Minicourse on exchangeable coalescents

by Jean Bertoin , UniversitÚ Paris VI

Dates: September 11-15

This course will consist of 2x45 minute lectures each day, 10 lectures in total.

The lectures will be 15:15-17:00 each day in the room Euler.
(There is a small chance the last Friday lecture will be moved to 13:15
but this will be decided much later).

There will be some background lectures by JS given on
August 31 10:15-12:00 and September 1 13:15-15:00 (both in MVL14).

Abstract

In a celebrated work published in 1982, Kingman constructed a remarkable coalescent process which is related to the genealogy of certain large population models. More recently, M÷hle, Pitman, Sagitov and Schweinsberg introduced a natural extension of Kingman's coalescent, which we shall refer to here as exchangeable coalescents. Roughly speaking, an exchangeable coalescent describes the evolution of a particle system in which particles coagulate as time passes. The coagulations can be both multiple (i.e. involve more than two particles) and simultaneous (i.e. several clumps of particles can merge at the same time), and enjoy an important property of exchangeability. The purpose of this mini-course is to introduce the main aspects of this theory and to present some recent developments. It is largely based on chapters 2 and 4 of my monograph "Random fragmentation and coagulation processes" (Cambridge Studies in Advanced Mathematics no. 102, 2006).

Program of the course

1. Various notions of partitions
2. Kingman's theory of exchangeable random partitions
3. Poisson-Dirichlet random partitions
4. Kingman's coalescent
5. Exchangeable coalescents: definition and first properties
6. Rates of coagulation and construction of exchangeable coalescents
7. Characterization of coagulation rates
8. Masses in exchangeable coalescents
9. Simple coalescents and generalized Fleming-Viot processes
10. The Bolthausen-Sznitman coalescent

For further information, email me at steif@math.chalmers.se