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A projective structure on a Riemann surface is determined by a holomorphic quadratic differential via the Schwarzian differential equation. The monodromy of this equation, or equivalently the holonomy of the projective structure, defines a representation of the fundamental group of the surface to PSL(2,ℂ), determined up to conjugation. In this talk, we shall describe some recent work concerning the space of projective structures on a punctured Riemann surface, corresponding to meromorphic quadratic differentials. In particular, we shall describe a geometric parametrization of that space which is an analogue of Thurston’s grafting theorem for closed surfaces, and discuss results about the monodromy map to the moduli space of framed PSL(2,ℂ)-representations, which was first defined by Allegretti-Bridgeland. This represents work in several papers, some written in collaboration with Gianluca Faraco, Spandan Ghosh and Mahan Mj. |