Georgia Institute of Technology College of Sciences

The study of the electronic properties of twisted bilayer graphene (TBG) has garnered much attention from the condensed matter community recently. TBG is obtained by stacking two graphene monolayers on top of each other, and rotating one of them with respect to the other. Theoretical and experimental analyses have found that the electronic properties of TBG depend very strongly on the angle between the layers. In fact, a handful of “magic” angles have been predicted at which TBG becomes a superconductor, and this has even been verified experimentally.

We present joint work with Artur Avila on delocalizing Schr\"odinger operators in arbitrary dimensions via arbitrarily small perturbations of the potential. As a consequence we obtain an analog of Simon's Wonderland Theorem for the case of dynamically defined potentials. We will discuss a mechanism based on the Feynman-Hellmann Theorem, whose infinite volume limit is instrumental in establishing delocalization in infinite volume. Furstenberg's correspondence principle then yields the desired delocalization statement in finite volume.

The local spacing of zeros of orthogonal polynomials is studied using scaling limits of Christoffel--Darboux kernels. Different limit kernels are associated with different universality classes, e.g. sine kernel with bulk universality and locally asymptotically uniform zero spacing. In recent years, new results have been obtained by using the de Branges theory of canonical systems.

We characterize global minimizers below the so-called first critical field of the inhomogeneous version of the Ginzburg-Landau energy functional in a three-dimensional setting. Minimizers of this functional describe the behavior of type-II superconductors exposed to an external magnetic field, which is characterized by the presence of codimension 2 singularities called vortices where superconductivity is locally suppressed.

The Walsh-quantized baker's maps are models for quantum chaos on the torus. We show that for all baker's map scaling factors D\ge2 except for D=4, typically (in the sense of Haar measure on the eigenspaces, which are degenerate) the distribution of the matrix element fluctuations for a randomly chosen eigenbasis looks Gaussian in the semiclassical limit N\to\infty, with variance given in terms of classical baker's map correlations. This determines the precise rate of convergence in the quantum ergodic theorem for these eigenbases.

The hard sphere model is a simple to define and long studied mathematical model of gas, in which the only interactions are the hard-core constraint that two spheres cannot overlap in space.  In three dimensions it is expected to exhibit a gas-to-crystal phase transition.  Despite its simplicty, rigorous results on the model are rather sparse.  I will introduce the model, discuss some of the main open questions, and present some results new and old.

We rigorously prove the filamentation phenomenon for a class of weak solutions to the Euler equations known as vortex caps. Vortex caps are characteristic functions representing time-evolving sets of Lagrangian type, with energy preserved at all times. The filamentation of vortex caps is characterized by L^1 -stability alongside unbounded growth of the perimeter of their interfaces. We recall the existence and stability results for vortex caps on the sphere, based on Yudovich theory.

We study differential inclusions of the type $A v=0$ and $v \in K$, where $v$ is a vector field satisfying a linear PDE system $A$ and $K$ is a compact set. We are particularly interested in the case when $K$ consists of two vectors (\textit{two-state problem}). We consider Dirichlet boundary conditions for $v$, in which case the differential inclusion typically has no solutions. We study a suitable relaxation of the system, in which we penalize the surface energy required to switch between the two states.

Random matrix products perhaps among some of the most extensively studied examples of random dynamical systems, and moreover are central to the study of one-dimensional disordered systems. We discuss recent results by the author (joint with S.