Title: The Kingman Coalescent as a Density on a Space of Trees
Abstract: Randomly pick n individuals from a population and trace their genealogy backwards in time until you reach the most recent common ancestor. The Kingman n-coalescent is a probabilistic model for the tree one obtains this way. The common definitions are stated from a stochastic point of view. The Kingman Coalescent can also be described by a probability density function on a space of certain trees. For the space of phylogenetic trees Ardila and Klivans showed, that this space is closely related to the Bergman fan of the graphical matroid of the complete graph. Using its fine fan structure, similar things can be said about the space we are considering.
I will describe the density and report on work in progress with Christian Haase about relations of population genetics and polyhedral and algebraic geometry.
Title : Some volume identities for $W$-permutahedra
Lovasz (2001) constructed a weighted adjacency matrix $A_G$ for the vertex-edge-graph of a given $3$-polytope $P$ containing the origin such that a basis of its kernel yields vertices for $P$ up to a linear transformation. In 2010, Izmestiev extended the construction to polytopes of arbitrary dimension. We study Izmestiev’s construction for generalized $W$-permutahedra $P_W$ were $W$ denotes a finite reflection group. This yields an interesting relation between volumes of faces of the dual polytope $P_W^\\Delta$ and the volume of simplices spanned by vertices of $P_W$.
Joint works with Ioannis Ivrissimtzis, Shiping Liu and Norbert Peyerimhoff