Co-sponsored by Chautauqua Programs

For the faculty members whose main focus is teaching undergraduate students, PCMI offers the opportunity to renew excitement about mathematics, talk with peers about new teaching approaches, address some challenging research questions, and interact with the broader mathematical community. Each year the theme of the UFP bridges the research and education themes of the Summer Session.

MARKOV CHAINS AND MIXING TIMES

David Levin, University of Oregon

Over the past twenty years, research in discrete probability – and finite Markov chains in particular – has enjoyed a great renaissance. The aim of the Undergraduate Faculty Program (UFP) is to communicate how these recent research developments can invigorate the undergraduate probability curriculum.

As part of the UFP, we will present the rudiments of a course, based on the forthcoming book Markov Chains and Mixing by D. Levin, Y. Peres, and E. Wilmer, on bounding the convergence time for finite Markov chains. (See below for a description of this topic.) Many of the examples presented will come from statistical echanics, the topic area of the Institute this year. This course will serve as a model for a second college-level course in probability, and the only background required of participants is an undergraduate course in probability. Lectures will be supplemented by discussions and examples of interactive exercises which might be incorporated into a college-level course.

About Mixing Times
How many times must a deck of cards be shuffled before it is wellmixed? If a walker explores a network by moving at each step to a randomly chosen neighboring location, how many steps must she take before the distribution of her location is spread over the entire network with little dependence on her starting position? These are both examples of Markov chains, randomly evolving systems which under mild assumptions settle down into steady-state distributions. The time required for a Markov chain to converge to its equilibrium distribution is is called the mixing time, and estimating mixing times is an active area of research in probability theory. During the UFP, we will show how exciting developments in the study of mixing times – with connections to physics and computer science – can be given an elementary exposition accessible to motivated undergraduates.

The Coordinator of PCMI's Undergraduate Faculty Program is William Barker, Bowdoin College.