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From CD to RCD spaces

Lecturer: 
Course Type: 
PhD Course
Academic Year: 
2022-2023
Period: 
February-April
Duration: 
60 h
Description: 

Aim of the course is to provide an introduction to the world of synthetic description of lower Ricci curvature bounds, which has seen a tremendous amount of activity in the last decade: by the end of the lectures the student will have a clear idea of the backbone of the subject and will be able to navigate through the relevant literature.

We shall start by studying Sobolev functions on metric measure spaces and the notion of heat flow. Then following, and generalizing, the intuitions of Jordan-Kinderlehrer-Otto we shall see that such heat flow can be equivalently characterized as gradient flow of the Cheeger-Dirichlet energy on L2 and as gradient flow of the Boltzmann-Shannon entropy w.r.t. the optimal transportation metric W2. This provides a crucial link between the Lott-Villani-Sturm (LSV) condition and Sobolev calculus on metric measure spaces and, in particular, it justifies the introduction of `infinitesimally Hilbertian' spaces as those metric measure structures for which W1;2(X) is a Hilbert space. By further developing calculus on these spaces we shall see that on infinitesimally Hilbertian spaces satisfying the LSV condition (these are called Riemannian curvature dimension spaces, or RCD for short) the Bochner inequality holds.

We shall then discuss more sophisticated calculus tools, such as the concept of differential of a Sobolev function, that of vector field on a metric measure spaces and the notion of Regular Lagrangian Flow on RCD spaces.

We shall finally see how these are linked to the lower Ricci curvature bound - most notably we shall prove the Laplacian comparison theorem - and finally how they can be used to prove a geometric rigidity result like the splitting theorem for RCD spaces. It is worth to notice that such statement gives new information - compared to those available through Cheeger-Colding's theory of Ricci-limit spaces - even about the structure of smooth Riemannian manifolds.

Prerequisites: 
Interested students would benefit from following also the lectures in Optimal Transport given earlier by Gigli as well as De Ponti’s class.
Next Lectures: 
Monday, February 13, 2023 - 09:00 to 11:00
Tuesday, February 14, 2023 - 09:00 to 11:00
Wednesday, February 15, 2023 - 09:00 to 11:00
Monday, February 20, 2023 - 09:00 to 11:00
Tuesday, February 21, 2023 - 09:00 to 11:00
Wednesday, February 22, 2023 - 09:00 to 11:00
Monday, February 27, 2023 - 09:00 to 11:00
Tuesday, February 28, 2023 - 09:00 to 11:00
Wednesday, March 1, 2023 - 09:00 to 11:00
Monday, March 6, 2023 - 09:00 to 11:00
Tuesday, March 7, 2023 - 09:00 to 11:00
Wednesday, March 8, 2023 - 09:00 to 11:00
Monday, March 13, 2023 - 09:00 to 11:00
Tuesday, March 14, 2023 - 09:00 to 11:00
Wednesday, March 15, 2023 - 09:00 to 11:00
Monday, March 20, 2023 - 09:00 to 11:00
Tuesday, March 21, 2023 - 09:00 to 11:00
Wednesday, March 22, 2023 - 09:00 to 11:00
Monday, March 27, 2023 - 09:00 to 11:00
Tuesday, March 28, 2023 - 09:00 to 11:00
Wednesday, March 29, 2023 - 09:00 to 11:00
Monday, April 3, 2023 - 09:00 to 11:00
Tuesday, April 4, 2023 - 09:00 to 11:00
Wednesday, April 5, 2023 - 09:00 to 11:00
Monday, April 17, 2023 - 09:00 to 11:00
Tuesday, April 18, 2023 - 09:00 to 11:00
Wednesday, April 19, 2023 - 09:00 to 11:00
Monday, April 24, 2023 - 09:00 to 11:00
Tuesday, April 25, 2023 - 09:00 to 11:00
Wednesday, April 26, 2023 - 09:00 to 11:00

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