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.

In this talk I will introduce Quantum Field Theory, its Path Integral formulation, and Segal’s Axioms. Then I will indicate some recent progress on rigorous construction of some concrete models using probability. After that I slightly extend Segal’s framework and show how it can be related to a useful physical quantity called “entanglement entropy”, where I suggest a geometric way of rigorously deriving relevant formulae based on recent works of the speaker and B. Estienne (2501.19014).

Nous développons une nouvelle technique pour minorer des sommes d’exponentielle de la forme $\sum f(n) e^{2i\pi\alpha n}$ pour tout $\alpha.$
Nous montrerons en particulier que la somme $\sum_{p\le x} e^{2i\pi\alpha p}$ est non bornée pour tout $\alpha,$ et plus précisément diverge au moins comme $x^{1/6-\varepsilon}$ pour une suite de $x$ tendant vers l’infini, uniformément en $\alpha.$

The geometry of the flag manifold G/P associated to a parabolic subgroup P of a semisimple Lie group G gives rise to a G-equivariant complex of elliptic operators (the BGG complex) which satisfy the property of maximal hypoellipticity, as shown by the work of Dave and Haller. In the case of the group SU(n,1), the BGG complex is the Rumin complex associated to the contact structure on the sphere S^{2n-1}. We shall describe the quaternionic analogue (i.e. for G=Sp(n,1)), and compute the bundles and operators involved in the BGG complex thanks to the Kostant theorem of 1961 (generalized by Cap and Slovak) on the cohomology of the maximal nilpotent subalgebra of a parabolic subalgebra of a semisimple Lie algebra.
We shall also explain that, in the context of Non Commutative Geometry, the BGG complexes are a crucial ingredient for the construction of the so called Kasparov gamma-element, which is the obstruction to the subjectivity of the Baum-Connes map.

The phenomenon known as ‘the Mackey analogy’ is a correspondance between the tempered dual of a Lie group $G$ and the unitary dual of its associated motion group $G_0$.  The precise statement, formulated and proven by Higson in the case of complex groups, and by Afgoustidis in the general case of real groups, has long been known to be intimately connected to the Connes-Kasparov isomorphism relating the K-theory of the reduced C$^\ast$-algebras of $G$ and $G_0$ via a deformation argument.
In this talk, I will report on joint work with Higson and Román, aimed at understanding the Mackey analogy directly at the level of the group C$^\ast$-algebras.  The main result is an embedding of the C$^\ast$-algebra of $G_0$ into the reduced C$^\ast$-algebra of $G$, which is proven to characterize the bijection of Afgoustidis and to induce the Connes-Kasparov isomorphism.

Anton Kapustin, Nikita Sopenko and others have studied the group of “locally generated automorphisms” of a UHF C*-algebra whose matrix algebra factors are labelled by the points of a coarse space.  This talk will be about a related Lie algebra of “locally generated derivations.”  Specifically it will be about the problem of computing the homology of this Lie algebra.  I shall describe a set of techniques that may be employed to solve the problem, at least for lattices in Euclidean space and similar spaces. Some are from the coarse Baum-Connes conjecture, and some, from homological algebra, are much older. This is joint work with Tsuyoshi Kato.

We construct a Poisson algebra bundle whose distributional sections are suitable to represent multilocal observables in classical field theory. To do this, we work with vector bundles over the unordered configuration space of a manifold M and consider the structure of a 2-monoidal category given by the usual (Hadamard) tensor product of bundles and a new (Cauchy) tensor product which provides a symmetrized version of the usual external tensor product of vector bundles on M.

In “$L_\infty$-algebras and higher analogues of Dirac structures and Courant algebroids” (arXiv:1003.1004), Marco Zambon constructed an explicit $L_\infty$-morphism between the Courant $r=1$-Lie algebra of a smooth manifold $M$, twisted by a closed 2-form $\sigma$, and the untwisted Courant $r=2$-Lie algebra of $M$. He left open the question of whether similar canonical $L_\infty$-morphisms exist in higher degrees — that is, between the Courant $r$-Lie algebra twisted by a closed $(r+1)$-form $\sigma$ and the untwisted Courant $(r+1)$-Lie algebra. In this talk, we present a general framework that naturally accounts for the existence of such morphisms for arbitrary $r$. This is joint work with Domenico Fiorenza.

