Exchangeable coalescents

The purpose of the talk is to explain some recent developments in the
theory of coalescent processes. More precisely, we consider the class of
exchangeable coalescents (a.k.a. Xi-coalescents) and study their small
time behavior in the sense of LLN for the number of blocks process. The
Kingman coalescent is the most famous and, in the sense to be explained,
rather singular exchangeable coalescent.  It will typically be assumed
that the coalescent either starts from a configuration containing
finitely many blocks, or that it comes down from infinity (CDI), meaning
that the initial number of block is infinite but at any future time this
number is finite. The asymptotic "speed of CDI" is a deterministic
function, given implicitly as a solution of a certain Cauchy problem or,
equivalently, as a definite integral (of a potentially complicated). In
special cases of interest, this integral can be computed, or estimated.
If time permits, I'll attempt to explain how speed can be used to find
the asymptotic sampling formulae analogous to the well-known Ewens'
sampling formula for the Kingman coalescent.