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Résumé : Considérons un système de particules aléatoires. quand peut-on dire qu'il est proche de l'équilibre ? Parfois, le système atteint rapidement l'équilibre de manière abrupte, ce que l'on qualifie de phénomène de cut-off. Le but de l'exposé est de présenter ce phénomène et d'expliquer quels renseignements fournit le comportement macroscopique d'un système sur le temps de mélange.

Résumé : Dans cette présentation, je rappellerai quelques résultats essentiels donnant des critères de compacité dans des espaces fonctionnels fondamentaux. Dans un second temps, nous verrons comment décrire la perte de compacité dans des espaces bien connus des analystes et edpistes, que sont les espaces de Sobolev.

Résumé : This work is devoted to the study of dimension reduction techniques for multivariate spatially indexed functional data and defined on different domains. We present a method called Spatial Multivariate Functional Principal Component Analysis (SMFPCA), which performs principal component analysis for multivariate spatial functional data. In contrast to Multivariate Karhunen- Loève approach for independent data, SMFPCA is notably adept at effectively capturing spatial dependencies among multiple functions. SMFPCA applies spectral functional component analysis to multivariate functional spatial data, focusing on data points arranged on a regular grid. The methodological framework and algorithm of SMFPCA have been developed to tackle the challenges arising from the lack of appropriate methods for managing this type of data. The performance of the proposed method has been verified through finite sample properties using simulated datasets and sea-surface temperature dataset. Additionally, we conducted comparative studies of SMFPCA against some existing methods providing valuable insights into the properties of multivariate spatial functional data within a finite sample.

Résumé : I will discuss a microlocal analysis approach to spectral theory on asymptotically Minkowski spaces both for scalar wave operators and also for Dirac type operators. This in turn gives rise to complex powers of the operators, allowing for the analysis of a spectral zeta function, relating its residues to geometric information. This is joint work with Nguyen Viet Dang and Michal Wrochna, with ongoing work on extensions also with Mikhail Molodyk.

Résumé : Piecewise Deterministic Markov Processes (PDMPs) constitute a family of Markov processes characterized by deterministic motion interspersed with random jumps. When controlled through discrete-time interventions, this leads to an impulse control problem. In the fully observed setting with known dynamics we develop a numerical method to compute an optimal strategy. In real-world applications, however, full observability is rarely available. Under partial observation, the impulse control of a PDMP can be reformulated as a Partially Observed Markov Decision Process (POMDP), which we address using deep reinforcement learning techniques. A major limitation of existing approaches is the assumption that the underlying PDMP dynamics are known or can be accurately simulated. This assumption is unrealistic in applications such as patient monitoring, where data may be scarce and disease dynamics may vary across individuals. To address this issue, we introduce a Bayesian Adaptive POMDP (BAPOMDP) framework, in which the unknown PDMP parameters are modeled probabilistically and updated through Bayesian inference. The resulting continuous-state BAPOMDP is solved using deep reinforcement learning methods adapted to high-dimensional belief spaces. This work thus combines stochastic control theory, Bayesian modeling, and deep reinforcement learning to provide a unified framework for decision-making under partial observability and model uncertainty. The proposed methodology is thoroughly illustrated and validated on a medical application : the adaptive follow-up and monitoring of patients diagnosed with multiple myeloma.

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Résumé : Lors de cet exposé, je présenterai des résultats récents obtenus en collaboration avec Debayan Maity concernant l'interaction entre un solide rigide et un fluide visqueux compressible dans un domaine extérieur. Notre analyse porte sur l'existence de solutions fortes et leur comportement asymptotique en temps long. En combinant un changement de variables approprié avec des estimations de décroissance $L_p-L_q$, nous établissons le caractère bien posé global du système couplé pour des données initiales petites. Dans cette démonstration, nous obtenons un taux de décroissance pour les vitesses du fluide et du solide lorsque $t \to \infty$, qui nous permet en particulier de montrer que le solide rigide tend vers une position finale.

Résumé : Double shuffle and regularization relations is a distinguished set of relations among multiple zeta values. Consider those relations formally, G. Racinet introduced the double shuffle Lie algebra (dmr). The Kashiwara-Vergne Lie algebra (krv) was introduced by A. Alekseev and C. Torossian when they reproved the Kashiwara-Vergne conjecture using Drinfeld associators. In this talk, I will present a topological characterization of the skew symmetric part of dmr and symmetric krv. As an application, we prove the inclusion of skew-symmetric part of dmr with IHX relations to symmetric krv. It is based on the preprints: 2509.20275, arXiv:2601.19455 with collaborators and some ongoing work.

Résumé : Dans cet exposé, je vous propose de partir à la découverte du jeu de Hex, un jeu de plateau à cases hexagonales pour 2 joueurs, dont la richesse mathématique n’a d’égale que sa simplicité. La première apparition du jeu de Hex revient à Piet Hein en 1942, puis il sera redécouvert de manière indépendante en 1948 par John Nash, qui en étudiera la mathématique. Après avoir présenté les règles, nous répondrons à plusieurs questions : Existe-t-il une stratégie gagnante ? Si oui, pour quel joueur ? Peut-il y avoir égalité ? Existe-t-il une configuration où les deux joueurs sont gagnants en même temps ? Toutes ces réponses nous permettront de nous servir du jeu pour démontrer un célèbre théorème de topologie : le théorème de Brouwer.

Résumé : Consider the eigenvalue problem for the discrete Laplacian on finite regular graphs. One can relate three types of phase space distributions associated to the resulting eigenfunctions, namely Patterson–Sullivan, invariant Ruelle, and Wigner distributions. It has been shown that, on hyperbolic surfaces, these three phase-space distributions are asymptotically equivalent, naturally raising the question of whether a similar relationship holds for finite graphs. In this talk, I will show how discrete analogues of these distributions can be defined and how they relate to each other.

Résumé : Let A be an abelian variety defined over a number field k. The connected monodromy field k(eA) is the minimal extension of k over which every \ell-adic Galois representation attached to A has connected image, or equivalently, the minimal extension over which all Tate classes on all self-products A^r are defined. When k(eA)/k(End A) has positive degree, "exotic" Tate classes arise on certain powers A^r — classes not explained by the endomorphism algebra alone. I will explain how to compute k(eA) when A is the Jacobian of a curve with complex multiplication. We will exploit CM theory to describe the algebra of Tate classes and make the Galois action on this algebra explicit in terms of periods — suitable integrals of algebraic differential forms. Though periods are generally transcendental, those attached to Tate classes are algebraic, so computing k(eA) reduces to identifying these periods as exact algebraic numbers.

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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.

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