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  Let where is a smooth function on and the usual Laplace operator on . Show that with respect to the measure , the operator is essentially self-adjoint on . Compute for the previous operator. Show that if, as a … Continue reading →

I am currently looking for a postdoc starting in August 2017 in the Mathematics department at the University of Connecticut. The position is described here: Assistant Research Position Please apply through mathjobs.org and contact me for further details.    

Exercise 1. Let be a locally subelliptic and essentially self-adjoint diffusion operator. Let be the semigroup generated by . By using the maximum principle for parabolic pdes, prove that if is in , then . Exercise 2: Let be an essentially self-adjoint … Continue reading →

Exercise: Let be an elliptic diffusion operator with smooth coefficients that is essentially self-adjoint with respect to a measure . Let be a multi-index. If is a compact set of , show that there exists a positive constant such that for … Continue reading →

In the previous lectures, we have proved that if L is a diffusion operator that is essentially self-adjoint then, by using the spectral theorem, we can define a self-adjoint strongly continuous contraction semigroup with generator L and this semigroup is … Continue reading →

This lecture is an introduction to the regularity theory of diffusion operators. Most of the statements will be given without proofs. For a good and easy  introduction to the theory in the elliptic case, we refer to the book : … Continue reading →

From September 1 to December 12, the Institut Henri Poincare will organize a thematic semester: Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds. I look forward to this event. I will be in sabbatical this Fall and it will be the … Continue reading →

An important part of my research from the past few years has been to try to understand the notion of Ricci curvature lower bound in sub-Riemannian geometry. On this problem, my point of view is more the one of an … Continue reading →