See the syllabus on my web page.

Textbook: No required textbook but the material is loosely based on

_Optimal Transport, Old and new, by C. Villani (ISBN 978-3-540-71050-9)

_Gradient Flows in metric space and in the space of probability measures, by L. Ambrosio, N. Gigli and G. Savare (ISBN-13: 978-3764387211).

General plan of the class:

- Chapter 1 The basics of Riemannian geometry in finite dimension: gradients, geodesics, the exponential map. Applications and examples with gradient flows in finite dimension.
- Chapter 2, An introduction to the Monge-Kantorovich optimization question: Basics on probability measure, the Monge formulation, Kantorovich formulation and Brenier's polar factorization.
- Chapter 3, Otto calculus I: Understanding optimal transport as a pseudo-Riemannian structure on the space of probability measures.
- Chapter 4, Otto calculus II: How to define and understand gradient flows, application to the heat equation as a gradient flow.
- Chapter 5, Processes in infinite dimension: Basic definitions of measures of measures, the limit of simple linear processes and PDE's in infinite dimension.
- Chapter 6, Non-linear processes in infinite dimension: The gradient of a measure of measure, applications to non-linear PDE's in infinite dimension.