The main topic of the address will be the study of isoperimetric inequalities and sets of finite perimeter using heat kernel techniques. The first part of the talk will be elementary with an historical perspective and then I will be presenting more recent research directions. Here is a more detailed abstract.
H-type manifolds are natural structures that arise as generalizations of H-type groups. Below is a talk I gave on that topic that took place during the conference : Sub-Riemannian Geometry and Interactions Paris, September 7–11, 2020.
This Fall, I am teaching a graduate course on Einstein manifolds.
In this course we will study some topics in Riemannian and pseudo-Riemannian geometry. We will mostly focus on Ricci curvature and its applications. The course will start with basics about Riemannian and pseudo-Riemannian geometry. We will assume familiarity with differential manifolds and basic calculus on them.
We will cover the following topics:
Linear connections on vector bundles: Torsion, Curvature, Bianchi identities Riemannian and pseudo-Riemannian manifolds Get the feel of Ricci curvature: Volume comparison theorems, Bonnet-Myers theorem Ricci curvature as a PDE Einstein manifolds and topology Homogeneous Riemannian manifolds Kahler and Calabi-Yau manifolds Quaternion-Kahler manifolds
The main reference for the class will be: A.L. Besse: Einstein manifolds, Springer, 1987.
Due to the Covid pandemic those lectures are online and the videos are publically posted on a dedicated webpage.
The purpose of these notes is to provide a new probabilistic approach to the Gauss-Bonnet-Chern theorem (and more generally to index theory). They correspond to a five hours course given at a Spring school in France (Mons) in June 2009.
These notes are the basis of a course given at the Institut Henri Poincare in September 2014. We survey some recent results related to the geometric analysis of hypoelliptic diffusion operators on totally geodesic Riemannian foliations. We also give new applications to the study of hypocoercive estimates for Kolmogorov type operators.
In those lecture notes, we review some applications of heat semigroups methods in Riemannian and sub-Riemannian geometry. The notes contain parts of courses taught at Purdue University, Institut Henri Poincaré, Levico Summer School and Tata Institute.
Those are the notes of a course on rough paths theory taught at Purdue University in Spring 2013. We develop the theory according to its founder Terry Lyons’ point of view and rely on the book by P. Friz and N. Victoir.