TGiZ-Seminar: Tropical geometry in Zoom (First meeting Winter Semester 2022/23)

Benjamin Schröter (Universität Frankfurt)

Abstract:
Valuations on polytopes are maps that combine the geometry of polytopes with relations in a group. It turns out that many important invariants of matroids are valuative on the collection of matroid base polytopes, e.g., the Tutte polynomial and its specializations or the Hilbert–Poincaré series of the Chow ring of a matroid.

In this talk I will present a framework that allows us to compute such invariants on large classes of matroids, e.g., sparse paving and elementary split matroids, explicitly. The concept of split matroids introduced by Joswig and myself is relatively new. However, this class appears naturally in this context. Moreover, (sparse) paving matroids are split. I will demonstrate the framework by looking at Ehrhart polynomials, relations in Chow rings of combinatorial geometries, and further examples.

This talk is based on the preprint `Valuative invariants for large classes of matroids’ which is joint work with Luis Ferroni.