The main subject of this course is the relation between optimaltransport theory and lower curvature bound. If we define the optimality of transport, then an optimal transport path becomes a minimal geodesic. The behavior of minimal geodesics tells us how a space is curved. In this course, among curvartures, we mainly discuss Ricci curvature, which controls the growth of the volume of metric ball.
We first study how to formulate an optimal transport problem as a variational problem on the space of probability measures. Then we briefly review Riemannian geometry. Finally, we investigate how Ricci curvature appears in optimal transport world.
・State the mathematical formulation of optimal transport problem
・Determine the optimality of transport plan
・Be familiar with Riemannian geometry
・Provide examples of Riemannian manifold with positive Ricci curvature
・Understand the relation between optimal transport theory and Ricci curvature
optimal transport theory, variatinal problem, metric space, complete, separble, geodesic, metric measure space, Riemannian geometry, Ricci curvature, Jacobi field, entropy, convexity, Brunn--Minkowski inequality
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
This is a standard lecture course. There will be some assignments.
Course schedule | Required learning | |
---|---|---|
Class 1 | The following topics will be covered in this order : -- optimal transport theory on a finite set -- optimal transport theory on Euclidean space -- manifold -- Riemannian manifold -- curvature -- Riemannian distance function -- Riemannian volume form -- Jacobi field -- entropy -- curvature-dimension condition -- Brunn--Minkowski inequality | Details will be provided during each class session. |
None required
Cédic Villani, Optimal Transport: Old and New, Springer, 2000.
Assignments (100%)
No prerequisites.