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Graduate Student Seminar: "Analysis on Manifolds via Li-Yau Gradient Estimates"

Xavier Ramos Olivé, UC Riverside

Friday, November 17, 2017
  1:10–2 p.m.

Location: Surge Building 284
  Parking Information

Category: Seminar

Description: We are always told that the motivation for defining a smooth structure on a manifold is to be able to do calculus and analysis on manifolds. But how exactly is this done, and why? Will analysis give us information about our manifold? In this talk we will see how to define some natural differential equations on Riemannian manifolds, and how studying their solutions we can get topological information of the underlying manifold. We will do this via an example: by studying the so called Li-Yau gradient estimates of the heat kernel, with a particular focus to their relationship to the Ricci curvature. These estimates can be used to derive some bounds on the Betti numbers of the manifold. If time permits, we will explore some different strategies to derive the gradient estimate under different curvature assumptions, although to protect our sanity, we will skip all the messy computations. No previous knowledge about the concept of curvature will be required for the talk.

Additional Information: Math Seminars

Open to: General Public
Admission: Free
Sponsor: Mathematics

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