The geometry of the realization spaces of matroids

Fatemeh Mohammadi, Katholieke Universiteit Leuven

Online Talk 

I will talk about matroids and their realization spaces, which provide a stratification of the Grassmannian. The realization space of a matroid is a parameter space of hyperplane arrangements of fixed combinatorial type. I will focus on the geometric properties of this space and its Zariski closure (matroid variety) such as their reducibility and defining equations. In particular, the reducibility of a matroid variety is related to the topology of the complement of its associated hyperplane arrangement. For example, Rybnikov has constructed two combinatorially equivalent hyperplane arrangements with non-homeomorphic complements, which implies the reducibility of the realization space of the corresponding matroid. Finally, I will mention the application to conditional independence models in statistics and will present some computational challenges around this problem.

The talk is based on joint works with Kevin Grace and Oliver Clarke.