Abstract: Magic angles are a hot topic in condensed matter physics:
when two sheets of graphene are twisted by those angles the resulting
material is superconducting. I will present a very simple operator
whose spectral properties are thought to determine which angles are
magical. It comes from a 2019 PR Letter
by Tarnopolsky--Kruchkov--Vishwanath.
The mathematics behind this is an elementary blend of representation theory
(of the Heisenberg group in characteristic three), Jacobi theta functions and
spectral instability of non-self-adjoint operators (involving Hörmander's
bracket condition in a very simple setting). Recent mathematical
progress also includes the proof of existence of generalized magic
angles and computer assisted proofs of existence of real ones
(Luskin--Watson, 2021).
The results will be illustrated by colourful numerics which suggest many
open problems (joint work with S Becker, M Embree, J Wittsten in
2020 and S Becker, T Humbert and M Hitrik in 2022).