Canonical coordinates for moduli spaces of rank two irregular connections on curves

I will present our recent joint work with A. Komyo, F. Loray and M-H. Saito. Given a generic meromorphic connection over a vector bundle of rank 2 and degree 2g-1 over a genus g curve, with unramified irregular singularities, we use elementary transformations to bring it to a normal form. This procedure produces a rational map from the moduli space of such connections to a symmetric power of the total space of the bundle of meromorphic 1-forms on the curve twisted by a cohomology class. We show that this map is birational and preserves symplectic structures. Time permitting, we point out on an example that this approach is compatible with confluence of singular points.