The Riemann-Hilbert Correspondence for Rank 2 Meromorphic Connections on Curves

The goal of this text is to introduce the Riemann-Hilbert correspondence in the rank two case and irregular setting over \(\mathbb C\) . This establishes a one-to-one correspondence between isomorphism classes of rank two vector bundles with meromorphic linear connections on curves, and some representation of fundamental groups or groupoids on the corresponding Riemann surface with punctures, up to some natural equivalence. The goal of this text is to provide a self-contained approach with proofs accessible for a master student, in particular including the ramified case. We omit a great part of the huge literature on this beautiful subject.