Institut de recherche mathématique avancée

Résumé : This talk will be centered around some finite dimensional complex representations of mapping class groups of surfaces. More precisely, we will discuss quantum representations of mapping class groups arising from Witten-Reshetikhin–Turaev Topological Quantum Fields Theories. These are projective unitary finite dimensional complex representations of mapping class groups indexed by an integer called level. When the level is a prime number, the image of the representation lands in the integral points of an algebraic group G and is Zariski dense in G. One important open problem is to know if this image has finite index in G(Z). As I will explain, for a fixed prime level p, knowing the kernel of the representation might help knowing if the image is arithmetic or not. I will explain a joint work with Renaud Detcherry where we can give some information about this kernel, more precisely we compute the two first terms of the so-called h-adic approximation of the representation (which is a sequence of finite groups approximation of the representation).

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 : 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.