La semaine de l’IMB

Based on the mechanical viewpoint that living tissues exhibit a fluid-like behavior, PDE models inspired by fluid dynamics are now well established as one of the main mathematical tools for the macroscopic description of tissue growth. Depending on the type of tissue, these models link the pressure to the velocity field using either Brinkman’s law (viscoelastic models) or Darcy’s law (porous medium equations, PME). Furthermore, the stiffness of the pressure law plays a crucial role in distinguishing density-based (compressible) models from free-boundary (incompressible) problems, in which the density is saturated.

In this talk, I will show how to connect different mechanical models of living tissues through singular limits. In particular, I will discuss the inviscid limit toward the PME, the incompressible limit of the PME leading to free-boundary problems of the Hele-Shaw type, and finally the joint limit.

Locally symmetric spaces arise in several areas of mathematics and are among the most structured spaces, exhibiting special harmony between geometry, analysis, arithmetic, algebra, and topology. Building on his celebrated arithmeticity theorem, Margulis conjectured that torsion-free cocompact arithmetic lattices of semisimple Lie groups are uniformly discrete. Geometrically, this means a uniform lower bound on the lengths of all closed geodesics for arithmetic locally symmetric spaces. This conjecture is widely open, even in the simplest case of compact arithmetic hyperbolic surfaces obtained as quotients of the hyperbolic plane. In joint work with M. Fraczyk, we proved that this is enough to prove Margulis' conjecture for all locally symmetric spaces of higher rank simple Lie groups. In fact, we establish uniform lower bounds on the lengths of ``most'' closed geodesics and prove that the main difficulty lies in rank one. The proof exploits in an essential way the arithmetic structure of these spaces.

En français :

Le but de cet exposé est d’expliciter les liens entre les propriétés analytiques de la fonction zêta de Riemann et la répartition des nombres premiers. Nous examinerons en particulier les conséquences du produit eulérien et de l’influence des zéros non triviaux, aussi bien sur les résultats asymptotiques que sur les formules exactes.

In english:

The aim of this talk is to make explicit the links between the analytic properties of the Riemann zeta function and the distribution of prime numbers. In particular, we will examine the consequences of the Euler product and the influence of the nontrivial zeros, both on asymptotic results and on explicit formulas.

This talk presents computational and theoretical advances/experiments in Mixed Integer Nonlinear Programming across two complementary themes. The first focuses on emerging MINLP techniques — online learning for pseudo cost estimation, ReLU-based neural methods for cut separation, and AlphaEvolve-style modelling — that aim to modernize the MINLP solver. The second focuses on aggregation-based cutting planes, highlighting the practical importance of Complemented Mixed Integer Rounding (CMIR) cuts in modern MILP solvers. A sparsity-driven aggregation framework is introduced that models aggregation as an MILP and a two-stage LP heuristic that produces sparse, strong aggregated rows with measurable gains on MIPLIB2017. Theoretical results show CMIR cuts frequently define faces (and empirically facets) of the convex hull; Fenchel-style normalization is proposed to strengthen them. Finally, we give a prospect on the MINLP solving.

I will first discuss a problem of high-dimensional probability known as the ellipsoid fitting conjecture. I will present recent progress on this conjecture using both non-rigorous analytical tools of statistical physics, and rigorous methods based on the combination of universality results in statistics and extensions of classical approaches in random convex geometry. In a second part, I will discuss how the techniques developed to analyze ellipsoid fitting can be used to sharply characterize optimal learning in a wide neural network with a quadratic activation function, as well as in a model of learning from long sequences of high-dimensional tokens. 

This talk concerns the numerical approximations of a class of linear dispersive Initial Boundary Value Problems (IBVP) which describe the wave propagation in dispersive media. The considered class of IBVP is defined by assumptions on the symmetrization and on the boundary conditions. Under these assumptions, a reformulation of the IBVP with non-local conservation laws will be established. This reformulation will lead to H^1 estimates for the solutions of the dispersive IBVP. A numerical strategy will be proposed to enforce a fully discrete version of such estimates and numerical experiments will be done to assess its relevancy.

Nous discuterons le résultat suivant. Supposons que l’on dispose de deux familles d’applications de Hénon $(f_t)_t$ et $(g_t)_t$, paramétrées par une courbe algébrique, définies sur un corps de nombres, et que l’une d’entre elles soit dissipative. Alors il existe une constante positive $C$ et deux entiers strictement positive $N$ et $M$ telles que, pour tout paramètre $t$, soit le nombre de points périodiques communs à $f_t$ et $g_t$ est inférieur à $C$, soit, $f_t^N = g_t^M$. C'est un travail en cours, en collaboration avec Marc Abboud.

Nous expliquons l’importance de l’estimation d’une moyenne de série singulière dans l’étude de la répartition des nombres premiers dans les petits intervalles.

Suivant la conjecture d’Hardy et Littlewood, elle nous permet d’estimer le moment de certaines sommes sur les nombres premiers. Nous expliquons ensuite les idées nouvelles de la méthode mise en place pour améliorer les résultats de Montgomery et Soundararajan et Kuperberg sur ces sommes de séries singulières relatives à trois nombres premiers.