Matroids and splits

Andreas Dress studied several discrete structures, among them matroids and (finite) metric spaces with a focus on their interrelation. His famous work includes the introduction of valuated matroids, and the split decomposition theorem for metric spaces, both of which are highly relevant in phylogenetics. From a polyhedral point of view, valuated matroids are lifting functions on matroids (base) polytopes that subdivide the polytope into matroidal cells. Furthermore, splits in this language are subdivisions into exactly two maximal cells. It follows that split decomposable subdivisions are dual to trees and the matroids occurring in the case of a valuated matroid are well structured. These matroids, called split matroids, have been intensively studied in the last years and have remarkable features, in particular with respect to matroid invariants and specially the Tutte polynomial a important matroid invariant. In this talk I will report on the latest developments around split matroids.