Lecturer: Jean-Pierre Ramis (Université Paul Sabatier, Toulouse, and Académie des Sciences).

Title: Gevrey asymptotic expansions, summability of divergent series and applications.

Abstract: We will first present a short history of the summation of some divergent series from Euler to Hardy and Ramanujan, and of the beginning of the theory of asymptotic expansions due to Poincaré (and Stieltjes). We will detail the work of Stokes on the fringes of the supernumerary rainbows.

The classical theory of asymptotics (Poincaré asymptotics) has some flaws, we will explain why and why it is better to replace it by Gevrey asymptotics. Using Gevrey asymptotics it is easy to re-interpret the Borel-Laplace method of summation of divergent series, more generally we will present the notion of k-summability and we will apply it to the study of analytic linear differential equations. We will insist on Ramis-Sibuya theorem which is an extremely powerful tool to get Gevrey asymptotics and k- summability "at a small price". The k-summability is not sufficiently powerful to deal with arbitrary analytic linear O.D.E. and it is necessary to use the notion of multisummability that we will shortly describe.

We will follow with Stokes phenomena and we will give some hints of the relations with resurgence (in the "generic" case).

We will finish by some applications of Gevrey asymptotics to singular perturbations problems (canards, Ackerberg-O'Malley resonance,...).