In this course I will discuss the problem of the heat diffusion in a sub-Riemannian manifold. The program of the course will be divided among the lectures (approximatively) in the following way:

Lecture 1: Definition of sub-Riemannian manifold. Basic properties and examples. The problem of the heat diffusion.

Lecture 2: Sub-Riemannian manifolds as a metric spaces (the Chow Theorem).

Lecture 3: Geodesic equations. Normal and abnormal extremals.

Lecture 4: Examples: Grushin, Heisenberg Martinet.

Lecture 5: The heat equation in a Riemannian manifold. The heat equation in a sub-Riemannian manifold. The problem of the intrinsic definition of a volume.

Lecture 6: The Hormader theorem. The support theorem of Stroock and Varadhan.

Lecture 7: Construction of the heat kernel in the Heisenberg group.

Lecture 8: Diffusion interpreted as limit of random walks: The Riemannian case.

Lecture 9: Diffusion interpreted as limit of random walks: The sub-Riemannian case. **Location:**

- A-136 on 24/04, 26/04 and 03/05 (afternoon)
- A-134 on 02/05, 03/05 (morning)
- A-133 on 23/04, 27/04, 30/04 and 04/05