Institut de recherche mathématique avancée

Résumé : The Riemann-Hilbert problem requires the existence of regular linear ordinary differential equations with prescribed singularity and monodromy. In higher dimensions, Deligne formulated it as a correspondence between regular meromorphic flat connections and local systems. Later, Kashiwara generalized it to a correspondence between regular holonomic D-modules and perverse sheaves on a complex manifold. Removing the regularity condition, Deligne and Malgrange proposed a correspondence between meromorphic connections on a curve and local systems endowed with a Stokes filtration. A few years ago, in joint work with Kashiwara, we dealt with general holonomic D-modules in any dimension. The target category of our correspondence is built on Kashiwara-Schapira's theory of ind-sheaves and Tamarkin's work on symplectic geometry. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya. In this lecture, I will give a brief overview of the above.

Résumé : Résumé : Etant donné une famille de variétés compactes rotationnellement symétriques indexée par la dimension et une fonction poids, nous étudierons le phénomène de cut-off pour le mouvement brownien. Notre preuve est basée sur la construction d'un processus entrelacé et sur la concentration de la mesure. Nous verrons une transition de phase concernant l'existence ou non de phénomènes de cut-off, qui dépend de la forme de la fonction poids.

Résumé : Résumé: The Butcher trees and series form a famous algebraic tool for the manipulation of Taylor expansions of flows of differential equations, with fundamental applications in numerical analysis. In particular, the backward error analysis studies the properties of a numerical flow by writing it as the exact solution of a modified system, which is then described with a Hopf algebra of trees.
After introducing Butcher series, their exotic and aromatic extensions for stochastic systems, and some of their numerical applications, we will show that exotic aromatic B-series satisfy a universal geometric property of equivariance. This will confirm that B-series are a natural geometric object that is interesting beyond their numerical applications.
If time allows, we shall give an overview of the combinatorial Hopf algebra framework linked to the backward error analysis of stochastic differential equations, and related open problems.