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Category Archives: Uncategorized
Log-Sobolev inequalities
Survey talk about log-Sobolev-inequalities
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Heat flow and sets of finite perimeter
I will be giving an invited address at the Spring Central Sectional Meeting on Sunday, April 18, 11.45am US Central time. The main topic of the address will be the study of isoperimetric inequalities and sets of finite perimeter using … Continue reading
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H-type manifolds
H-type manifolds are natural structures that arise as generalizations of H-type groups. Below is a talk I gave on that topic that took place during the conference : Sub-Riemannian Geometry and Interactions Paris, September 7–11, 2020.
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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 … Continue reading
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Caccioppoli sets, Part I
In the next two posts, longer than usual, I will explain some ideas of recent works written in collaboration with Patricia Alonso-Ruiz, Li Chen, Luke Rogers, Nageswari Shanmugalingam and Alexander Teplyaev about the study of bounded variation functions in the context of Dirichlet … Continue reading
HW5. MA3160, Due 10/07/2021
Exercise 1. Three balls are randomly chosen with replacement from an urn containing 2 blue, 3 red, and 2 yellow balls. Let denote the number of red balls chosen. What are the possible values of ? What is the density … Continue reading
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H-type structures
In two works in collaboration with Erlend Grong, Gianmarco Molino and Luca Rizzi, we introduced the notion of H-type structure: H-type foliations Comparison theorems on H-type sub-Riemannian manifolds Consider a triple where is a manifold, a constant rank sub-bundle of … Continue reading
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Lecture 8. Sobolev inequalities on Dirichlet spaces: The Varopoulos approach
In this lecture, we study Sobolev inequalities on Dirichlet spaces. The approach we develop is related to Hardy-Littlewood-Sobolev theory The link between the Hardy-Littlewood-Sobolev theory and heat kernel upper bounds is due to Varopoulos, but the proof I give below I … Continue reading
Lecture 6. The Lp theory of Markovian semigroups
Our goal, in this lecture, is to define, for , on . This may be done in a natural way by using the Riesz-Thorin interpolation theorem that we recall below. Theorem: [Riesz-Thorin interpolation theorem] Let , and . Define by … Continue reading
Research and Lecture notes 