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. 

We introduce a nonlinear interpolation framework for parametric fields that relies on a variational mapping approach to track and align coherent structures across parameter values. Starting from high-fidelity simulations, we employ scalar sensors to extract point clouds representing key solution features—such as shocks, shear layers, or other coherent structures—and use registration techniques to construct bijective domain mappings that allow accurate nonlinear interpolations.

Within the parametric model order reduction setting, these variational procedures exploit solution snapshots to identify coordinate transformations that improve the approximation of the solution set. Optimization-based methods minimize a target function measuring the alignment of coherent structures across the parameter domain, over a family of bijections defined on a bounded domain. We consider diffeomorphisms generated as vector flows of velocity fields with vanishing normal component on parts of the domain boundary; we rely on a sensor to extract point clouds from the collected solution snapshots and develop an expectation–maximization strategy to simultaneously solve the point cloud matching problem and determine the mapping. We then combine the resulting registration with convex displacement interpolation [Iollo, Taddei, J. Comput. Phys., 2022] to obtain accurate interpolations of fluid-dynamic fields in the presence of shocks. Numerical results for a two-dimensional inviscid transonic flow past a NACA airfoil and a three-dimensional viscous transonic flow past an ONERA M6 wing illustrate the key components of the methodology and demonstrate the effectiveness of nonlinear interpolation for shock-dominated regimes.

Work with Jean-Baptiste Chapelier, Jon Labatut and Tommaso Taddei

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.