Next Fall, I will teach a graduate course on curvature dimension inequalities, and, as usual, the Lectures will be posted on this blog.
The theory of curvature dimension inequalities and of their applications to the geometric analysis of manifolds is, at that time, my main research interest. I am therefore quite enthusiastic about the perspective of preparing the set of Lecture notes.
Curvature dimension inequalities were first introduced and extensively used by Dominique Bakry. The original motivation was to study hypercontractivity criteria for diffusion semigroups and boundedness properties of Riesz transforms. They nowadays play a crucial role in the geometric analysis of manifolds, because they provide a very robust and synthetic way to analyse the impact of curvature bounds on the global geometry of a space.
We will mainly focus on curvature dimension inequalities in Riemannian geometry and, at the end of the course will cover more recent applications and generalizations to sub-Riemannian manifolds. Only very basic notions of differential geometry will be required.
The course will cover the following topics:
1. Symmetric diffusion semigroups;
2. The Laplace-Beltrami operator on a Riemannian manifold, Bochner’s formula;
2. The heat semigroup on a Riemannian manifold;
3. Li-Yau type inequalities and Harnack estimates;
4. The heat kernel proof of Bonnet-Myers theorem;
5. Sub-Riemannian manifolds with transverse symmetries;
6. Sub-Riemannian Li-Yau inequalities;
7. Open problems and recent developments: Geometric analysis of contact manifolds.