## Spring/Summer 2020: Fridays 2-3 pm Zoom link

The Cardiff Informal Analysis & PDE seminar is meant to be a slow-paced group discussion, led by one speaker. This is meant to allow members of the group to learn new topics, and to give external guests an opportunity to expand on their formal seminar talks in an informal setting.

The schedule of the Cardiff Analysis Seminar can be found here.

May 22: Sabine Bögli (Durham) - On Lieb-Thirring inequalities for one-dimensional non-self-adjoint Jacobi and Schroedinger operators

In the first part of this talk I will give an introduction to spectral theory of Schroedinger operators and their discrete analogues, the Jacobi operators. Lieb-Thirring inequalities give estimates on the discrete eigenvalues and their accumulation rate to the essential spectrum. Such information is useful in quantum mechanics, but the estimates are well understood only in the self-adjoint setting. In the second part of the talk I will present recent results concerning possible extensions of the Lieb-Thirring inequalities to the non-self-adjoint setting (based on joint work with Frantisek Stampach).

May 29: Marco Marletta (Cardiff) - A Laplace operator with boundary conditions singular at one point

In this talk I will present some work with Rozenblum from 2009. While it has been known for more than half a century that the Laplace operator on a smooth, bounded domain may have essential spectrum if the boundary conditions are suitably chosen, typical choices involved non-local operators. In this talk I will show, with very elementary arguments, that even local boundary conditions, singular even just at a single point - can have a huge impact on the spectrum and eigenfunctions. The example we consider, first proposed by Berry and Dennis. still has empty essential spectrum and compact resolvent. However Weyl’s law fails completely because the spectrum becomes unbounded below. The positive eigenvalues still obey Weyl asymptotics, to leading order; however the (absolute values of the) negative eigenvalues do not obey a power law distribution.

June 5: Ian Wood (Kent) - Introduction to Dirichlet to Neumann maps and boundary triples

Dirichlet-to-Neumann maps play an important role in the analysis of many boundary problems for partial differential equations, in particular inverse problems and interface problems. We will introduce the map for second order elliptic differential operators and look at its cousin for ordinary differential equations, the Weyl-Titchmarsh m-function. We will then introduce the abstract framework of boundary triples, where an analogue of the Dirichlet-to-Neumann map, the M-function, can be introduced, and look at the question of how much spectral information the M-function contains.

June 12: Jean Lagacé (UCL) - Homogenisation in geometric spectral theory --- a quick introduction

In this talk, I will describe an homogenisation construction for the Steklov problem on manifolds. While we do not assume that there is a periodic structure, the construction is nevertheless deterministic. I will make explicit what is different from the usual periodic setting, and what we can learn even in the periodic setting from this construction. The audience of this talk will not be supposed to be familiar with either homogenisation theory nor the Steklov problem. With time permitting, I will also discuss some applications to differential geometry. The talk is based on joint work with Alexandre Girouard, Antoine Henrot, and Mikhail Karpukhin.

June 26: Laura Monk (Strasbourg) - Geometry and spectrum of random hyperbolic surfaces

- how one can consider and study random hyperbolic surfaces,
- the geometry of typical random surfaces: their diameter, Cheeger constant, ...,
- how to deduce from the geometry of a hyperbolic surface information about the spectrum of the Laplacian.

In this talk, I will explain: