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Probability and its Applications / Théorie des probabilités et leurs applications (Sponsored by the National Programme Committee (CRM, Fields, PIMS) / Parrainée par le Comité du Programme Nationale (CRM, Fields, PIMS))
(Martin Barlow, Rick Durrett, Claudia Neuhauser and Edwin Perkins, Organizers)

DAVID BRILLINGER, California-Berkeley

PETER CALABRESE, Cornell University, Ithaca, New York  14853, USA
Microsatellite models

Microsatellites are a type of genetic marker: specifically they are simple sequence tandem repeats in DNA, for example AT repeated 25 times in a row. Numerous models for microsatellite mutation have been proposed. We use entire genome sequence data and the likelihood ratio test to sort out which model is best.

DON DAWSON, Carleton University, Ottawa, Ontario  K1S 5B6
Some aspects of the infinitely many Alleles model

The infinitely many alleles model plays a significant role in population genetics. In this lecture we describe the analysis of this and some related models. In particular, we describe recent joint results with Shui Feng on large deviations and joint work with Andreas Greven on mean-field migration models with selection based on the infinitely many alleles model.

JOE FELSENSTEIN, University of Washington, Seattle, Washington, USA
Constructing a robotic evolutionary geneticist

Much of the data collected by evolutionary geneticists consists of population samples of molecular sequences or microsatellite genotypes. Many of the past papers in the field have involved choosing a somewhat arbitrary statistic, obtaining its expectation as a function of some parameters by great heroics, and then vaguely hoping that someone would use this to estimate the parameters. Using coalescent approaches, we are using Markov Chain Monte Carlo methods to approximate the likelihood functions for data involving a variety of evolutionary forces, including genetic drift, mutation, migration, population growth, and natural selection. Work on LAMARC, a unified computer program to allow analysis of complicated combinations of these forces, is well advanced in our laboratory. The program will use object-oriented methods and will be available for free. We hope that this will free evolutionary geneticists to spend their time on data collection and on the interpretation of the resulting likelihood surfaces, so that they will not be too offended at having their former activities replaced by a black box.

STEVE KRONE, Department of Mathematics, University of Idaho, Moscow, Idaho  83844, USA
When can one detect overdominant selection in the infinite alleles model?

Gillespie (1999), using simulations of the infinite alleles model with heterozygote advantage, noticed that when the scaled mutation rate and selection intensity get large together the behavior appears to be indistinguishable from that of the corresponding neutral model. He explains how these results for the infinite alleles diffusion imply something about the difficulty of detecting selection in large populations with fixed mutation and selection parameters.

We attempt to explain and refine these somewhat surprising observations by analyzing the limiting behavior of the likelihood ratio of the stationary distributions for the model under selection and neutrality, as the mutation rate and selection intensity go to infinity in a specified manner. In particular, we show that the likelihood ratio tends to one (i.e., the selective model cannot be distinguished from the neutral model) as the mutation rate goes to infinity, provided the selection intensity grows like the mutation rate raised to any power less than 3/2. When the power is greater than 3/2 selection can be detected. These results rely on Gaussian limit theorems for the Poisson-Dirichlet distribution, which appear to be of interest in their own right. (Joint work with Paul Joyce and Thomas G. Kurtz.)

NEAL MADRAS, Department of Mathematics and Statistics, York University, Toronto, Ontario  M3J 1P3
A model of segregation

In 1998, the New York Times ran a story about mathematical models that try to explain why racial segregation is such a persistent problem. In one version of the model, particles of two different colours live at the sites of a large grid. A fixed fraction of sites are empty. Each unit of time, some particle chooses an empty site at random and moves there if the eight nearest sites are occupied by at least two particles having the same colour as the particle that wants to move. Simulations show that large clusters occur (i.e., segregation). A weakness in this model led me to modify the rule so that there is a nonzero probability of moving to a square with too few like neighbours. Simulations and heuristics indicate a certain critical value of this nonzero probability above which large scale segregation does not occur. This has implications about the adequacy of the original model.

To be announced

JEREMY QUASTEL, University of Toronto, Toronto, Ontario
Internal DLA

Internal DLA is a stochastic growth model in which particles are produced at the origin of a multidimensional lattice and then perform independent random walks until they reach the first site which has not been previously visited. Then they stop at that forever. We show how methods of hydrodynamic limits can be applied to study the asymptotic shape, growth rate and profile of live particles in appropriate regimes.

To be announced
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