Master Class 2026
probabilités Master Class 2026
probability

Résumé : À venir Abstract: Coming soon

Résumé : This talk explores probabilistic methods for modeling and managing natural catastrophe (NatCat) risks, with a focus on two interconnected challenges. Abstract: This talk explores probabilistic methods for modeling and managing natural catastrophe (NatCat) risks, with a focus on two interconnected challenges.

First, Risk pooling and dependence structures: We analyze the spatio-temporal aggregation of NatCat losses, illustrating how stochastic models—calibrated to current and projected climate conditions—can quantify the benefits of risk diversification across regions and time. Case studies demonstrate the sensitivity of claim distributions to climate trends (e.g., increased frequency/severity of events). First, Risk pooling and dependence structures: We analyze the spatio-temporal aggregation of NatCat losses, illustrating how stochastic models—calibrated to current and projected climate conditions—can quantify the benefits of risk diversification across regions and time. Case studies demonstrate the sensitivity of claim distributions to climate trends (e.g., increased frequency/severity of events).

Second, optimal control under climate uncertainty from two different viewpoints: For insurers, we look into optimal dividend strategies using stochastic control, balancing shareholder returns against the long-term solvency constraints imposed by evolving catastrophe risks. For climate policy, we adapt these techniques to optimize carbon emission trajectories under budget constraints, incorporating behavioral biases (e.g., present-bias) and policy instruments (e.g., carbon taxes). The framework reveals trade-offs between immediate abatement costs and future risk exposure. Second, optimal control under climate uncertainty from two different viewpoints: For insurers, we look into optimal dividend strategies using stochastic control, balancing shareholder returns against the long-term solvency constraints imposed by evolving catastrophe risks. For climate policy, we adapt these techniques to optimize carbon emission trajectories under budget constraints, incorporating behavioral biases (e.g., present-bias) and policy instruments (e.g., carbon taxes). The framework reveals trade-offs between immediate abatement costs and future risk exposure.

Methodological tools include dependence modeling for extreme events (e.g., copulas, point processes) and dynamic programming approaches to solve control problems with path-dependent constraints. Methodological tools include dependence modeling for extreme events (e.g., copulas, point processes) and dynamic programming approaches to solve control problems with path-dependent constraints.

Résumé : La méthode dite de "Propagation de la rumeur" aussi appelée "Belief propagation" ou encore "Message Passing Algorithm" a été introduite en 1982 par Judea Pearl et a récemment été très utilisée en statistiques dans des problèmes de détection de communauté. Abstract: The so-called "rumor spreading method", also known as Belief Propagation or Message Passing Algorithm, was introduced in 1982 by Judea Pearl and has recently been widely used in statistics for community detection problems.

Nous montrerons dans ce cours son utilisation dans des problèmes d'appariements optimaux dans des graphes aléatoires simples, et évoquerons la manière dont Aldous a pu démontrer rigoureusement en 2000 un résultat fascinant obtenu dans les années 80 par Mézard et Parisi. We will show how it can be applied to optimal matching problems in simple random graphs, and explain how Aldous rigorously proved in 2000 a fascinating result first obtained in the 1980s by Mézard and Parisi.

Si le temps le permet nous parlerons également de son utilisation dans la compréhension de la géométrie des graphes aléatoires. If time permits we will also discuss how it helps us understand the geometry of random graphs.

Résumé : Dans la première partie de ce mini-cours, nous partirons du constat que l'intelligence artificielle repose fondamentalement sur les mathématiques. Je présenterai le fonctionnement des réseaux de neurones artificiels profonds dans leur architecture la plus simple, et montrerai comment les utiliser pour construire un système capable de reconnaitre automatiquement des chiffres manuscrits. Nous étudierons ensuite ces objets d'un point de vue plus théorique, en nous intéressant à leur comportement lorsque la taille des couches cachées devient très grande et en mettant en évidence un théorème limite qui décrit ce qu'il se passe. Nous utiliserons pour cela la méthode de Stein, un outil central en théorie des probabilités. Abstract: The first part of the mini-course starts from the fact that artificial intelligence fundamentally relies on mathematics. I will present how deep artificial neural networks operate in their simplest architecture and show how to train them to automatically recognize handwritten digits. We will then study these objects from a theoretical angle, focusing on their behavior when the hidden layers grow very large and deriving a limit theorem that describes this regime, using Stein's method—a central tool in probability theory.

Dans la seconde partie, nous changerons de perspective : grâce aux avancées spectaculaires des IA génératives, il devient désormais possible d'utiliser des modèles de langage (LLM) comme assistants de recherche en mathématiques. In the second part we change perspective: thanks to the spectacular progress of generative AI, it is now possible to use language models (LLMs) as research assistants in mathematics.