# Sub-Riemannian manifolds: from geodesics to hypoelliptic diffusion

### September 1 - 5, 2014

Four courses of 6 hours each will be organized.

## Courses

1. Geometric control and sub-Riemannian geodesics (L. Rifford, University of Nice).

This will be an introduction to sub-Riemannian geometry from the point of view of control theory. We will define sub-Riemannian structures and prove the Chow Theorem. We will describe normal and abnormal geodesics and discuss the completeness of the Carnot-Carathéodory distance (Hopf-Rinow Theorem). Several examples will be given (Heisenberg group, Martinet distribution, Grusin plane).

Filmed lectures
2. The geometry of subelliptic diffusions (A. Thalmaier, University of Luxembourg).

We discuss hypoelliptic and subelliptic diffusions; the lectures include the following topics: Malliavin calculus; Hormander's theorem; smoothness of transition probabilities under Hormander's brackets condition; control theory and Stroock-Varadhan's support theorems; hypoelliptic heat kernel estimates; gradient estimates and Harnack type inequalities for subelliptic diffusion semi-groups; notions of curvature related to sub-Riemannian diffusions.

Filmed lectures
3. Differential forms and the Hölder equivalence problem (P. Pansu, University of Paris Sud).

In 1993, Gromov asked for which exponents $\alpha$ there exist $C^\alpha$-Hölder continuous (local) homeomorphisms of Euclidean spaces to sub-Riemannian Carnot groups. I will explain some of Gromov's partial results on this question. They rely on Rumin's theory of differential forms adapted to sub-Riemannian spaces. If time permits, other approaches to the Hölder equivalence problem will be discussed. Recommended reading : Gromov's book "Carnot-Caratheodory spaces seen from within".

Filmed lectures
4. Hypoelliptic operators and analysis on Carnot-Carathéodory spaces (N. Garofalo, University of Padova).

In this course, we will define the sub-Laplacian associated with a sub-Riemannian structure, and we will describe its hypoellipticity under the Hormander condition. We will introduce the main tools for the study of sub-elliptic PDEs.

Filmed lectures

## Joint afternoon with the "Séminaire Commun d'Analyse Géométrique"

A seminar session will be organized on Friday afternoon, with the following talks:
1. Hausdorff volumes in sub-Riemannian geometry (Frédéric JEAN, ENSTA Paris Tech).
2. Hypoelliptic random walks (Laurent MICHEL, LJAD - Nice).
3. Wasserstein space of a negatively curved space (Jérôme BERTRAND, Institut de Mathématiques de Toulouse)