La semaine de l’IMB

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.