Local wild mapping class groups

Connection link: https://hu-berlin.zoom.us/j/61686623112
Contact: Gaetan Borot (gaetan.borot@hu-berlin.de)

The standard mapping class groups can be constructed as fundamental groups of moduli spaces/stacks of genus-g Riemann surfaces: they thus encode much information about the topology of the deformations of such surfaces. This can be extended to pointed Riemann surfaces, adding the braid groups into the picture (related to the fundamental group of the configuration space of the marked points). Recently this story has been extended to wild Riemann surfaces, which generalise pointed Riemann surface by adding local moduli at each marked point --- the irregular classes. The new parameters control the polar parts of meromorphic connections with arbitrary singularities, defined on principal bundles over Riemann surfaces, and importantly provide an intrinsic viewpoint on the times of isomonodromic deformations. In this talk we will explain how to compute the fundamental groups of spaces of deformations of irregular classes, related to cabled versions of braid groups, which thus play the role of `wild' mapping class groups. This is joint work with P. Boalch, J. Douçot and M. Tamiozzo.