In this talk I will describe a way to implement the resonance method in problems of analytic number theory which are not necessarily multiplicative in nature.
This extension of the method not only produces improved extreme results wherever Dirichelt’s approximation theorem has been usually employed but it also highlights its connection to Bohr’s and Jessen’s proof of Kronecker’s approximation theorem.

En se laissant guider par l’exemple des ensembles de Sidon (ensembles de nombres dont les sommes de deux éléments sont uniques, très étudiés en combinatoire additive), je présenterai des résultats récents, en collaboration avec R. Riblet, où des techniques de théorie des ensembles permettent de construire des ensembles “grands” en certains sens (cardinalité, mesure ou dimension) tout en étant “épars” car évitant des configurations prescrites (pas de relation linéaire, ou ne contenant pas de parallélogramme, etc.). Des questions subtiles en lien avec l’axiome du choix seront évoquées.

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

Inspiré par la physique des films de savon, le problème de Plateau est un problème classique des mathématiques qui consiste à minimiser l’aire d’une surface s’appuyant sur un bord. Il admet en fait de nombreuses formulations, certaines ne permettant pas toujours de représenter fidèlement les films de savon. Dans cet exposé, on s’intéresse à un modèle issu des travaux d’Almgren et de David, particulièrement bien adapté pour décrire ces films. Il s’agit du «problème de Plateau glissant», où l’on minimise les déformations continues d’une surface sans la détacher du bord. Bien qu’assez intuitive, c’est l’une des formulations qui résiste le mieux aux mathématiciens car on ne connait toujours pas de théorème d’existence général. On présentera un résultat d’existence dans un cas particulier, obtenu avec G. David.

Attention, ce séminaire aura lieu en salle Döblin.

Water-wave models play a crucial role in understanding, predicting, and controlling the dynamics of surface water waves across various real-world scenarios, including oceanic waves, waves in lakes and rivers, and those affecting man-made structures. These models integrate mathematical, physical, and numerical frameworks with wide-ranging applications in environmental science, engineering, and maritime industries. In this talk, we will explore key mathematical results for several water-wave models, highlighting their relevance to real-world applications.

In this talk, I will present an asymptotic expansion for the minimal Dirichlet energy of  $\mathbb{S}^2$-valued maps outside a finite number of particles of size $\rho$ in $\mathbb{R}^3$ under general anchoring conditions at the particle boundaries as $\rho\to 0$. The first two terms of this expansion consist of the minimal energy after zooming in at scale $\rho$ around each particle and a Coulomb-like interaction that agrees with the electrostatics analogy depending on the far-field behavior of the corresponding single-particle minimizer. This approximation is commonly used in the physics literature for colloid interactions in nematic liquid crystals and for the first time a precise estimate of the energy error introduced by that linearization is derived, by developing new tools that address the lack of convergence rate when zooming in at scale $\rho$.

The talk is based on joint work with L. Bronsard, X. Lamy and R. Venkatraman.

We survey interesting properties of some nonlocal operators that have no analogue for linear second order elliptic PDE.

Je présenterai un travail en collaboration avec Thibault Lacombe, où
nous démontrons que les solutions Lipschitz $u$ de $\mathrm{div}\,
G(\nabla u)=0$ dans un domaine du plan sont $C^1$, pour des champs de
vecteurs strictement monotones $G$ dans $C^0(\mathbb R^2;\mathbb R^2)$
satisfaisant une condition d’ellipticité très générale. Lorsque le champ
de vecteurs $G$ est le gradient d’une fonction strictement convexe,
notre résultat généralise des résultats de De Silva et Savin (Duke Math.
J. 2010). Lorsque $G$ n’est pas un gradient, l’hypothèse d’ellipticité
doit être interprétée correctement, et nous produisons un exemple qui
montre l’effet non trivial de la partie antisymétrique de $\nabla G$.

