Toric compactifications and discrete periodic operators

Toroidal compactifications of Bloch varieties and Fermi surfaces for operators on
discrete periodic graphs were used explicitly in the 1990's and implicitly
recently.  In this work, the graphs all had a similar (but common and important)
structure.  The theoretical foundations for toroidal compactifications---toric
varieties has advanced considerably since the 1990's, including their
structures as real algebraic varieties.

  I will explain how to associate a pair of projective toric varieties to any
discrete periodic graph G such that the Bloch variety and Fermi surfaces of any
operator on G are naturally hypersurfaces in these toric varieties.  Not only does
this provide a uniform construction of compactifications, but these toric
varieties naturally admit a non-standard algebraic anti-holomorphic involution.
When the operator is self-adjoint, the Bloch variety and Fermi surfaces become
real algebraic hypersurfaces in their ambient non-standard real toric variety.