Talks and presentations

The Steklov spectrum of cuboids

July 28, 2017

Talk, Mathematical Congress if the Americas, Spectrum and Dynamics session, Montréal, Québec

Almost nothing is known in general about the Steklov spectrum of domains or manifolds with singularities on the boundary. In this joint work with A. Girouard, I. Polterovich and A. Savo, we use right cuboids as a model for such domains and obtain various spectral properties: two terms spectral asymptotics, characterisation of the eigenfunctions and scarring sequences, bottom of the spectrum behaviour and shape optimisation for the first eigenvalue. I will formulate more precisely those results and I will make some remarks as to how they would help us understand the general spectral properties of domains with singular boundaries.

Lattice point counting in spectral theory

July 27, 2017

Talk, Mathematical Congress of the Americas, CMS-Studc student session, Montréal, QC

The spectral asymptotics of elliptic operators (e.g. the Laplacian) and analytic number theory are linked together via lattice point counting problems. As an example, counting eigenvalues of the Laplacian on a flat square torus corresponds to counting integer points in a disk of large radius, which is precisely the Gauss circle problem, a longstanding problem in analytic number theory. In this presentation, I will explain how lattice counting methods are applied to study spectral asymptotics for Schrödinger operators on waveguides and resonators, and also for the Steklov eigenvalue problem on a cube. These two settings will illustrate different flavors of lattice counting problems arising in spectral theory. The talk is based on joint works with L. Parnovski (UCL), as well as with A. Girouard (Laval), I. Polterovich (Montréal) and A. Savo (Rome).

The Steklov spectrum of hyperrectangles

June 20, 2017

Talk, Université de Neûchatel, Neûchatel, Switzerland

While the computation of the spectrum of the Laplacian on a hyperrectangle in Euclidean space is a standard exercise for PDE students, understanding the Steklov spectrum on such a domain is more subtle. I will present recent results, joint work with Alexandre Girouard, Iosif Polterovich and Alessandro Savo, about both high energy asymptotics and bottom of the spectrum characterisation. More precisely, I will give a complete characterisation of the eigenfunction and the eigenvalues then show that we have two term asymptotics for the counting function of the eigenvalues that determine the (d−1)-volume of the boundary of the hyperrectangle and the (d−2)-volume of 2-boundary. I will also show that the hypercube maximises the first eigenvalue when normalised either by d-volume of the hyperrectangle or (d−1)-volume of the boundary.

Symmetry and Dirac points in graphene spectrum

June 06, 2017

Talk, Spectral Geometry, Graphs, Semiclassical Analysis and Dynamics, Peyresq, France

In this workshop talk, I present Gregory Berkolaiko and Andrew Comech’s paper Symmetry and Dirac points in graphene spectrum . I overview the results presented therein, with more attention given specifically to the geometric aspect of the proofs.