Lecturer: Teresa Martínez-Seara (Universidad Politécnica de Cataluña, Barcelona, Spain).

Title: Exponentially small phenomena and resurgence

Abstract: Exponentially small phenomena in some parameter are phenomena which occur for analytical dynamical systems (either discrete or continuous, finite or infinite dimensional) in which the dominant part of some relevant magnitude behaves like and, hence, it is below any given power of for sufficiently small. They are present in problems of normal forms, splitting of separatrices (or, in general, distance between invariant manifolds), bifurcations, unfoldings, slow-fast systems, perturbations of the identity, averaging, either in maps or in flows, etc. The main difficulty in detecting and computing such phenomenon is that they can not be treated using classical perturbation methods, like power expansions in the parameters of the system. The source of this phenomenon relies on the existence of singularities in the complex phase space and/or time. Therefore its treatment needs to use tools of complex analysis.

In this course we will relate these phenomena with resurgence theory in a natural way. They occur when the series in power of the parameter of the solutions is a Gevrey series, so is divergent but Borel summable. The tools needed to detect such phenomena will be approximation theory in suitable complex domains which need to deal with some equations, called inner equations, that can be treated using Borel summability methods and resurgence theory.