Toric Geometry of Periodic Operators

The Laplacian of a Z^d-periodic  graph, or more generally of
a   Schroedinger   operator,   has   spectrum   (generalized
eigenvalues) that is a union  of intervals in the real line.
More   structure   is   revealed   from   the   vantage   of
representations of Z^d:  The operator becomes a  map of free
modules over the  character ring and the  spectrum becomes a
real algebraic hypersurface in a torus, with a natural toric
compactification.  Many natural questions about the spectrum
become   accessible   using   the  tools   of   our   trade:
computational, combinatorial,  and real  algebraic geometry.
I will sketch  the background and discuss  some results from
this perspective.