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Équations différentielles motiviques et au–delà

# Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids

IHP
11, rue Pierre et Marie Curie

I'll start by recalling that the sixth Painlevé equation is the simplest non-abelian Gauss--Manin
connection, and how this motivates the notion of {\em wild Riemann surface}, in order to
explain'' the other Painlevé equations (and their higher dimensional friends) in a similar
fashion. Then I'll describe recent results (joint with J. Doucot and G. Rembado) studying the local
wild mapping class groups in the twisted setting for arbitrary formal structure in type A. In
concrete terms we study the spaces of admissible deformation parameters (times'') for the
irregular isomonodromy connections, and the braid groups that occur as their fundamental groups. In
simple examples we obtain the braid groups of the complex reflection groups known as the
generalised symmetric groups, showing how they appear naturally in 2d gauge theory. This study also
enables us to define skeleta classifying deformation classes of wild Riemann surfaces and to write
down the dimensions of the (global) moduli spaces of rank n, trace-free wild Riemann surfaces for
any n, a generalisation of Riemann's count''.