Asymptotic Theories and Painlevé Equations

After the spectacular discovery of the Painlevé transcendents at the beginning of the XXth century, the study of the Painlevé Equations underwent a long period of stagnation.

Today, due mostly to their relations with isomonodromic deformations of differential systems and their arising in questions such as the Ising model and solvable models of Statistical Physics, they have sparked renewed interest among a growing number of mathematicians and of theoretical physicists as well. The Painlevé transcendents are the special functions of the XXI-st century.
In addition, the recent developments in asymptotic analysis (Gevrey, Resurgent or Hyper-asymptotic theories both in the continuous and the discrete fields) have provided new tools whose efficiency in questions strongly related to Painlevé Equations such as new geometrical or algebraic criterions of integrability (new Painlevé tests, as it were) has been proved.

There can be no doubt that such an interaction will be the source of a highly rewarding strong cooperation in future. The Conference should provide the opportunity for everyone to exchange their experience to common benefit; it will be the occasion to have a number of natural problems displayed, together with their known solutions and also to show some new and very efficient tools at work.

The Conference will be organized around three main axes and their interconnections:

Complex Asymptotics with applications to Painlevé equations and their discrete analogs,

Geometrical and Algebraic Approaches,

Complete or Partial Integrability.

The present situation is ideal for a young researcher to start research work in these domains. A general introduction to each field will be given in order to help the non-specialists and so that the young participants in particular, can derive full benefit from the more specialized lectures.