: transferance plan, characterization of geodesics, positive curvature