The Critical Point Degree of a Periodic Graph

Given an operator on a Z^d-periodic graph, its Bloch variety encodes
its spectrum with respect to the unitary characters of Z^d.  Finer
questions about the spectrum involve understanding the critical points
of the projection to R.  Previous work with Faust gave a bound for the
number of complex critical points in terms of the volume of the Newton
polytope of the dispersion polynomial.

This talk will present background and then describe refined bounds on
the number of critical points that are combinatorial in nature and
involve an analysis of asymptotic behavior of the Bloch variety.  This
is joint work with Faust and Robinson.