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Mathematics Colloquium

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Date:  Tuesday, February 25, 2014

Title:  Stochastic Analysis and Its Applications to Geometric Problems

Abstract:  Recent popularity of probability theory and stochastic analysis beyond their traditional confine is to a large extent due to successful applications of their methods and results in other areas of pure and applied mathematics. In this talk I will explain how probability theory and stochastic analysis can be applied to certain problems from analysis and differential geometry. The central object of these applications is Brownian motion on a Riemannian manifold, a diffusion process generated by the Laplace-Beltrami operator. Its transition density function is the fundamental solution of the attendant heat equation. This connection between stochastic analysis and classical analysis and differential geometry makes it possible to study certain geometric properties of Riemannian manifolds by techniques from stochastic analysis. In this context, the concept of stopping times play an important role. I will explain the basic framework of the theory and showcase some of the most interesting results from this fruitful union of stochastic analysis and differential geometry, including eigenvalue estimates, heat kernel asymptotics, harmonic functions on manifolds, escape rate of Riemannian Brownian motion, and diffusion and geometric models from financial mathematics.


Speaker:  Elton Hsu
Institution:  Northwestern University

 

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