# PDE and Applied Mathematics Seminar[ Edit ]

*Abstract:*

The spectrum of the Laplacian on graphs which have certain symmetry properties can be studied via a decomposition of the operator as a direct sum of one-dimensional operators which are simpler to analyze. In the case of metric graphs, such a decomposition was described by M. Solomyak and K. Naimark when the graphs are radial trees. In the discrete case, there is a result by J. Breuer and M. Keller treating more general graphs. We present a decomposition in the metric case which is derived from the discrete one. By doing so, we extend the family of (metric) graphs dealt with to also include certain symmetric graphs that are not trees. In addition, our analysis describes an explicit relation between the discrete and continuous cases. This is joint work with Jonathan Breuer.

*Abstract:*

We address the question of convergence of Schrödinger operators on metric graphs with general self-adjoint vertex conditions as lengths of some of graph's edges shrink to zero. We determine the limiting operator and study convergence in a suitable norm resolvent sense. It is noteworthy that, as edge lengths tend to zero, standard Sobolev-type estimates break down, making convergence fail for some graphs. The failure is due to presence of what we call "exotic eigenvalues": eigenvalues whose eigenfunctions increasingly localize on the edges that are shrinking to a point.

We establish a sufficient condition for convergence which encodes an intricate balance between the topology of the graph and its vertex data. In particular, it does not depend on the potential, on the differences in the rates of convergence of the shrinking edges, or on the lengths of the unaffected edges. In some important special cases this condition is also shown to be necessary. Moreover, when the condition fails, it provides quantitative information on exotic eigenvalues.

Before formulating the main results we will review the setting of Schrödinger operators on metric graphs and the characterization of possible self-adjoint conditions, followed by numerous examples where the limiting operator is not obvious or where the convergence fails outright. The talk is based on a joint work with Yuri Latushkin and Selim Sukhtaiev, arXiv:1806.00561 and on work in progress with Yves Colin de Verdiere.

*Abstract:*

Graphene is an allotrope of carbon consisting of a single layer of carbon atoms that are densely packed in a honeycomb crystal lattice. Suppose that one take a graphene sheet with holes and start to change the magnetic flux through the holes. When the changes of the magnetic fluxes become integer, the energetic spectrum as a whole should return to its initial state. However, the individual eigenenergies are not necessarily periodic; they can cross the boundary between hole and electron states. The number of such crossings (counted with sign) is called the spectral flow. The situation of non-zero spectral flow is important for physicists and is called the Aharonov-Bohm effect.

Single-layer graphene is described by the Dirac operator acting on a two-or four-component spinor. There is also a bilayer form of graphene, which is described by self-adjoint differential operators of non-Dirac type. In the talk I will show how the spectral flow can be computed, using topological methods, for both single-layer and bilayer graphene (and, more generally, for a one-parameter family of arbitrary first order self-adjoint elliptic operators over a compact surface with classical boundary conditions). The talk is based on my papers arXiv:1108.0806 and arXiv:1703.06105.

*Abstract:*

During the last 20 years there has been a considerable literature on a collection of related mathematical topics: higher degree versions of Poncelet’s Theorem, certain measures associated to some finite Blaschke products and the numerical range of finite dimensional completely non-unitary contractions with defect index 1. I will explain that without realizing it, the authors of these works were discussing Orthogonal Polynomials on the Unit Circle (OPUC). This will allow us to use OPUC methods to provide illuminating proofs of some of their results and in turn to allow the insights from this literature to tell us something about OPUC. This is joint work with Andrei Martínez-Finkelshtein and Brian Simanek. Background will be provided on the topics discussed.

*Abstract:*

Estimating a manifold from (possibly noisy) samples appears to be a difficult problem. Indeed, even after decades of research, there are no (computationally tractable) manifold learning methods that actually "learn" the manifold. Most of the methods try, instead, to embed the manifold into a low-dimensional Euclidean space. This process inevitably introduces distortions and cannot guarantee a robust estimate of the manifold.

In this talk, we will discuss a new method to estimate a manifold in the ambient space, which is efficient even in the case of an ambient space of high dimension. The method gives a robust estimate to the manifold and its tangent, without introducing distortions. Moreover, we will show statistical convergence guarantees.

It is on a work in progress, joint with Barak Sober.