UVA Probability Seminar
Wednesdays, 3:30 - 4:30pm
Kerchof 326
Organizers: Christian Gromoll & Tai Melcher
Mathematics Other Seminars Maps & Directions

Spring 2013

16 Jan Probability Seminar Organizational Meeting
30 Jan Crash Course in Markov Chains I
6 Feb Crash Course in Markov Chains II
13 Feb Crash Course in Markov Chains III
20 Feb Scott McKinley (U Florida)
Fluctuating hydrodynamics of microparticles in biological fluids: modeling, simulation and analysis
27 Feb Crash Course in Markov Chains IV
20 Mar Shishi Luo (Duke)
Multiscale evolutionary dynamics: A measure-valued process perspective
3 Apr Paul Jung (UAB)
Symmetry breaking in some Coulomb systems
10 Apr Bryce Terwilliger
Intro to extreme value theory I
17 Apr Nate Eldredge (Cornell)
Widder's representation theorem for Dirichlet forms
24 Apr Dan Dobbs* (meets in Kerchof 317)
Properties of measures and processes related to BMs on infinite-dimensional Heisenberg-like groups
1 May Bryce Terwilliger
Intro to extreme value theory II
* please note time, date, and/or location change

Abstracts

Fluctuating hydrodynamics of microparticles in biological fluids: modeling, simulation and analysis

Scott McKinley (U Florida)

Recent progress in advanced microscopy reveals that foreign microparticles in biological fluids exhibit anomalous diffusive behavior. Intrinsic to particle trajectories are time and length scale correlations that challenge conventional probability frameworks.

This talk will consist of two parts. In the first half we will look at microparticle tracking data (joint work with David Hill and Greg Forest among others), looking in particular at the statistical signals that endorse using fractional Brownian motion as our base model. In the second half I will present the Landau-Lifshitz formulation of thermally fluctuating viscous fluids with an eye toward describing particle-particle interactions. The stochastic PDEs associated with these models pose numerous computational and analytical challenges though. I will give one example of analytical work (joint with Jonathan Mattingly and Natesh Pillai) in which we establish geometric ergodicity of a bead-spring pair with stochastic Stokes forcing. The method employs control theoretic arguments, Lyapunov functions and hypoelliptic diffusion theory to prove exponential convergence via a Harris chain argument.
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Multiscale evolutionary dynamics: A measure-valued process perspective

Shishi Luo (Duke)

Evolution by natural selection can act at multiple biological levels, often in opposing directions. This is particularly the case for pathogen evolution, which occurs both within the host it infects and via transmission between hosts, and for the evolution of cooperative behavior, where individually advantageous strategies are disadvantageous at the group level. In mathematical terms, these are multiscale systems characterized by stochasticity at each scale. We show how a simple and natural formulation of this can be viewed as a ball-and-urn (measure-valued) process. This equivalent process has very nice mathematical properties, namely it converges weakly to either (i) the solution of an analytically tractable integro-partial differential equation or (ii) a Fleming-Viot process. We can then study properties of these limiting objects to infer general properties of multilevel selection.
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Symmetry breaking in some Coulomb systems

Paul Jung (UAB)

The jellium is a system of positive point charges placed in a neutralizing background of negative charge (with Lebesgue density). The Gibbs measures governing the point charges are determined by Coulomb interactions. In one dimension, translation symmetry breaking in the Gibbs measures was shown by Kunz (74), Brascamp and Lieb (75), Aizenman and Martin (80), and Aizenman, Goldstein and Lebowitz (01) using varied techniques.

Motivated by a connection with Laughlin states in two-dimensional strips, Jansen, Lieb and Seiler (09) showed symmetry breaking for the jellium in such strips whenever the inverse temperature 1/T is an even integer. We extend this result to all temperatures. The quantum version, using Fermi-Dirac statistics, was also considered by Brascamp and Lieb (75) for which it was shown that symmetry breaking occurs at low temperatures. We also extend this result to all temperatures. This talk is based on joint work with M. Aizenman and S. Jansen.
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Widder's representation theorem for Dirichlet forms

Nate Eldredge (Cornell)

In classical PDE theory, a well-known annoyance is that solutions of the initial-value problem for the heat equation on R^n need not be unique. There is an "obvious" solution given by convolution with the Gaussian heat kernel, but there are other pathological solutions as well. Widder's Theorem asserts that by restricting our attention to nonnegative solutions, we can exclude these pathological solutions and recover uniqueness. This is a reasonable restriction to make, since, for instance, temperatures below absolute zero don't make physical sense.

I will discuss an extension of this theorem to the context of a metric measure space equipped with a local Dirichlet form. This provides a weak sort of geometry, and gives us a "Laplacian" that we can use to define a notion of a "solution of the heat equation", as well as a continuous Markov process that plays the role of Brownian motion. I'll begin with a primer on Dirichlet forms with some examples, describe how Widder's theorem looks in this context, and give a sketch of the proof and some of the ingredients it requires.

This is joint work with Laurent Saloff-Coste.
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Mathematics Other Seminars Maps & Directions