A basic property of a meromorphic connection is that its solutions (horizontal sections) can be infinitely analytically continued outside of its singularities. This gives rise to a monodromy representation of the fundamental group of the punctured Riemann surface. The moduli space of connections with at most simple fixed poles (Fuchsian singularities) can be locally identified with the space of monodromy representations: this the Riemann--Hilbert correspondence. However, for connections with higher order poles (irregular singularities), the monodromy alone does not contain enough information. One has to consider additional local invariants of the singularities, called Stokes data. These no longer have a topological meaning but instead they have to do with existence of natural filtrations on the solution space by the growth rate on certain sectors at the singularities and with their mismatch on different sectors: this is known as the Stokes phenomenon.
In the talk I'll try to give a gentle introduction to this classic topic by focusing on connections on rank 2 vector bundles. In this case the underlying geometry is related to a horizontal foliation of a meromorphic quadratic differential: the determinant of the connection matrix. If time permits I'll talk about confluences of singularities and how Stokes data arise out of confluence of monodromy.
