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Author Archives: Fabrice Baudoin
HW2 MA5311: Due February 3
Exercise 1. Let , . Show that is a 2-dimensional smooth manifold homeomorphic to the torus . Exercise 2. Let be the stereographic projection from the north pole , and be the stereographic projection from the south pole . Show that for … Continue reading
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MA5161. HW 1 due Wednesday 1/25
Exercise 1. Show that the -algebra is also generated by the following families: where and where are Borel sets in . where and where is a Borel set in . Exercise 2. Show that the following sets are in :
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MA5311. About John Milnor
John Milnor is a renowned mathematician who made fundamental contributions to differential topology and was awarded the Fields medal in 1962. One of his most striking result is the existence of several distinct differentiable structures on the 7 dimensional sphere (!). The following … Continue reading
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MA5311: Homework 1, due Wednesday 1/25
Exercise 1. Show that the sphere is a -dimensional smooth manifold. Exercise 2. We consider the following two subsets of the plane and . Are and smooth manifolds ? Of course, justify your answer with a proof. Exercise 3. Let be a n-dimensional … Continue reading
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MATH 5311: Differential Topology
During the Spring, I will be teaching a class on differential topology. Lecture Notes will not be posted on this blog since I will be explicitly using several books. The course will mainly be organized in two parts. Part 1. Introduction … Continue reading
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Take home questions
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
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Lecture 13. The Bochner’s formula
The goal of this lecture is to prove the Bochner formula: A fundamental formula that relates the so-called Ricci curvature of the underlying Riemannian structure to the analysis of the Laplace–Beltrami operator. The Bochner’s formula is a local formula, we therefore only need to prove it … Continue reading
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Lecture 12. The distance associated to subelliptic diffusion operators
In this lecture we prove that most of the results that were proven for Laplace-Beltrami operators may actually be generalized to any locally subelliptic operator. Let be a locally subelliptic diffusion operator defined on . For every smooth functions , … Continue reading
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Lecture 11. Laplace-Beltrami operators on Rn
In this lecture we define Riemannian structures and corresponding Laplace–Beltrami operators. We first study Riemannian structures on Rn to avoid technicalities in the presentation of the main ideas and then, in a later lecture, will extend our results to the manifold case. … Continue reading
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Assistant Research Position at Uconn
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.
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Research and Lecture notes 