From Magnus' expansion to Butcher's B-series

Lien vers la page Web de l'orateur Abstract: In this talk we'll start by reviewing Magnus' expansion and its fine structure. The Magnus expansion is a peculiar infinite series involving Bernoulli numbers, and iterated Lie brackets together with integrals. It results from the recursive solution of a particular differential equation, which was introduced by Wilhelm Magnus in 1954, and which characterizes the logarithm of the solution of linear initial value problems for operators. We will analyze it in the context of Gian-Carlo Rota's proposition of the notion of integration algebras, complementary to the already existing theory of differential algebras. It turns out that pre-Lie algebras play a crucial role in this approach. They allow to link the Magnus expansion to Butcher's B-series, the latter being an important tool in the theory of numerical methods for differential equations. We will also present recent work on the Magnus expansion using rooted trees. We will provide a short review of background material to make the talk reasonably self-contained. This talk is based on joint work with Frederic Patras (CNRS, Nice, France) and Dominique Manchon (CNRS, Clermont-Ferrand, France).