Matroids and Algebra

A matroid is a combinatorial object based on an abstraction of linear independence in vector spaces and forests in graphs. I will discuss how matroid theory interacts with algebra via the so-called von Staudt constructions. These are combinatorial gadgets to encode polynomials in matroids.

I will discuss generalized matroid representations as arrangements over division rings, subspace arrangements, and probability space representations together with their relation to group theory. As an application, this yields a proof that the conditional independence implication problem from information theory is undecidable.

Based on joint work with Rudi Pendavingh and Geva Yashfe.