Oberseminar Stochastik

04.02.2025

Fabio Frommer (JGU Mainz)

Solutions of the Kirkwood-Salsburg equations at negative activity

It is well-known that the correlation functions of grand-canonical Gibbs measures satisfy the Kirkwood-Salsburg equations. If the activity is small enough it can be shown that this solution is unique and an analytic function of the activity. We show that in this case the solution of the Kirkwood-Salsburg equations at negative activity correspond to the Janossy densities of the so-called Kirkwood-closure process. This is an extension of the existence result of Kuna et al.

Einladung

On the reliability of Maximum Parsimony for encoding and reconstructing phylogenetic trees

Phylogenetic trees play a major role in the reconstruction and representation of evolutionary relationships among different species. Maximum parsimony (MP) is one of the oldest and simplest phylogenetic tree reconstruction criteria. While it is not based on a nucleotide substitution model but works in a purely combinatorial fashion, it is "folklore knowledge" amongst biologists that it works well whenever the number of substitutions is relatively small. Proving this assertion, which in some regard can be viewed as an extension of the famous Buneman theorem in mathematical phylogenetics, is mathematically quite intriguing. In my talk, I will provide some first steps in this regard, and I will make use of some beautiful combinatorial properties of MP. The results presented in my talk can be regarded as an important step towards proving that MP is justified whenever the number of substitutions is sufficiently small. I will also highlight how these findings on MP impact Maximum Likelihood, another famous tree reconstruction criterion. I will conclude my talk by pointing out some areas of ongoing and future research.

Einladung