Institut de recherche mathématique avancée

Résumé : The context of this talk is the study of inverse problems for wave propagation phenomena. The goal is to formalize, analyze, and discretize sequential data assimilation strategies, in which measurements are incorporated as they become available. The resulting system, called an observer, stabilizes on the observed trajectory and hence reconstructs the state and possibly some unknown parameters of the system. In the first part of the talk, I focus on source reconstruction, an estimation problem intermediate in complexity between state estimation and the more general problem of parameter identification. In this setting, we define, in a deterministic infinite-dimensional framework, a so-called Kalman estimator that sequentially estimates the source term to be identified. Using tools from dynamic programming, we show that this sequential estimator is equivalent to the minimization of a functional. This equivalence allows us to establish convergence results under observability conditions. We demonstrate these observability inequalities for different source types by combining functional analysis, multiplier methods, and Carleman estimates. The second part concerns discretization issues and their analysis. We propose a discretize-then-optimize approach, where the observability inequalities need to be extended to discretized systems. In particular, in this context, we extend uniform exponential stabilization results to the discretization for high-order finite element discretizations of a 1D wave equation. Joint work with S. Imperiale and P. Moireau.

Résumé : En 1917, S. Kakeya pose la question suivante: quelle est l'aire minimale d'un domaine du plan à l'intérieur duquel on peut faire tourner (d'un tour complet) un segment de longueur 1? La réponse, apportée par A. Besicovitch, est qu'il est possible de le faire dans un domaine d'aire arbitrairement petite. Une question similaire se pose en dimension supérieure, et porte de le nom de conjecture de Kakeya: l'année dernière, cette conjecture a été résolue en dimension 3 par Hong Wang et Joshua Zahl, dans des travaux qui ont eu un grand retentissement. J'essayerai d'expliquer l'intérêt pour cette question qui semble à première vue seulement récréative, et son impact dans différents domaines des mathématiques.

Résumé : I will present an overview of semi-group techniques that allow us to obtain, for certain EDPs, results showing the system’s return to equilibrium—sometimes at an optimal rate—and the existence of global solutions in a perturbed regime around the equilibrium.

Résumé : The aim of this talk is to present the different questions that arise in the Langlands program. To make this very obscure subject a bit less mysterious, I will start with an illustration of how the methods used in the Langlands program, namely modularity, yield interesting arithmetic results. Then, I will highlight the different generalisations that give rise to the various branches of the area. If time permits, I would like to give more details about whatever the audience is more interested in.

Résumé : Extreme meteorological events often occur in complex temporal configurations, where the impacts of one hazard may depend on the prior occurrence of others. Characterising such temporal dependencies is essential for understanding compound climate risks, yet remains challenging due to the discrete, heterogeneous, and clustered nature of extreme events. In this study, we apply temporal point process methods to characterise dependencies among extreme meteorological events occurring within appropriately defined spatial regions across Europe, focusing exclusively on their temporal structure.
We introduce an event-based framework in which extreme events are represented as marked temporal point processes, with marks describing key characteristics such as intensity or duration. Global first- and second-order temporal statistics are used to quantify clustering, co-occurrence, and directional dependencies between different types of extremes. In particular, we rely on directional cross-$K$ functions to assess whether the occurrence of one type of extreme event systematically modifies the short-term probability of subsequent events of another type.
Two complementary applications illustrate different facets of compound event analysis. First, we demonstrate the relevance of the framework for preconditioned compound events through a temporal analysis of wildfire-related meteorological extremes. Second, we examine temporal dependence between extreme precipitation, extreme wind, and extreme atmospheric instability across all European NUTS-2 regions.
Building on these second-order statistics, we develop formal tests of temporal independence to assess the significance of observed directional interactions between different types of extreme events. Overall, this temporal point process framework provides a rigorous and interpretable approach to the analysis of compound and preconditioned climate extremes, with direct applications to climate risk assessment and early-warning systems.