Module 1: Basic multisummability
Instructor: F.Sanz, J.P. Rolin, P. Speissegger

- Gevrey asymptotics, Borel and Laplace transforms, k-summability.
- Cauchy-Heine transforms and decomposition theorems.
- Multisummability, iterated Borel and Laplace transforms, singular directions.
- Application: strong analytic transcendence from multisummability.
- Braaksma's theorem for nonlinear meromorphic ODEs.

Module 2: Resurgent functions
Instructor: D. Sauzin

- Reminders on Gevrey asymptotics and Borel-Laplace transform (cf. Schäfke's course). The Nevanlinna theorem. Examples (Airy, Ei(z), Erf (z), Stirling).

- The definition of "resurgence". Analytic continuation in the Borel plane and stability by convolution. Application to non-linear dynamics. The example of the saddle-node.

- Ecalle's "Alien calculus". The definition of "alien derivations". The "bridge equation".=

Module 3: Non-oscillatory trajectories
Instructor: F. Sanz

The course deals with the qualitative study of oscillatory and non- oscillatory trajectories of real analytic vector fields, mainly in dimension three. In the first part of the course, we describe several kinds of asymptotic behaviour that such transcendental objects can have: axial spiraling, asymptotic linking, separation by projection. In the second part, we will study non-oscillatory trajectories that belonging to new o-minimal structures, an application of the contents of the courses given by J.-P. Rolin, R. Schäfke and F. Cano.