Georgia Institute of Technology College of Sciences

I'll review a few known results about quantum graphs that maximize or minimize eigenvalues or combinations of eigenvalues, and will then concentrate on ratios of eigevalues under topological constraints on the graph.  In particular a new discovery with James Kennedy and Gabriel Ramos is that the largest ratio of the first two eigenvalues of the Laplacian on a finite tree graph with Dirichlet conditions at the ends is achieved by equilateral stars. Some related, Weyl-sharp estimates of arbitrary eigenvalue ratios can be obtained using similar ideas.  If time permits, I will also describe some optimal results about differences and other combinations of eigenvalues.

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. The model commonly used to study TBG is an effective one, and was derived by Bistritzer and MacDonald. In this talk, I will present recent results on developing a framework to study TBG from first principles. To be more exact, we consider a tight-binding model for the electrons, but make no further approximations. Using a renormalization group technique, we construct a perturbative expansion to study TBG that is convergent when the twisting angle satisfies certain diophantine conditions. This is joint work with V. Mastropietro.

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).