The Stokes phenomenon in exact asymptotics

As an introduction we present a new, elementary and constructive proof of the multisummability properties of formal solutions of linear ODE's at irregular singular points. This serves to illustrate the geometric approach to multisummation. Basic properties of multisums and the associated sheaves are derived. Next, we study Cauchy-Heine transforms in relation to multisummation and the Stokes phenomenon. We show how to construct multisums with a prescribed Stokes phenomenon, using the Malgrange-Sibuya isomorphism. Starting from the Stokes automorphisms we introduce the alien derivations of J. Ecalle and derive Ecalle's bridge equation for the general integral of linear ODE's. The main ideas are illustrated with some very simple examples.