For a real hyperplane arrangement A, Varchenko--Gelfand ring is the ring of functions from the chambers of A to the integers with pointwise addition and multiplication. Varchenko and Gelfand gave a simple presentation for this ring, along with a filtration whose associated graded ring has its Hilbert function given by the coefficients of the Poincaré polynomial. Their work was extended to oriented matroids by Gelfand—Rybnikov, who gave an analogous presentation and filtration. We extend this work first to pairs (A,K) consisting of an arrangement A in a real vector space and open convex set K, and then to conditional oriented matroids. Time permitting, we will discuss an interesting special case arising in Coxeter theory: Weyl cones of Shi arrangements. We find that the coefficients of the cone Poincaré polynomial of a Weyl cone are described by antichains in the root poset. This talk contains joint work with Christian Stump, Nicholas Proudfoot, and Jayden Wang.

Understanding the irreducibility of Bloch and Fermi varieties for discrete periodic operators is important in the study of the spectrum of periodic operators, providing insight into the structure of spectral edges, embedded eigenvalues, and other applications. In this talk we will present several new criteria for obtaining irreducibility of Bloch and Fermi varieties for infinite families of discrete periodic operators.