Institut de recherche mathématique avancée

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.

Résumé : Les variétés de caractères sont des espaces classifiant les représentations d’un groupe dans un autre, à conjugaison près. Ils apparaissent naturellement dans de nombreux problèmes d’origines géométriques et dynamiques.Ces variétés de caractères ont un groupe de symétries naturelles (le groupe modulaire pour les groupes fondamentaux des surfaces), et on peut s’intéresser à l’action de ce groupe. Cette action est riche d’un point de vue dynamique. Nous expliquerons d’abord que cette dynamique est très chaotique, et assez bien comprise, quand le groupe d’arrivé est compact, puis nous présenterons un programme et des résultats qui visent à comprendre cette dynamique quand le groupe d’arrivé est PSL(2,R), en utilisant des techniques qui viennent de la renormalisation des échanges d’intervalles.

Résumé : For a graph $G$ and integer $k$, the Ramsey number $R(G; k)$ is the smallest integer $n$ such that any edge-coloring of a complete graph $K_n$ on $n$ vertices with $k$ colors results in a monochromatic copy of $G$. Determining Ramsey numbers even in the case of two colors has been a stubborn problem since its introduction in 1930. Except for finding the classical Ramsey number $R(t) = R(K_t;2)$ for cliques, one of the main open problems in the area is to determine multicolour Ramsey numbers for cycles. For positive integers $k$ and $\ell$, we show that $R(C_{2 \ell + 1}; k) \leq (4 \ell)^k \cdot k^{k/\ell}$, where $C_{2\ell +1}$ is a cycle of length $2\ell+1$. This is the first improvement for fixed $\ell$ and large $k$ since the bound $2\ell (k+2)!$ by Erd\H{o}s et al. from 1973. This is a joint work with Wouter Cames van Batenburg, Oliver Janzer, Lukas Michel, and Mathieu Rundstr\"{o}m.

Résumé : Traditionally, Langlands functoriality refers to the identification of automorphic forms whose parameters take a special shape. In this talk, we explain how to ask analogous questions on the Hitchin moduli space using perspectives from the relative Langlands program. We gain, in this setting, the advantage of working with a version of Langlands duality which is readily computable, and if time permits we will discuss the ramifications of these calculations for automorphic periods and L-functions.

Résumé : Non-associative algebras arise in many mathematical settings, with the Lie algebra being a significant example. A powerful tool to study a Lie algebra is its universal enveloping algebra, which is an associative algebra, whose representation coincides with the representation of that Lie algebra. The classical Poincaré-Birkhoff-Witt (PBW) theorem plays a role by giving a vector space basis of the enveloping algebra. We will generalise the notion of enveloping algebra to arbitrary (non-associative) algebra and present the generalised PBW theorem via giving examples. If time permits, we could talk about a more generalised perspective — universal enveloping Operads.