Courses

Discrete Spatial Processes in Probability, Geoffrey Grimmett (University of Cambridge, UK)

Math 609E section 921 - catalogue #222075

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Many of the most beautiful and important problems of probability theory involve random processes associated with networks. The inspirations for such problems come often from areas of applied science stretching from physics to epidemiology and biology, and their solutions feed back into these fields. The primary target of this lecture series will be to present a coherent theory of discrete spatial processes emerging from a number of areas including: random walk, random trees, percolation, models for ferromagnets and spin glasses, and interacting particle systems. In each case, we will progress from the basics to the principal open problems. Special emphasis will be placed on connections between topics, and on generic methodology including correlation inequalities and concentration.

Brownian Motion and Analysis, Chris Burdzy (University of Washington, USA)

Math 608D section 921 - catalogue #222077

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Brownian motion is arguably the central concept of stochastic calculus. It is used as a model for real-world phenomena studied by physicists, biologists and economists. The mathematical challenge is to develop methods providing effective computational tools. While each specific application of might appear to need its own approach, several major general ideas have been invented and developed over the years.

The course will not be a list of theorems on Brownian motion. I will review a few current research problems involving Brownian motion and show how one can attack these problems using various ideas. In this way, I will introduce techniques of stochastic analysis by showing how they are applied to interesting and challenging questions. I will discuss a number of open problems to give students the taste of current research and suggest topics for their own projects.

Guest Lecture (Fri 4 July, 14:00-15:30):
Maximum overhang and the black cat problem, Yuval Peres (Microsoft Research, Redmond, USA)

A black cat starts at the origin of the lattice Z^d. In each step you can give the cat a command to jump if it is at a specific vertex v, and then it will jump to a neighboring vertex, all choices being equally likely. You do not see the cat. How many such commands are needed to ensure that the cat is at distance greater than n from the origin with probability at least 1/2?

We relate this question to the overhang problem: How many blocks are needed to build a stack that reaches distance n from the edge of a table? (This part is joint with Paterson, Thorup, Winkler and Zwick).

In the second part of the talk (Joint with L. Levine), I will discuss the rotor-router model, a deterministic analogue of random walk proposed by Jim Propp. We prove that the asymptotic shape of this model in the Euclidean lattice is a ball, and discuss open problems on the patterns of directions of particle last exits and on the related abelian sandpile model. An open problem on the sandpile model brings us back to the black cat problem.

Participants will also have the opportunity to give short presentations on their own work.