Institut de recherche mathématique avancée

Résumé : This study introduces a nonlinear reduced order model (ROM) for fluid dynamics, which combines proper orthogonal decomposition (POD) with deep learning error correction. Our approach merges the interpretability and physical adherence of classical POD Galerkin ROMs with the predictive capabilities of deep learning. The hybrid model addresses errors within and outside the POD subspace. Firstly, POD generates part of the reduced state, complemented by an autoencoder compressing only the unretained POD modes. Thus, the most energetic modes are computed interpretably, while the least energetic are handled with a superior reduction method. Secondly, the time integration employs a hybrid neural Ordinary Differential Equation (neural ODE). A POD ROM estimates part of the dynamics, and a deep learning model corrects its error. Using Neural ODE aligns the model with underlying physics for enhanced stability and accuracy. The proposed method differs from current hybrid methods operating solely in the POD subspace and using Mori-Zwanzig time dependency, posing potential initialisation issues. Our model is applied to the viscous Burgers' equation, the parametric circular cylinder flow, and the fluidic pinball test case. Accuracy and numerical complexity are compared to classical POD Galerkin ROMs, fully data-driven models, and concurrent hybrid methods.

Résumé : A Drinfeld associator is a certain Lie series deeply related to braids on a disk, which is a genus 0 surface. On the other hand, a solution to the Kashiwara–Vergne (KV) problem, originated from Lie theory, corresponds to a solution of the formality problem of the Goldman–Turaev Lie bialgebra associated with a pair-of-pants by the result of Alekseev, Kawazumi, Kuno and Neaf. These objects are first related by Alekseev and Torossian, and Massuyeau constructed an explicit map from the set of Drinfeld associators to the solution set of the KV problem. In this talk, we extend their method to higher genera to obtain a similar map based on Gonzalez’ definition of higher genus Drinfeld associators.

Résumé : Un théorème des années 70 nous dit qu'il n'existe pas un algorithme pour résoudre les équations polynomiales à coefficients entiers. Dans le désespoir, les arithméticiens ont commencé à développer des méthodes pour essayer de le faire au moins dans certaines situations. Puisqu'il est facile de trouver des solutions modulo n, on se demande: quand peut-on remonter ces solutions pour en obtenir une dans Z? Cela nous conduira au théorème de Hasse-Minkowski et à l’obstruction de Brauer-Manin.

Résumé : The first non-zero eigenvalue, or spectral gap, of the Laplacian on a closed hyperbolic surface encodes important geometric and dynamical information about a surface. In this talk, I will discuss the typical size of the spectral gap for a random surface with large genus sampled with respect to the Weil-Petersson probability measure. In particular, I will explain joint work with Will Hide and Davide Macera where we obtain a spectral gap with a polynomial error rate. Our result uses a fusion of the polynomial method used in recent breakthroughs on the strong convergence of group representations with the trace formula for hyperbolic surfaces.

Résumé : Dans cet exposé, on s’intéresse à la cohomologie étale à coefficients dans G_m d’un espace rigide analytique X lisse sur C_p. Dans le cas algébrique, on sait que si X est un schéma noethérien normal régulier alors en degré plus grand que 2, ces groupes sont de torsion. Ce résultat découle du fait que pour un point x de X et un faisceau abélien F_x sur x, les groupes de cohomologie de F_x sont eux-mêmes de torsion. La même stratégie ne peut pas être appliquée aux espaces rigides analytiques puisque ceux-ci n’ont pas assez de points. Une solution pour contourner ce problème est de remplacer le faisceau G_m par son quotient par le sous-groupe des unités principales U. Dans le cas où X est un espace de Stein, la cohomologie de U peut être calculée par des méthodes p-adiques en passant par le site proétale de X. La cohomologie du quotient G_m/U peut ensuite être calculée par des méthodes similaires à celles du cas algébrique, i.e. en calculant la cohomologie au dessus d’un point. Il s’agit d’un travail en collaboration avec Damien Junger.

Résumé : A common assumption in classical risk theory is the independence between claim frequency and severity. However, this often fails in practice, especially when subtle or nonlinear dependencies are present. This study analyzes a real-world dataset comprising 15,665 claims from housing and liability insurance contracts, recorded between 01/01/2015 and 31/12/2019, provided by an anonymous insurer. Unlike most literature focused on automobile insurance, our data allow us to explore dependence in less-studied lines of business, which pose unique challenges. We investigate the presence and nature of dependence between frequency and severity, and how this relationship evolves over time and across insurance types. Our approach combines parametric and nonparametric methods: we fit Poisson-Inverse Gaussian, Negative Binomial, Weibull, and Log-Normal distributions to the marginals, and apply copula-based techniques to assess joint behavior. Using pseudo-observations, we estimate empirical copulas, visualize joint densities, and perform statistical tests of independence and equality (KcopTest), which reveal a structural break in housing insurance in 2016. Results indicate strong positive dependence in liability insurance. In housing insurance, we find near-independence in most years, but a weak and significant negative dependence in 2016. Finally, GAMLSS models confirm diverging patterns: in liability insurance, severity increases with frequency; in housing insurance, it decreases, in contrast to findings by Garrido (2016). We discuss implications for pricing, reserving, and solvency assessment under dependence.