# Northwest Probability Seminar

### and Pacific Institute for Mathematical Sciences

The Birnbaum Lecture in Probability will be delivered by Ofer Zeitouni (University of Minnesota) in 2004.

Northwest Probability Seminars are one-day mini-conferences held at the University of Washington and organized in collaboration with the Oregon State University, the University of British Columbia, the University of Oregon, and the Theory Group at the Microsoft Research. There is no registration fee. Participants are requested to contact Chris Burdzy (burdzy@math.washington.edu ) in advance so that adequate facilities may be arranged for.

The Scientific Committee for the NW Probability Seminar 2004 consists of Chris Burdzy (U Washington), Zhenqing Chen (U Washington), Ed Perkins (U British Columbia), Qi-Man Shao (U Oregon) and Ed Waymire (Oregon State U).

The talks will take place in Savery 239 and 241. See the map of north-central campus for 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 after 12:00 (noon). More information is available at a parking Web site provided by UW.

### Schedule

• 10:30 Coffee - Savery Hall 239/241
• 11:00 Mina Ossiander, Oregon State University (photo)
• A Probabililistic Representation of Solutions to Incompressible Navier-Stokes Equations
The Navier-Stokes equations governing the velocity of incompressible fluids in 3 dimensions have been studied extensively over the last century. Important open questions still remain concerning the existence and uniqueness of smooth solutions for given initial velocity. The goal of this talk is to describe a representation of solutions in physical space as scaled expectations of functionals of a Markov branching process. If the forcing and initial data are jointly small enough in a certain function space this representation can be used to show that a weak solution to the Navier-Stokes equations exists and is unique for all time.

• 12:00 Rick Kenyon, University of British Columbia (photo)
• Asymptotics of random crystalline surfaces
We consider a natural family of models of random crystalline surfaces in R^3 arising in the planar dimer model (domino tiling model). For fixed boundary conditions, the law of large numbers leads to a PDE for the limit shape (when the lattice spacing tends to zero) of the surfaces. This PDE is a variant of the complex Burger's equation and can be "solved" analytically. This is surprising since the surfaces generically have both smooth parts and facets. The interplay between analytic (even algebraic) functions and facet formation in the surfaces leads to some interesting questions in real algebraic geometry.

• 1:00 - 2:30 Lunch - Student Union Building 209A ("HUB")

• 2:45 Ofer Zeitouni, University of Minnesota (photo)
• "Birnbaum Lecture": Results and challenges in the study of multi-dimensional random walks in random environments
Random walks in random environments on Z^d represent a perfect playground for probabilists: there are many interesting phenomena, several outstanding open problems, and recent progress. I will describe the model, some predictions made about it, and some partial answers. For the closely related problem of diffusions in random environments, I will describe recent results (obtained with A.-S. Sznitman) that prove diffusivity in the perturbative regime for d\geq 3.

• 3:45 Oded Schramm, Microsoft Research (photo)
• Sensitivity, complexity, harmonic analysis and exceptional times of Boolean functions
Let $f:\{-1,1\}^n\to\{-1,1\}$ be a Boolean function on the discrete cube. If $f$ is likely to change when a small collection of random bits of its input are flipped, then we say that $f$ is noise sensitive. The sensitivity of $f$ is easily read off from its Fourier expansion. We describe results by several authors relating the Fourier expansion to various measures of complexity of $f$. Some interesting conclusions are:

(1) If $f$ is monotone, then any randomized algorithm that determines $f$ has some bit that it examines with probability at least $c n^{-1/3}$. This is sharp, up to $\log n$ terms.
(2) A dynamic version of critical percolation on the triangular grid in the plane has exceptional times in which the origin is in an infinite cluster (but the exceptional times have measure zero).

• 5:30 No host dinner at Cedars Restaurant on Brooklyn. Click on the restaurant name to go to its Web page.