Kris Shaw (UiO): Oriented matroids and real toric varieties

Matroids are an axiomatic formulation of the notion of independence in mathematics, coming for example from graph theory or linear algebra. Oriented matroids are matroids with additional structure, which sometimes arise from directed graphs or linear algebra over the real numbers.

In this talk, I will describe how to extend recently established connections between matroid theory and algebraic geometry to the realm of oriented matroids. For ordinary matroids, this connection comes from associating polyhedral fans to matroids, which in turn yield toric varieties, Minkowski weights, and tropical varieties all having very special properties. In joint work with Johannes Rau and Arthur Renaudineau, we define "real phase structures" on matroid fans and prove that such a structure is equivalent to providing an orientation of the underlying matroid. Then using a generalisation of Viro's patchworking, we first recover Folkman and Lawrence's famous topological representation theorem for oriented matroids. In addition, a matroid fan together with a real phase structure determines a homology class in a real toric variety and the conditions for real phase structures can be thought of as the real analogues of Minkowski weights for toric varieties. Lastly, I will formulate some homological obstructions to matroid orientability. Matroid orientability is known to be an NP-complete problem by work of Richter-Gebert.

This talk is partially based on joint work with Johannes Rau and Arthur Renaudineau.