The Moduli Space of Matroids

Lecture Series Framework:  A unifying framework for F1-geometry, tropical schemes and matroid theory. In this series of 3 lectures, I will present a recent approach towards F1-geometry and its links to tropical geometry, matroid theory, Lorentzian polynomials, buildings and cluster algebras.  Most of this is joint work with Matthew Baker, among several other collaborators including June Huh and Mario Kummer.

This first lecture is geared to a general audience. Its purpose is to give an impression of what the moduli space of matroids is and use derived invariants to prove results about realizations of matroids over fields.

In more detail, I will give a primer to Baker-Bowler's theory of matroids with coefficients and explain how to define a scheme theory from exotic algebraic objects (so-called bands, which generalize rings, similar to how pastures generalize fields). This allows us to construct Grassmannians over F1, which turn out to be moduli spaces of matroids.

The residue fields of these moduli spaces control the representation theory of the corresponding matroids. A structural study of these invariants leads to results in the representation theory of matroids. We will showcase this in a simple example.