Bubbles: symmetries, deformations and an application to monodromy of ODEs.

Complex projective structures with branch points arise naturally in the study of a certain class of linear rank 2 ODEs on Riemann surfaces. We focus on the genus 2 case, where these structures can be conveniently described in terms of bubbling, i.e. connected sum with the Riemann sphere. This geometric point of view allows for an effective description of the symmetries and deformations of these geometric structures. We will present an application to the study of the Riemann-Hilbert map for the aforementioned class of ODEs.