The standard mapping class groups are fundamental groups of moduli spaces/stacks of pointed Riemann surfaces. The monodromy properties of
a large family of nonlinear differential equations, the tame isomonodromy connections, are encoded as the action of the mapping class group on the character varieties of the surface. Recently this story has been extended to wild Riemann surfaces, which generalise pointed Riemann surfaces by adding local moduli at each marked point: the irregular classes. These new parameters control the polar parts of meromorphic connections with wild/irregular singularities, defined on principal bundles, and importantly provide an intrinsic viewpoint on the `times' of irregular isomonodromic deformations. The monodromy properties of the wild/irregular isomonodromy connections are then encoded as the action of the resulting wild mapping class group on the wild character varieties of the surface.
In this talk we will explain how to compute the fundamental groups of (universal) spaces of deformations of irregular classes, which bring about cabled versions of (generalised) braid groups. The case of generic meromorphic connections has been understood for some time (and known to underlie the Lusztig symmetries of the quantum group since 2002) so the focus will be the new features such as cabling that occur on the general setting. This is joint work with P. Boalch, J. Douçot and M. Tamiozzo (arXiv:2204.08188, 2208.02575, 2209.12695).
If time allows we will sketch a relation with bundles of irregular conformal blocks in the Wess--Zumino--Witten model, in joint work with G. Felder (arXiv:2012.14793) and G. Baverez (in progress).