## Einstein manifolds

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.

## Lecture notes: Dirichlet spaces

Those lecture notes are associated to a course I taught at the University of Connecticut in Spring 2019. The focus is on the theory of Dirichlet spaces and heat kernels in metric measure spaces.

## Lecture Notes: Brownian Chen series and Gauss-Bonnet-Chern theorem

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.

## Lecture notes: Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations

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.

## Lecture notes: Heat semigroups methods in Riemannian geometry

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.

## Lecture notes: An introduction to the geometry of stochastic flows

Those are the notes corresponding to my book on stochastic flows. Most of them were written in 2003 during my stay as a postdoc at the Technical University of Vienna.

## Lecture notes: Rough paths theory

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.

## Lecture Notes: Stochastic differential equations driven by fractional Brownian motions

Those are lecture notes on stochastic differential equations driven by fractional Brownian motions. It only deals with the case $H >1/2$, so that the equations are understood in the sense of Young’s integration.

Those notes correspond to a mini course given during the Finnish Summer School in Probability 2012.

## Lecture Notes: Stochastic calculus and Diffusion semigroups

Those are the lecture notes of the stochastic calculus course I have been teaching at the University of Toulouse (2003-2008) and then at Purdue University. Some parts of this book grew out of the lectures posted on this blog.