Periodic Graph Operators for Algebraic Geometers

Understanding the spectrum of the Schr\"odinger operator in a periodic
medium is a  fundamental problem in mathematical physics  that is best
approached using methods from analysis.   The discrete version of this
concerns operators on periodic graphs.   In this discrete version, the
primary  objects  are real  algebraic  varieties,  and thus  algebraic
geometry  becomes   relevant  for  the  study   of  discrete  periodic
operators.   The purpose  of  this talk  will be  to  explain this  to
algebraic  geometers, and  describe  some results  obtained from  this
perspective, as  well as some computational  and combinatorial aspects
of this study.