Algebra and Combinatorics Seminar

Date: October 7, 2022

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Matthew Faust, TAMU

Title: On the Irreducibility of Bloch and Fermi Varieties

Abstract: Given an infinite ZZ^n periodic graph G, the Schrodinger operator acting on G is a graph Laplacian perturbed by a potential at every vertex. Complexifying and choosing an M-periodic potential for some full rank free module M of ZZ^n fixes a representation of our operator as a finite matrix whose entries are Laurent polynomials. The vanishing set of the characteristic polynomial yields the Bloch variety, the vanishing set for fixed eigenvalues gives the Fermi variety. Questions regarding the algebraic properties of these objects are of significant interest in mathematical physics. We will focus our attention on the irreducibility of these varieties. Understanding the irreducibility of Bloch and Fermi varieties 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. This is joint work with Jordy Lopez.