Kris Shaw: Oriented matroids and real toric varieties
Time: Thu 2021-12-02 14.00 - 15.00
Lecturer: Kris Shaw (University of Oslo)
Abstract: Matroids axiomatise the notion of independence across different fields of mathematics, such as graph theory and linear algebra. Given a matroid one can construct a polyhedral fan which produces a class in the Chow ring of a permutahedral toric variety. Recently, many matroid invariants have been obtained and better understood thanks to this connection to algebraic geometry. Oriented matroids axiomatise independence in oriented graphs or linear algebra over the real numbers. Not all matroids are orientable and deciding matroid orientability is an NP-complete problem. Using the notion of real phase structures on fans, we can associate to an oriented matroid homology classes in real toric varieties. This connection to real toric geometry provides a homological obstruction to the orientability of a matroid. The definition of a real phase structure easily extends to non-singular tropical varieties, which are polyhedral spaces that are locally matroidal fans. With this language, the bounds obtained by Renaudineau and myself on the Betti numbers of real algebraic hypersurfaces constructed by Viro's patchworking can be generalised to real algebraic subvarieties of toric varieties near the non-singular tropical limit. This is partially based on joint work in progress with Johannes Rau and Arthur Renaudineau.