In this talk, we are interested in a transmission problem between a dielectric and a metamaterial. The question we consider is the following: does the limiting amplitude principle hold in such a medium? This principle defines the stationary regime as the large time asymptotic behavior of a system subject to a periodic excitation.

An answer is proposed here in the case of a plane interface between a metamaterial represented by the Drude model and the vacuum, which fill respectively complementary half-spaces. In this context, we reformulate the time-dependent Maxwell’s equations as a conservative Schrödinger equation and perform its complete spectral analysis. This permits a quasi-explicit representation of the solution via the ”generalized diagonalization” of the associated unbounded self-adjoint operator. As an application of this study, we show finally that the limiting amplitude principle holds except for a particular fequency characterized by a ratio of permittivities and permeabilities equal to −1 across the interface. This frequency is a resonance of the system and the response to this excitation blows up linearly in time.

Joint work with Christophe Hazard (CNRS, Poems team) and Patrick Joly (INRIA, Poems team)

Un processus auto-régressif multivarié (VAR) de dimension p est défini par l’équation Xt+1=ΘXt+Zt​ où Θ est une matrice réelle de taille p*p et (Zt)_t est un bruit blanc gaussien. Dans cet exposé, je vous présenterai deux axes de recherche visant à étudier les processus VAR dans un contexte de grande dimension.

Dans un premier temps, on analysera les réalisations d’un processus VAR en supposant que la dimension p du processus (Xt)t est élevée. Sous l’hypothèse que la matrice Θ est de rang faible, l’objectif sera de déterminer si elle subit un changement au cours du temps ou non. Pour détecter cette rupture, nous proposerons un test statistique dont l’efficacité sera évaluée à l’aide de simulations numériques.

Dans un second temps, on s'intéressera à la question de savoir si Θ est nulle ou non, autrement dit, si notre processus est un bruit blanc ou non. Pour cela, nous examinerons la plus grande valeur propre de la matrice d’autocovariance empirique du processus VAR. Cette partie de l’exposé s’attachera donc à l’étude des valeurs propres de matrices aléatoires, avec une première partie sur les théorèmes de base de la théorie des matrices aléatoires hermitiennes. Ensuite, nous étudierons le cas des matrice aléatoires non hermitiennes, c'est le cas de la matrice d'autocovariance empirique par exemple . La plus grande difficulté dans l’étude des valeurs propres d’une matrice non hermitienne est leur instabilité : une perturbation, même minime, peut avoir un très fort impact dans le comportement des valeurs propres.

Gaussian fields with logarithmically decaying correlations, such as branching Brownian motion and the two-dimensional Gaussian free field, are conjectured to form universality class of extreme value statistics (notably in the work of Carpentier & Le Doussal and Fyodorov & Bouchaud). This class is the borderline case between the class of IID random variables, and models where correlations start to affect the statistics. In this talk, I will describe a general approach based on rigorous works in spin glass theory to describe features of the Gibbs measure of these Gaussian fields. I will focus on the two-dimensional discrete Gaussian free field. At low temperature, we show that the normalized covariance of two points sampled from the Gibbs measure is either 0 or 1. This is used to prove that the joint distribution of the Gibbs weights converges in a suitable sense to that of a Poisson-Dirichlet variable. (with L.-P. Arguin, 2015).

In a second work (with M. Pain, 2021), we prove absence of temperature chaos for the two-dimensional discrete Gaussian free field using the convergence of the full extremal process, which has been obtained by Biskup and Louidor. This means that the overlap of two points chosen under Gibbs measures at different temperatures has a nontrivial distribution. Whereas this distribution is the same as for the random energy model when the two points are sampled at the same temperature, we point out here that they are different when temperatures are distinct: more precisely, we prove that the mean overlap of two points chosen under Gibbs measures at different temperatures for the DGFF is strictly smaller than the REM's one. Therefore, although neither of these models exhibits temperature chaos, one could say that the DGFF is more chaotic in temperature than the REM.

Finally, I will discuss in detail (depending on the time left) recent works with B. Bonnefont (ex-PhD student, now Post-Doc at University of Geneva) and M. Pain (CR @Toulouse), on questions suggested by B. Derrida.

In this talk we will explain a computation describing Hochschild-Kostant-Rosenberg isomorphism theorems as exponential maps. This computation uses the construction of a filtered circle obtained in collaboration with Moulinos and Toën. As applications we will describe motivic Donaldson-Thomas invariants in positive characteristic and an extension of Hochschild homology for elliptic curves.

A classical topic in spectral theory is Weyl’s law describing the asymptotics of the eigenvalues of the Laplacian on a bounded open set. We are interested in these asymptotics in low regularity situations. Both in the Dirichlet and in the Neumann case we show two-term asymptotics for Riesz means of any positive order under the assumption that the boundary is Lipschitz continuous. For convex sets we obtain universal, nonasymptotic bounds. Tools in our proof are universal heat kernel bounds, as well as Tauberian Remainder Theorems.

Les surfaces d'Alexandrov à courbure intégrale bornée sont les surfaces les plus générales telles que le théorème de Gauß-Bonnet existe et admettant des coordonnées isothèrmes. Une fois cette introduction faite nous discuterons, suivant la théorie de Sturm, l'existence d'inégalité de Poincaré, de mesure doublante, d'inégalité d'Harnack et enfin de noyau de la chaleur.

Cette présentation explore les variétés toriques, où géométrie et combinatoire se rencontrent. Nous montrons comment des éventails (objets géométriques simples) définissent des variétés algébriques riches, en établissant un dictionnaire précis entre leurs propriétés. La correspondance est illustrée par des exemples classiques (espaces projectifs, cônes) puis étendue aux variétés horosphériques, révélant des applications en géométrie birationnelle. Un pont entre visualisation concrète et théorie profonde.

In this presentation, we will construct regular solutions to linear and nonlinear elliptic-parabolic equations in which the natural direction of parabolicity reverses along a critical line. To prevent the emergence of singularities, we will impose orthogonality conditions on the source terms, and follow them during the execution of the nonlinear schemes.

This is a joint work with Anne-Laure Dalibard and Jean Rax, motivated by recirculation problems in boundary layer theory for fluid mechanics, and based on the preprint https://arxiv.org/abs/2203.11067