Bloch Discriminants

Frank Sottile

Abstract: For  a given a  periodic graph  G, there is  a finite-dimensional
parameter space  encoding all  possible edge  weights and  potentials.  For
each  point in  this parameter  space, one  may consider  the corresponding
Bloch variety  and its various  features (critical points and  values, band
gaps,  singularities, etc.)   Understanding/classifying how  these features
vary and change  at different points in the parameter  space is referred to
as determining the geography of parameter  space, and is a notoriously hard
yet fundamental question considered in real algebraic geometry.

In this talk, I will discuss this classification for a few diatomic graphs,
and  why  this  question  for  Bloch  varieties  is  interesting  from  the
perspective of real algebraic geometry.  For spectral theory, this recovers
the  well-known counterexample  of  Filonov and  Kachkovskiy to  optimistic
hopes for critical  points of Bloch varieties, while  providing the context
of the  operators with other parameters  on the same graph.   This is joint
work with Margaret Regan and Simon Telen.