On the residue map

Consider the space of meromorphic differentials on Riemann surfaces with given orders at the zeroes and the poles. From this space, there is a residue map which associates to a differential the residues at its poles. The structure of this map has interesting consequences for exemple in mathematical physics and algebraic geometry. In this talk, I want to present the few results that we know about this map.
The talk is based on joint works with Guillaume Tahar and an ongoing paper with him, Dawei Chen and Miguel Prado.