# Metric Geometry and Optimal Transport

### February 7, 14, 2018

### Speaker: Thibaut Le Gouic (Ecole Centrale de Marseille, Marseille)

This mini-course aims to introduce the theory of optimal transport. In order to present the theory, we will briefly recall basic notions of topology and in particular metric spaces; then introduce length spaces in order to study Wasserstein spaces. Then, we will talk about several recent works.

0) Topology and metric spaces.

I) Length spaces

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

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

3. Hopf-Rinow Theorem

4. Spaces of bounded curvature

II) 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: definition, triangular inequality, topology of the Wasserstein space

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

**Venue**: The mini-course will be held at the **HSE faculty of Computer Science** (3 Kochnovsky Proezd building 1), **room 509.**

**Time: **18:10-20:00