Title: Analytic hypoellipticity and classically forbidden regions for
twisted bilayer graphene

Abstract: The study of twisted bilayer graphene is a topic of current
interest in condensed matter physics: when two sheets of graphene are
twisted by certain, coined magic, angles, the resulting material displays
unusual electronic properties, such as superconductivity. In this talk, we
shall discuss a simple periodic Hamiltonian describing the chiral limit of
twisted bilayer graphene, which displays some striking spectral properties
occurring at magic angles. 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.

Degenerate Diffusion Equations and Applications in the Modelling of Biofilms

In this course I will give an introduction to degenerate diffusion equations motivated by models for biofilm growth. Biofilms are dense aggregations of bacterial cells attached to a surface and held together by a self-produced slimy matrix. They affect many aspects of human life and play a crucial role in natural, medical and industrial settings. 

We will discuss important properties and mathematical difficulties of degenerate diffusion equations. To this end we will look at the porous medium equation as a prototype example and compare the qualitative behavior of solutions with the qualitative behavior of solutions of the classical heat equation. In particular, we will discuss self-similarity, finite speed of propagation, regularity and travelling wave solutions. Furthermore, we will connect the theory with applications looking at models for bacterial biofilms where degenerate diffusion effects play an important role.

This is an introduction to the diffusive Hamilton-Jacobi equation $u_t-\Delta u=|Du|^p$, 

which plays an important role in optimal stochastic control as well as in the KPZ surface growth model.

The qualitative analysis of this equation has undergone numerous developments in the past 15 years and reveals a rich variety of phenomena. 

We will focus on questions of gradient blowup (GBU) for the initial-Dirichlet problem, which occurs in the superquadratic case $p>2$,

and on the related topic of Liouville type theorems. 

More in detail, the following questions will be discussed:

– Motivations from stochatic optimal control and surface evolution equations.

– Local and global existence: classical and viscosity solutions
– Sufficient conditions for GBU or for global classical existence
– Gradient estimates of Bernstein and Li-Yau types. 

– Localization of singularities on the boundary, single-point GBU and final space profiles

– Liouville type theorems and their applications to the description of singularities

– Time rates of GBU
– Post GBU behavior for viscosity solutions: loss and recovery of boundary conditions, multiple time singularities.

Global weak & generalised solutions of wave equations on singular spacetimes

In this series of lectures I will look at global solutions to the wave equation and Klein–Gordon equation on singular spacetimes and spacetimes of low regularity. In the three lectures I will describe three approaches to obtaining solutions. The first involves the use of smoothing operators to regularise the coefficients, the second uses the Galerkin approximation of the PDE by a system of ODEs, while the third uses numerical viscosity methods. In all three cases we show how to extract solutions lying in suitable Sobolev spaces.

Lecture 1

In this lecture I will begin by motivating the study of wave type equations on singular spacetimes. These are important to an approach to the study of gravitational singularities as obstructions to the evolution of test fields (described, for example, by the Klein–Gordon equation) rather than the standard description as obstructions to the evolution of test particles (described by timelike geodesics). They are also important for looking at quantum fields on non-smooth spacetimes. In the first lecture we will look at the use of smoothing operators to regularise the spacetime (M, g) to obtain a family of smooth spacetimes (M, g_ε). In order to construct global solutions we will need to do this in a way that also controls the causal structure of the spacetime. By constructing various higher-order energy estimates one can show that one can extract from the generalised solution φ_ε a solution φ in a Sobolev space H^s(M), where the order depends on the regularity of the spacetime metric.

Lecture 2

In the second lecture I will look at solutions to the wave equation on spacetimes with cosmic string-like singularities. I will start by looking at “very weak solutions” and show how these are related to Colombeau generalised solutions. I will then go on to look at the Galerkin method, where one writes the wave equation in first-order form and then approximates this by a system of ODEs. This has the advantage, compared to smoothing, that the approximate solution has naturally better regularity than the original hyperbolic equation. By using the Galerkin approximation together with energy estimates for the first-order system and its adjoint, one can extract global H^1(M) solutions for the wave equation on these spacetimes.

Lecture 3

In the third lecture we obtain general conditions under which the wave equation is well-posed in spacetimes with metrics of Lipschitz regularity. In particular, the results can be applied to spacetimes where there is a loss of regularity on a hypersurface, such as shell-crossing singularities, thin shells of matter, and surface layers. We will utilise the vanishing viscosity method to show the existence of a weak solution of the wave equation. This involves approximating the wave equation by a system of second-order parabolic equations with a parameter ε corresponding to the viscosity. As with the use of smoothing operators, we again obtain a one-parameter family of solutions but are now able to utilise results from parabolic regularity theory to have better control over the solutions and their convergence. This enables us to show that the one-parameter family converges to a weak solution of the zero-viscosity equation, which corresponds to an H^1 solution of the wave equation. The basic method we use follows that of Evans (Partial Differential Equations, American Mathematical Society, 2002), but the details differ and our results are also distinct from his since we assume less regularity and, as a result, our solutions also exhibit less regularity.