V.Fantini - Introduction to summability and resurgence for irregular singular ODEs - I

Trainer: Veronica Fantini
Title: Introduction to summability and resurgence for irregular singular ODEs
Abstract: Solving ODEs in complex domain with irregular singularities can be done either formally or analytically. At the formal level, a method by Poincaré allows to build a frame of formal (and divergent) solutions. Then, Borel-Laplace summation allows to reconstruct an analytic frame of solutions by "re-summing" the formal ones. Alternatively, we can build a frame of analytic solutions by turning the ODE into an integral Volterra type equation. In the first two lectures we will discuss both approaches for a class of ODEs called of level 1. Then, in the third lecture will discuss some basic aspects of the theory of resurgence introduced by Écalle. In particular, we will study the properties of resurgent series that arises from solutions of irregular singular ODEs.