Matroids, Incidence Theorems, and TilingsOberseminar Darstellungstheorie

by Lukas Kühne (Uni Bielefeld)

Friday Jun 19, 2026, 2:15 PM → 3:15 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

A matroid is a fundamental and actively studied object in combinatorics. Matroids generalize linear independence in vector spaces as well as many aspects of graph theory. After a short introduction to matroids, I will present parts of a new OSCAR module for matroids through several examples. I will focus on computing the moduli space of a matroid, which is the space of all arrangements of hyperplanes with that matroid as their intersection lattice.

 

Fomin and Pylyavskyy describe how to obtain incidence theorems from tilings of an orientable surface; they call this result the "master theorem." Since most classically known incidence theorems, such as Pappus’s and Desargues’s theorems, are instances of the master theorem, they ask whether this holds for all incidence theorems. As an application of the presented OSCAR module, we provide an explicit example of an incidence theorem involving 13 points, based on a matroid with an exotic moduli space, that is not an instance of the master theorem.

 

Based on joint work with Matt Larson.