# Jeunes chercheurs

**Eldar Akhmetgaliyev
**

*Integral equation methods for spectral problems*

**Résumé: **In this talk we present a range of numerical methods which, based on
use of Green functions and integral equations, can be applied to
produce solution of Laplace eigenvalue problems with arbitrary
boundary conditions (including, e.g., Dirichlet/Neumann mixed
boundary conditions) and in arbitrary domains (including e.g. domains
with corners and multiply connected domains).

As part of our presentation we present newly obtained characterizations of the singularities of solutions and eigenfunctions which arise at transition points where Dirichlet and Neumann boundary conditions meet; the numerical methods mentioned above rely on use of these characterizations in conjunction with the novel Fourier Continuation technique to produce solutions with a high order of accuracy. In particular, the resulting method exhibits spectral convergence for smooth domains (in spite of the solution singularities at Dirichlet/Neumann junctions) and prescribed high-order convergence for non-smooth domains.

A point of interest concerns the search algorithm in our eigensolver, which proceeds by searching for frequencies for which the integral equations of the problem admit non-trivial kernels. As it happens, the “minimum-singular-value” objective function gives rise to a challenging nonlinear optimization problem. To tackle this difficulty we put forth an improved objective functional which can be optimized by means of standard root-finding methods.

In addition, an integral-equation based methods for Steklov eigenvalue problem will be presented that exhibit spectral convergence for smooth domains (including multiply connected domains) and high order convergence for domains with corners and mixed boundary conditions (e.g. sloshing problem).

This work benefited from collaboration with Nilima Nigam.

Akhmetgaliyev, Eldar, Oscar Bruno, and Nilima Nigam, A boundary integral algorithm for the Laplace Dirichlet-Neumann mixed eigenvalue problem., arXiv preprint arXiv:1411.0071 (2014).

**Yaiza Canzani**

*Zero sets of monochromatic random waves*

**Résumé**: The motivation of this talk is the understanding of the zero sets of
eigenfunctions of the Laplace operator on a compact Riemannian
manifold. For example we would like to understand the size of the zero
sets, the number of connected components, and the topological
structure of these components. Since attacking such questions has
proved to be a very hard task, it is reasonable to randomize the
problem and study the average behavior for these zero sets. Therefore,
instead of working with actual eigenfunctions, we shall work with the
model of monochromatic random waves and study the aforementioned
questions in this setting.

**Semyon Dyatlov**

*Spectral gaps via additive combinatorics*

**Résumé: **A spectral gap on a noncompact Riemannian manifold is an asymptotic strip free of resonances (poles of the meromorphic continuation of the resolvent of the Laplacian). The existence of such gap implies exponential decay of linear waves, modulo a finite dimensional space; in a related case of Pollicott--Ruelle resonances, a spectral gap gives an exponential remainder in the prime geodesic theorem.

We study spectral gaps in the classical setting of convex co-compact hyperbolic surfaces, where the trapped trajectories form a fractal set of dimension $2\delta + 1$. We obtain a spectral gap when $\delta=1/2$ (as well as for some more general cases). Using a fractal uncertainty principle, we express the size of this gap via an improved bound on the additive energy of the limit set. This improved bound relies on the fractal structure of the limit set, more precisely on its Ahlfors- David regularity, and makes it possible to calculate the size of the gap for a given surface.

**Corentin Léna**

*On the number of nodal domains for flat tori*

[ pdf ]

**Résumé: **We are interested in the eigenvalues and eigenfunctions of the Laplacian on the flat tori obtained when quotienting the plane by a rectangular lattice. The eigenvalues, and an orthogonal basis of eigenfunctions, are of course explicit but complex nodal patterns can occur because of high multiplicities when the ratio of the sides in the lattice is rational.

We will consider some results concerning two questions. The first was initiated by Pleijel in 1956, and consists in finding eigenfunctions associated with the $k$-th eigenvalue and having $k$ nodal domains, which is the maximal number of nodal domains allowed by Courant's theorem. In the case of a square torus, we will see that this occurs only for the first and second eigenvalues.

We will also consider some results concerning a question raised by T. Hoffmann-Ostenhof in 2012: are there eigenfunctions having an odd number of nodal domains. We will see that the answer depends on the ratio of the sides in the rectangular lattice. The answer is in particular negative in the case of a square torus.

**Asma Hassannezhad**

*Counting function and multiplicity of the Laplacian eigenvalues*

**Résumé**: In this talk, we study geometric upper bounds for the multiplicity and counting function of the Laplacian eigenvalues. We discuss some of classical upper bounds due to Cheng, Gromov and Buser. Then we extend these results to domains with Dirichlet and Neumann boundary conditions. The idea of the proof and some interesting open questions will be discussed. This talk is based on joint work with G. Kokarev and I. Polterovich.

**Romain Petrides**

*Existence and regularity of maximal metrics for the Laplace eigenvalues on surfaces*

**Résumé**: Given a compact surface, we deal with an old question (since the works by Yang and Yau in the 80s) about the sequence of the eigenvalues of the Laplacian : Is there some regular Riemannian metric which maximises the k-th eigenvalue on this surface ? We also give the link between this problem and minimal immersions into spheres

**David Sher
**

*The Steklov spectrum of surfaces*

**Résumé**: In recent years, many authors have explored the spectrum of the Dirichlet-to-Neumann operator on a manifold with boundary, also called the Steklov spectrum. In this talk, we will consider the associated inverse problem. Specifically, we will show how to recover the number of boundary components, as well as each of their lengths, from the Steklov spectrum of a smooth surface with boundary. The proof relies on surprisingly sharp spectral asymptotics. This is joint work with A. Girouard (U. Laval), L. Parnovski (UCL), and I. Polterovich (U. Montreal).