Toric Geometry for Spectral Theory

The spectrum of a Schrodinger operator records the energy levels of
particles in a quantum system.  For a periodic medium, and after the
Fourier (Floquet) transform, the spectrum becomes an analytic
hypersurface in T^n x R (T := unit circle), which reveals more of its
structure.  Discretizing, we obtain an operator on a (weighted)
Z^n-periodic graph whose spectrum is an algebraic hypersurface in T^n
x R.  A number of physically important properties become questions in
algebraic geometry, including the non degeneracy of spectral edges,
embedded eigenvalues, and the density of states.

These questions were spectacularly addressed in the 1990's in a series
of papers by Baettig, Gieseker, Knoerer, and Trubowicz.  But this was
for a particuar operator on a particular periodic graph, using a
compactification in a torioidal embedding.  The ensuing 30 years have
seen a deepening of the theory of toric varieties along with a
development of spectral theory (and more open questions).  This talk
sketches some of this, in particular that these questions remain open
for operators on more general discrete periodic graphs.  It may serve
as an informal introduction to this emerging application area.