Dans le domaine des interactions fluide/solide, une question cruciale est la description de l'écoulement d'un fluide confiné entre deux surfaces solides en mouvement. Dans ce contexte, il est d'usage de considérer le problème de Stokes stationnaire de la mécanique de fluide complété par des conditions aux bords reproduisant le déplacement des deux surfaces et il s'agit d'obtenir un développement asymptotique de la solution quand la distance entre les deux surfaces tend vers 0.

Dans cet exposé, je présenterai une méthode variationnelle permettant de discuter l'asymptotique de la solution du problème de Stokes dans les cas bidimensionnels et tridimensionnels. Nous verrons que ces différents cas recèlent des difficultés différentes inhérentes aux propriétés des champs de vecteurs à divergence nulle dans chaque dimension. Le contenu de ce travail est le fruit de collaborations avec E. Bocchi, D. Bonheure, C. Patriarca et G. Sperone.

We prove the quilted Floer cochain complexes form A infinity n-modules over the Fukaya category of Lagrangian correspondences. Then we prove that when we restrict the input to mapping cones of product Lagrangians and graphs, the resulting bar-type complex can be identified with bar complex from ordinary Floer theory. As an application we use a family version of quilt unfolding argument to prove two long exact sequences conjectured by Seidel that relates the Lagrangian Floer cohomology of a collection of (possibly intersecting) Lagrangian spheres and the fixed point Floer cohomology of composition of Dehn twists along them.

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.