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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
Lecture 4. Markovian semigroups
Let be a measurable space. We say that is a good measurable space if there is a countable family generating and if every finite measure on can be decomposed as where is the projection of on the first coordinate and … Continue reading
Lecture 1. Semigroups in Banach spaces: The Hille-Yosida theorem
The first few lectures will be devoted to some elements of the general theory of operators in Banach and Hilbert spaces which are useful when studying Dirichlet forms. In this lecture, we focus on the Hille–Yosida theorem. Let be a … Continue reading
Sub-Laplacian comparison theorems
This is a talk about sub-Laplacian comparison theorems on Sasakian manifolds given at the Institut Fourier in Grenoble in October 2018. The associated paper, written in collaboration with Erlend Grong, Kazumasa Kuwada and Anton Thalmaier can be downloaded here. … Continue reading
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HW10. MA3160. Due 11/29
Exercise. Let X, Y have joint density if and 0 otherwise. (a) Find c that makes this a joint pdf: (b) Find E (X Y^2 ). (c) Find Var(X+Y)
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Research and Lecture notes 