Category Archives: Hypoelliptic operators

Lecture 6. Hypocoercivity

Hypocoercivity is a concept introduced by C. Villani. It aims to give quantitative estimates for the convergence to equilibrium of hypoelliptic models. In this last lecture, I present an approach to hypocoercivity which parallels the Bakry-Emery approach to hypercontractivity. It … Continue reading

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Lecture 5. The horizontal Bonnet-Myers

This lecture gives the proof of the horizontal Bonnet-Myers theorem for Riemannian foliations with totally geodesic leaves. The proof relies on diffusion semigroups methods. Part 1 covers sections 6.1 and half of section 6.2 in the Lecture Notes and part … Continue reading

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Lecture 4. The generalized curvature dimension inequality

In this Lecture 4, we use the Weitzenbock formula proved in the previous lecture to prove a generalized curvature dimension inequality, from which we will deduce Li-Yau type estimates for the horizontal heat kernel. Part 1 covers sections 4.3, 4.4 … Continue reading

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Lecture 3. Hopf fibration. Transverse Weitzenbock formulas

In this lecture we continue the study of the Hopf fibration and compute the horizontal heat kernel of this fibration. In the second part of the lecture, we come back to the general framework of Riemannian foliations and introduce a … Continue reading

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Lecture 2. Riemannian foliations. Model spaces

In this lecture, we first quickly review what was done in the previous one.  We then introduce the horizontal and vertical Laplacians and prove basic commutations results. We explain them the notion of Riemannian foliation. In the second part of … Continue reading

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Lecture 1. Introduction and Riemannian submersions

In this first lecture, I give a general introduction about the plan of the lectures and the materials to be covered and start with the study of Riemannian submersions. Thanks to  Ugo Boscain, the videos of the lectures are available … Continue reading

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Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations

I finally finished the lecture notes of my course for the Institut Henri Poincare trimester Geometry, Analysis and Dynamics on sub-Riemannian manifolds. The course which is entitled Hypoelliptic operators is divided into two parts of each 12 hours. Nicola Garofalo … Continue reading

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