Institut de recherche mathématique avancée

Résumé : L'identité dite standard donne une version faible de la commutativité pour un produit de k>2 éléments. Cette identité a été mise en lumière en 1950 dans l'article célèbre par Amitsur et Levitzki qui ont montré que l'algèbre de matrices carrées de taille n est faiblement commutative pour k=2n. Dans les années 1970, plusieurs personnes ont trouvé cette identité dans le contexte de la géométrie différentielle, plus précisément en regardant l'algèbre de Lie de champs de vecteur sur une variété lisse. L'année dernière j'ai trouvé deux autres situations surprenantes ou l'identité standard apparaît : les crochets de Rankin-Cohen, d'origine dans la théorie des formes modulaires, et les opérations bilinéaires utilisées dans les formules de Kontsevich de la quantification par déformation. Dans cet exposé, je vais donner une introduction très accessible à tous ces résultats.

Résumé : Given a sequence a_n, one can ask about its behaviour as n grows, i.e., its asymptotics. This question has been very well studied in a wide context, with general results developped, for instance, within analytic combinatorics. What happens when our sequence has two parameters that both go to infinity ? Now we're in the realm of bivariate asymptotics, for which there exists some results from analytic combinatorics, but the current knowledge is still much more limited than in the univariate case. Together with Andrew Elvey Price, Wenjie Fang and Michael Wallner, we developped new methods to study bivariate recurrences and obtain asymptotics from it, which we apply to several contexts as advertised in the title. I will explain the historical background as well as our new results, without going too much into the technicalities.

Résumé : The birational geometry of smooth projective complex varieties with trivial canonical bundle is a deep and intensively studied subject. While there has been a lot of work on birational maps between such varieties (for example, on birational automorphisms of Hyper-Kähler manifolds), less has been said about rational maps of degree at least two, which we will call rational covers. What restrictions do rational covers Y --> X (and especially their monodromy) impose on the geometry of X, when both X and Y have trivial canonical bundle? For a given X, when does there exists such a cover which is furthermore Galois?
In this talk, we will present and answer various questions around this theme, with a particular interest in the case where X is Hyper-Kähler. The main motivation is the following question of Laza: can it be that any Hyper-Kähler deforms to one birational to A/G, where A is an abelian variety and G a finite group acting on it?

Résumé : The goal of this talk is to define a p-adic analogue of 2πi by seeing it as the residue of a holomorphic function at zero, and hopefully convince you that the answer to this question is both deep and satisfying. We will firstly explain why the naive approach to this question doesn't work, and how algebraic geometers think of 2πi. Then we will try and copy the definition we found in the p-adic setting and see how it leads us to the construction of an element of a ring with a specific behavior with respect to the Galois action, and hence why the p-adic analogue of the complex numbers is not the first thing that comes to mind.

Résumé : Covariance estimation has many applications, such as brain connectivity, portofolio allocation, to cite a few. Following a Bayesian approach, a typical covariance prior is the Inverse-Wishart distribution. However, a well-known issue of this prior is that it concentrates too little mass over covariance matrices with small eigengaps. To rebalance the mass, Berger et al. (2020, Annals of Statistics) proposed a more generic family, Shrinkage Inverse-Wishart, which thus offers a more flexible prior choice for covariance matrices. However, sampling from it remains challenging. The existing algorithm relies on a nested Gibbs sampler, which is slow and lacks rigorous theoretical convergence analysis. We propose a new algorithm based on the Sampling Importance Resampling method, which is significantly faster and comes with theoretical convergence guarantees. In this talk, we first derive the new sampling algorithm of Shrinkage Inverse-Wishart. Then we apply it to the inference of a Bayseian model of covariance estimation. We show inference results over a real data set of fMRI signals of rats.

Résumé : Résumé : I will discuss the construction of symplectic Lefschetz fibrations using tools from classical Lie theory, and then explain how they were used to provide examples of new phenomena in Mirror Symmetry.