In a landmark paper, Scott Sheffield introduced a famous bijection, called the “hamburger-cheeseburger” bijection, to encode a certain model of random planar maps as a certain queue model in a kitchen selling hamburgers and cheeseburgers. Under this correspondence, natural geometric observables have a nice “burger” interpretation, for which Sheffield established scaling limit results. These scaling limits exhibit a phase transition in a special regime where the maps are believed to be “critical”. The goal of this talk is to present the analogue of these scaling limit results in the critical case, which can be thought of as the first convergence result of planar maps in the universality class of critical Liouville quantum gravity, in the so-called peanosphere sense. The talk is based on joint work with Xingjian Hu, Ellen Powell and Mo Dick Wong.

Many phenomena of interest in various applicative fields (epidemiology, neuroscience,…) can be idealized as interacting particle systems on random graphs. Various approaches have been proposed in recent years to develop tractable approximations of these dynamics that take the graph geometry and particle correlations into account. One of them, introduced by Lacker, Ramanan and Wu, focuses on dynamics on sparse graphs and their local limits. Analogously to mean-field models on complete and dense graphs, it is possible to establish so-called local-field equations on random trees that provide an autonomous description of the neighborhood of the root. In this talk, we will give a general overview of the local-field approach, as well as a recent result of quantitative propagation of chaos in this framework.

Exposé à Metz. Titre et résumé à venir.

Exposé à Metz. Titre et résumé à venir.

Date possible pour le colloquinte

La suite d’Oldenburger-Kolakoski est l’unique mot infini sur l’alphabet {1,2} qui commence par un “1” et est point fixe de l’opérateur de dérivation. En 1991, M.S. Keane conjecture que cette suite admet une fréquence d’1/2 pour la lettre “1”.

Les suites dites “auto-descriptives” sont une généralisation du mot d’Oldenburger-Kolakoski. Ces suites sont en bijection naturelle avec l’ensemble de toutes les suites sur l’alphabet {1,2} : une suite auto-descriptive est dite “dirigée” par son homologue naturelle sur {1,2}. Est-il possible d’inférer les fréquences de lettres de l’une à partir de l’autre ?

Je présenterai dans cet exposé deux approches à cette question : l’une probabiliste (Boisson, Jamet, Marcovici — 2024), l’autre analytique (Akiyama, Jamet, Marcovici, T.C. — 2024).

Many arguments in mathematics rely on the introduction of an auxiliary function that is compactly supported. Sometimes this compactly supported function is required to satisfy some additional regularity, but this cannot be pushed too far. It can for instance not be analytic as follows by the identity principle.

In this talk, we wish to present a technique that may bypass this obstruction. Namely, instead of considering a single function, we shall construct a sequence of function that are supported in the same compact, and satisfy some good uniform bounds on their derivatives.

This method is a powerful tool as it can, in some circumstances, turn a heuristic argument relying on a compactly supported analytic auxiliary function, into a rigorous proof. As an application, we shall discuss how this technique leads to improvements in some quantified Tauberian theorems.

Nous donnerons une construction de pavages de type Penrose dans le cas hyperbolique utilisant les pavages par substitutions classiques. Nous expliciterons les objets que l’on peut associer à  ces pavages et les relierons aux objets classiques de la théorie des pavages par substitutions dans l’optique du calcul de la K-théorie associée à  cette construction.

http://www.iecn.u-nancy.fr/~nombres/Les-Seminaires/

http://www.iecn.u-nancy.fr/~nombres/Les-Seminaires/

http://www.iecn.u-nancy.fr/~nombres/Les-Seminaires/

On a compact connected spin manifold the index theorem by Atiyah and Singer gives lower bounds for the dimension of the kernel of the Dirac operator. Metrics for which the lower bound is attained are called D-minimal. It is conjectured that generic metrics are D-minimal and that non-D-minimal metrics exist on any manifold of dimension at least 3. These conjectures go back to Hitchin’s article “Harmonic spinors” where first steps in this program were done, and they were clearly conjectured by C. Bär and M. Dahl. That D-minimal metrics are generic was proven by Bär, Dahl, Humbert and myself using Gromov-Lawson type surgery methods and bordism theory. The existence of non-D-minimal metrics in dimension at least 7 is current work with Bunke, Pilca and Nowaczyk and uses recent progress by Crowley and Schick about the topology of the space of metrics with positive scalar curvature.

http://www.iecn.u-nancy.fr/~nombres/Les-Seminaires/

Nous définissons la notion de famille exhaustive de représentations d’une C*-algèbre A. Si F est une telle famille de représentations de A, alors un opérateur D affilié à  A est inversible si, et seulement si, phi(D) est inversible pour tout phi dans F. Cette propriété caractérise les familles exhaustives. Ensuite nous appliquons ces résultats aux familles paramétriques d’opérateurs (pseudo) différentiels.

Je vais présenter un problème d’optimisation à frontière libre analogue au problème de Alt-Caffarelli pour les fonctions biharmoniques. Ce problème apparaît dans différentes questions d’optimisation de forme, dont la minimisation de la trainée d’un obstacle dans un fluide sous contrainte de mesure, la minimisation de la première valeur propre de l’opérateur de Stokes (ou de flambage) dans les domaines du plan, etc.. On s’attend à ce que la frontière libre obtenue soit généralement une union de courbes lisses, pouvant se rejoindre avec un angle d’environ 1.43pi, et je présenterai plusieurs résultats allant dans ce sens.

C’est un travail en collaboration avec Jimmy Lamboley.

Résumé à venir

Résumé à venir

Résumé à venir

Résumé à venir

Résumé

The Hartree equation is an important example of nonlinear Schrödinger evolution. It can be derived as the mean field limit of a system of many non-relativistic bosons with pair interaction. Such a limit is now well understood for a wide class of interaction potentials and initial quantum configurations. The model has many physical applications, e.g. in studying Bose-Einstein condensation. It is thus important to have a control of the rate of convergence towards the limiting dynamics, for it would give a quantification of the error caused by the approximation of many particles with an infinite number of them. In this talk I will present a recent result, obtained with Z. Ammari and B. Pawilowski, where we provide bounds for the rate of convergence towards the Hartree dynamics for generic many-body initial quantum states.

Résumé

Résumé

Résumé

We will talk about necessary and sufficient conditions for an entire solution $u$ of a biharmonic equation with exponential nonlinearity $e^u$ to be a radially symmetric solution. We need to know the asymptotic expansions of the solution $u$ and its laplacian at infinity in order to apply the standard Moving-Plane-Method (MPM) for obtaining the radial symmetry for a system of equations.

Plan de l’exposé : 1/ Introduction (historique,origine des équations et comparaison avec équations de Navier-Stokes incompressibles). 2/ Position du problème (Estimations de l’énergie anisotropiques).

Résumé

This lecture is devoted to address the problem of controlling uncertain systems submitted to parametrized perturbations. We introduce the notion of averaged control according to which the average of the states with respect to the uncertainty parameter is the quantity of interest. We observe that this property is equivalent to a suitable averaged observability one according to which the initial datum of the uncertain dynamics is to be determined by means of averages of the observations done. We will first discuss this property in the context of finite-dimensional systems to later consider Partial Differential Equations, mainly, of wave and parabolic nature. As we shall see, surprisingly, the averaging process with respect to the unknown parameter may lead a change of type ion the PDE under consideration from hyperbolic to parabolic, for instance, significantly affecting the expected control theoretical properties. We will also present some open problems and perspectives of future developments.