Pacific Northwest Probability Seminar

The Thirteenth Northwest Probability Seminar
October 15, 2011
Supported by the Pacific Institute for the Mathematical Sciences (PIMS) and Microsoft Research.

Speaker photographs

The Birnbaum Lecture in Probability will be delivered by Steven Evans (University of California, Berkeley) in 2011.

Northwest Probability Seminars are one-day mini-conferences held at the University of Washington or Microsoft Research and organized in collaboration with the Oregon State University, the University of British Columbia and the University of Oregon. There is no registration fee.

The Scientific Committee for the NW Probability Seminar 2011 consists of Omer Angel (U British Columbia), Chris Burdzy (U Washington), Zhenqing Chen (U Washington), Alexander Holroyd (Microsoft), Yevgeniy Kovchegov (Oregon State U), David Levin (U Oregon) and David Wilson (Microsoft).

The talks will take place in Savery Hall 260. See the map the location of Savery Hall and Padelford Hall (the Department of Mathematics is in the Padelford Hall). More campus maps are available at the UW Web site.

Parking on UW campus is free on Saturdays only after noon. See parking information.

### Schedule

• 10:40 Coffee Savery 260
• 11:10 - 11:50 Ori Gurel-Gurevich, University of British Columbia
Linear Cover Time is Exponentially Unlikely
We will show that the probability that a simple random walk will cover a finite, bounded degree graph in linear time is exponentially small. Joint work with Itai Benjamini and Ben Morris.
• 12:00 - 1:30 Lunch, catered, Walker-Ames Room, Kane Hall.
• 1:30 - 2:20 Steven Evans, University of California, Berkeley
"Birnbaum Lecture": Time and chance happeneth to them all: Mutation, selection and recombination
Many multi-cellular organisms exhibit remarkably similar patterns of aging and mortality. Because this phenomenon appears to arise from the complex interaction of many genes, it has been a challenge to explain it quantitatively as a response to natural selection. I survey attempts by me and my collaborators to build a framework for understanding how mutation, selection and recombination acting on many genes combine to shape the distribution of genotypes in a large population. A genotype drawn at random from the population at a given time is described in our model by a Poisson random measure on the space of loci, and hence its distribution is characterized by the associated intensity measure. The intensity measures evolve according to a continuous-time, measure-valued dynamical system. I present general results on the existence and uniqueness of this dynamical system, how it arises as a limit of discrete generation systems, and the nature of its equilibria.
• 2:30 - 3:10 Bartek Siudeja, University of Oregon
Heat kernels and spectral theory
I will discuss connections between Brownian motion, heat equation, and spectral theory of Laplace operator. We will see how probabilistic methods can lead to analytic and geometric results involving Dirichlet eigenvalues. The approach will later be generalized to other processes and their usually non-local generators.
• 3:20 - 3:40 Coffee break Savery 260
• 3:40 - 4:20 Jason Miller, Microsoft Research
Imaginary Geometry of the Gaussian Free Field
Fix constants $\chi >0$, $\theta \in [0,2\pi)$, and let $h$ be an instance of the Gaussian free field on a planar domain. We will describe the flow lines of the vector fields $e^{i(h/\chi+\theta)}$ starting at a fixed boundary point of the domain. Letting $\theta$ vary, one obtains a family of curves that look locally like ${\mathrm SLE}_\kappa$ processes with $\kappa \in (0,4)$ (where $\chi = \tfrac{2}{\sqrt{\kappa}} -\tfrac{ \sqrt{\kappa}}{2}$) which we interpret as the rays of a random geometry with purely imaginary curvature. In contrast to what happens when $h$ is smooth, flow lines of different angles cross each other at most once but subsequently bounce off of each other. Flow lines of the same angle started at different points may merge into each other, forming a tree structure. We also construct so-called counter flow lines (${\mathrm SLE}_{16/\kappa}$) within the same geometry as ordered light cones'' of points accessible by angle-restricted trajectories and develop a robust theory of flow and counterflow line interaction. We will explain how this theory leads to new results about ${\mathrm SLE}$. Based on joint work with Scott Sheffield.
• 4:30 - 5:10 Steffen Rohde, University of Washington
Random Quasiconformal Homeomorphisms
Quasiconformal maps are a natural generalization of conformal maps and arise frequently in geometry, analysis, and dynamics. I will illustrate how they arise naturally in the study of random curves (such as SLE), trees (such as CRT), and surfaces.
• 6:00 No-host dinner.
• Restaurant: Chiang's Gourmet. 7845 Lake City Way NE, Seattle, WA 98115, Tel: (206) 527-8888