Mathematical Physics and Harmonic Analysis Seminar

Date: March 7, 2025

Time: 1:50PM - 2:50PM

Location: BLOC 302

Speaker: Frank Sottile, Texas A&M University

Title: A Bestiary of Bloch Varieties

Abstract: To a Z^d-periodic weighted graph with vertices V, we may associated a periodic graph operator that acts on ℓ₂(V). After Floquet transform, we obtain its Bloch variety, which is an algebraic hypersurface in T^d×R whose projection to R is the spectrum of the operator. Features on Bloch varieties such as Dirac (double) points, critical points, and their level sets (Fermi varieties) reflect spectral properties of the operator.
    With students Faust and Robinison, and using the Brazos cluster, we investigated many thousands of small periodic graphs, recording invariants and features of their Bloch anf Fermi varieties. In this talk, I will briefly discuss the background and present some examples of interesting behavior of Bloch varieties that we uncovered.