Il s’agit d’un travail en collaboration avec Guedes-Bonthonneau, Chhaibi, Rivière et To.

Sur une surface riemannienne, nous construisons une mesure de Yang-Mills sur les connexions distributionnelles, avec le processus d’holonomie correspondant. Les ingrédients de notre construction sont: une nouvelle jauge de Morse et l’analyse d’équations de transport pour les flots de gradient en très faible régularité. Nous montrons que notre mesure retrouve les formules de la théorie de Yang-Mills en dimension 2, telles qu’on les trouve dans les travaux de Witten, Driver, Sengupta et Lévy. Enfin, nous expliquons en quel sens notre mesure converge vers le volume symplectique d’Atiyah–Bott–Goldman sur l’espace de modules des connexions plates dans la limite semi-classique.

The objective of this work is to study the stability of two systems of type Rao-Nakra sandwich beam in the whole line $R$ with a frictional damping or an infinite memory acting on the Euler-Bernoulli equation. When the speeds of propagation of the two wave equations are equal, we show that the solutions do not converge to zero when time goes to infinity. In the reverse situation, we prove some $L^2 (R)$-norm and $L^1 (R)$-norm decay estimates of solutions and theirs higher order derivatives with respect to the space variable. Thanks to interpolation inequalities and Carlson inequality, these $L^2 (R)$-norm and $L^1 (R)$-norm decay estimates lead to similar ones in the $L^q (R)$-norm, for any $q\in [1,+\infty]$. In our both $L^2 (R)$-norm and $L^1 (R)$-norm decay estimates, we specify the decay rates in terms of the regularity of the initial data and the nature of the control. Applications to some Cauchy Timoshenko type systems will be also given. The proof is based on the energy method combined with the Fourier analysis (by using the transformation in the Fourier space and well chosen multipliers).

A part of these results was obtained in collaboration with Salim Messaoudi (University of Sharjah, UAE).

For the details, see the following papers:

A. Guesmia, Some $L^q (R)$-norm decay estimates ($q\in[1,+\infty]$) for two Cauchy systems of type Rao-Nakra sandwich beam with a frictional damping or an infinite memory, J. Appl. Anal. Comp., 12 (2022), 2511-2540.
A. Guesmia, On the stability of a linear Cauchy Rao-Nakra sandwich beam under frictional dampings, Taiwanese J. Math., 27 (2023), 799-811.
A. Guesmia and S. Messaoudi, Some $L^2 (R)$-norm and $L^1 (R)$-norm decay estimates for Cauchy Timoshenko type systems with a frictional damping or an infinite memory, J. Math. Anal. Appl., 527 (2023), 127385.

It is well established that the L_1 optimal transport can be employed to characterize solutions of the Beckmann problem, where one looks for a vector-valued measure of minimal total variation with the constraint that fixes its divergence as the difference of two probabilities distributions. In turn, the Beckmann problem underlies optimal design of heat conductors.

I will talk about a second-order counterpart of this theory, where in the Beckmann problem we have a constraint on the double divergence. In 2D, this problem enjoys the interpretation of optimally designing a ceiling using a grillage structure. I will show that its solutions can be characterized through a new formulation where we look for a probability that dominates the data in the sense of convex order while attaining minimal variance. Afterwards, equivalent optimal transport formulations can be proposed for efficient numerical treatment.

Work in collaboration with Guy Bouchitté.

Kato’s well known distributional inequality for the magnetic Laplacian holds equally in the more general setting of non-relativistic quantum electrodynamics (QED), where the wave function is vector-valued and the vector potential is quantized. We give two new applications of this result: First, we show that eigenstates satisfy a subsolution estimate. Second, for general states, with energy distribution strictly below the ionization threshold, we give a short proof of pointwise exponential decay in the electronic configuration.

Joint work with Valentin Kußmaul