Institut de recherche mathématique avancée

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 goal of this talk is to describe joint work with Scholze where we study a version of algebraic geometry which is built, not out of spectra of commutative rings, but out of spectra of symmetric monoidal higher categories. The resulting higher geometry contains the usual category of qcqs schemes, but also provides a home to new and interesting objects which cannot be studied with more classical means. Time permitting, I will sketch the role that some of these objects play in ongoing work joint with Ben-Zvi and Nadler on various Langlands duality statements in the context of three dimensional topological field theory.

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.

Résumé : Résumé : (Universal) enveloping algebras of finite-dimensional Lie algebras are among the most well-understood noncommutative rings: in fact, many of the fundamental techniques of ring theory were developed in order to understand these enveloping algebras. However, when the Lie algebra becomes infinite-dimensional, its enveloping algebras becomes much more mysterious. This talk will survey what's known about enveloping algebras of infinite-dimensional Lie algebras, starting with the definition and focussing on noetherianity questions and applications to representation theory.