Moduli spaces of wild connections: deformations & quantisations

Moduli spaces of meromorphic connections on (principal bundles over) Riemann surfaces have a rich geometric structure. In the logarithmic case, they encompass the complex character varieties of pointed Riemann surfaces, which assemble into flat Poisson/symplectic fibre bundles upon deforming the surface; after geometric/deformation quantisation, this yields (projectively) flat vector bundles over the base space of deformations, famously including the Knizhnik--Zamolodchikov connection from 2d conformal field theory.

In this talk we will aim at a review of part of this story, and then present recent work about extensions involving deformations & quantisations of moduli spaces of irregular singular (`wild') meromorphic connections. These are joint work with (P. Boalch, J. Douçot, M. Tamiozzo) & (G. Felder, R. Wentworth).