Dynamics on Wild Character Varieties

TL;DR

This paper explores the natural dynamics on wild character varieties, illustrating the concept through the study of Painlevé VI and V differential equations, contributing to the understanding of complex geometric structures.

Contribution

It introduces a general research program on dynamics in wild character varieties and applies it specifically to Painlevé equations P_VI and P_V.

Findings

Characterizes dynamics on wild character varieties

Links dynamics to Painlevé differential equations

Provides new insights into geometric structures

Abstract

In the present paper, we will first present briefly a general research program about the study of the "natural dynamics" on character varieties and wild character varieties. Afterwards, we will illustrate this program in the context of the Painlev\'e differential equations $P_{\rm VI}$ and $P_{\rm V}$.