Asymptotics for the first Painlevè equation : a resurgent view point

This talk aims at showing on an example how one can deal with nonlinearities and resonances with resurgent methods. We will pay special attention to the first Painlevé equation, which plays a significant role in many physical models : for instance, this equation and its resurgent structure were particularly adressed recently in string theory thanks of its interpretation in the context of 2D quantum gravity in the so-called double scaling limit. We will show how to get the Painlevé transcendents by Borel-Laplace summation, and we will interpret the formal integral in terms of holomorphic conjugations. We will describe the nonlinear Stokes phenomena and the Ecalle's holomorphic invariants.