Magic angles are a topic of current interest in condensed matter physics and refer to a remarkable theoretical (Bistritzer–MacDonald, 2011) and experimental (Jarillo-Herrero et al, 2018) discovery: two sheets of graphene twisted by a certain (magic) angle display unusual electronic properties, such as superconductivity. In this talk, we shall discuss a simple periodic Hamiltonian describing the chiral limit of twisted bilayer graphene (Tarnopolsky-Kruchkov-Vishwanath, 2019), whose spectral properties are thought to determine which angles are magical. We show that the corresponding eigenfunctions decay exponentially in suitable geometrically determined regions as the angle of twisting decreases, which can be viewed as a form of semiclassical analytic hypoellipticity. This is joint work with Maciej Zworski.[-]

In this lecture I will describe a framework for the Fredholm analysis of non-elliptic problems both on manifolds without boundary and manifolds with boundary, with a view towards wave propagation on Kerr-de-Sitter spaces, which is the key analytic ingredient for showing the stability of black holes (see Peter Hintz' lecture). This lecture focuses on the general setup such as microlocal ellipticity, real principal type propagation, radial points and generalizations, as well as (potentially) normally hyperbolic trapping, as well as the role of resonances.[-]

The classical Riemann-Hilbert correspondence establishes an equivalence between the triangulated categories of regular holonomic D-modules and of constructible sheaves. In a joint work with Masaki Kashiwara, we proved a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular. The construction of our target category is based on the theory of ind-sheaves by Kashiwara-Schapira and is influenced by Tamarkin's work on symplectic topology. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya.[-]