Title: Multiple partition structures.
Abstract.
I will give an overview on multiple partition structures which are sequences of probability measures on families of Young diagrams subjected to a consistency condition. Multiple partition structures
are generalizations of Kingman's partition structures, and are motivated by a problem of population genetics.
They are related to harmonic functions and coherent systems of probability measures on a certain branching graph.
The vertices of this graph are multiple Young diagrams (or multiple partitions), and the edges depend on the Jack parameter.
I will present a representation theorem for multiple partition structures. In addition, I will discuss
a multiple partition structure which is expected to be relevant for
a model of population genetics for the genetic variation of a sample of gametes from a large population. I will explain that this multiple partition structure
can be represented in terms of a multiple analogue of the Poisson-Dirichlet distribution called
the multiple Poisson-Dirichlet distribution.