Georgia Institute of Technology College of Sciences

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. This includes necessary and sufficient conditions for a family of scaling limits corresponding to homogeneous de Branges spaces; this family includes bulk universality, hard edge universality, jump discontinuities in the weight, and other notable universality classes. It also includes local behaviors beyond scaling limits. The talk is based on joint works with Benjamin Eichinger, Brian Simanek, Harald Woracek, Peter Yuditskii.

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. We will talk about how to adapt the results from the standard Ginzburg-Landau theory into an inhomogeneous framework and present results from a recent work in collaboration with Carlos Roman (Pontificia Universidad Catolica de Chile).

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 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. We study the asymptotics of the regularized energy integral. We show that the asymptotics depend polynomially on the regularization parameter with a quantification which — somewhat surprisingly — depends on the order of the linear PDE system $A$. Joint work with A. R\”{u}land, C. Tissot, A. Tribuzio.