Institut de recherche mathématique avancée

Résumé : Graph learning is an active research domain in statistics, highlighted by well-known models such as Gaussian graphical models for i.i.d. data and autoregressive models for time series. In this talk, we will present two new models developed to address distinct analytical challenges. The first model tackles a non-classical data setting where the data points are probability distributions. Here, the graph is inferred to represent the dependence structure of a set of distributional time series. Leveraging Wasserstein space theory, we develop a novel autoregressive model, which is then applied to a demographic dataset. The second model is designed to meet an application-driven need: inferring a functional connectivity graph for a single subject’s brain fMRI time series while quantifying uncertainty. We adopt a Bayesian modeling approach to infer these graphs, with posterior distributions over edges providing uncertainty estimates. In particular, we introduce a prior for correlation matrices that facilitates the integration of expert knowledge. The model is applied to a rat fMRI dataset, where two follow-up analyses—edge detection and subject comparison—are conducted. The results highlight the robustness gained through uncertainty quantification.

Résumé : A large toolbox of numerical schemes for dispersive equations has been established, such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). These classical schemes are based on linearised time dynamics and in many situations allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene such as for problems at low regularity and high oscillations. Classical schemes fail to capture the oscillatory nature of the solution, and this may lead to severe instabilities and loss of convergence. In this talk I present a new class of resonance based schemes. The key idea in the construction of the new schemes is to tackle and deeply embed the underlying nonlinear structure of resonances into the numerical discretization. As in the continuous case, these terms are central to structure preservation and offer the new schemes strong geometric properties at low regularity.

Résumé : It is in general a hard problem to determine whether two given finitely generated groups are quasi-isometric, even among some "well behaved" classes. One such class, which has been the subject of intensive research in group theory, is the one of wreath products, as they often exhibit unexpected and interesting behaviours. The classification up to quasi-isometry of lamplighters over Z goes back to 2013, and in a recent work (2021), Genevois and Tessera extended it to all lamplighters over finitely presented one-ended groups. This raises the question of also classifying their permutational variants. In this context, strong rigidity phenomenon as the ones observed for standard wreath products do not hold, whence the need of another approach. After having introduced and discussed several re-inforcements of quasi-isometries, I will sketch the main guidelines of the proof of a quasi-isometric classification of some permutational wreath products, that covers a number of classical cases. If time permits, we will also discuss some applications and open problems.

Résumé : Résumé : Une marche aléatoire renforcée par ses pas est une marche aléatoire qui, à chaque instant, répète un de ses pas passés choisis uniformément au hasard ou fait un nouveau pas. Dans cet exposé, on présente une version modifiée où le choix du pas n’est plus uniforme mais dépend du passé de façon « amnésique ». On montre que dans un certain régime, on trouve toujours un résultat de type principe de Donsker où la limite est la somme de deux processus gaussiens dépendants et explicites, en lien avec la limite dans le cas d’une mémoire uniforme.