# IST Lecture Series in Algebraic Geometry & Physics

### Stokes Phenomenon and Dynamics on Wild Character Varieties of Painlevé Equations (III)

Third Lecture

Firstly I will discuss the problem of the definition of the wild monodromy for an arbitrary irregular singularity in the linear case, in relation with Stokes phenomena, $k$-summability, multisummability, Laplace transform and resurgence. The source of this topic is, a century ago, a R. Garnier paper 1919.

I will detail the basic example: the monodromy and Stokes phenomena in the case of Hypergeometic Equations (classical and confluent). I will explain the confluence of monodromy towards wild monodromy.

In the second part of the lecture I will describe some non-linear Stokes phenomena and the corresponding unfoldings: saddle-nodes, symplectic saddle-nodes.

I will end with the application of all the tools to the case of the confluence of PVI towards PV. As a byproduct, it is possible to get a proof of the rationality of the wild dynamics of PV (M. Klimes). It is extremely technical and I will give only the (simple!) basic ideas and the main lines.

It is a first step towards a proof of the following conjecture (Ramis 2012):

The (wild) dynamics on the (wild) character variety of each Painlevé equation is rational and explicitely computable.

Funded under FCT project UIDB/MAT/04459/2020.