Algebraic Geometry and Discrete Periodic Operators

When  viewed from  the  representation theory  of Z^d,  the
spectrum of an  operator on a Z^d-periodic  graph becomes a
real-algebraic hypersurface in (S^1)^d x R.  This opens the
door to methods from algebraic geometry for use in spectral
theory.   While  this  perspective  is not  new,  there  is
renewed interest  in it,  to the  benefit of  both spectral
theory and  algebraic geometry.  My talk  will discuss some
recent work in this direction.