In this talk, we investigate the Narrow escape problem in the unit disk, which is to find the first exit time and exit point for a Brownian particle confined within the unit disk with a reflecting boundary, except for small disjointed windows through which it can escape. This problem has practical applications in various fields. To solve this problem, we study the eigenvalue problem for the Laplacian operator with a Dirichlet boundary condition on a small absorbing part of the boundary and a Neumann boundary condition on the remaining reflecting part. We obtain rigorous asymptotic expansions of the first eigenvalue and the normal derivative of the associated eigenfunction. 

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).