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