Mathematical Physics and Harmonic Analysis Seminar

Date: February 25, 2022

Time: 1:50PM - 2:50PM

Location: BLOC 302

Speaker: Frank Sottile, TAMU

Title: Toric compactifications and discrete periodic operators

Abstract: 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.