Complex projective structures on surfaces

Abstract: Roughly speaking, a complex projective structure ona surface is way of locally identifying the surface with the complex plane in such a way that the transition maps are Mobius transformations. Despite their geometric description, these types of structures have a long history related to solutions to certain second order differential equations going back to the work of Poincare in the early 1900s. In this setting, the solution to the differential equation allows one to construct a complex projective structure. When the surface is closed the space of all such complex projective structures is well studied and understood, however, the non-compact case is still poorly understood. In this talk I give an overview of complex projective structures on surfaces, survey what is known in the closed setting, and describe some recent work (joint with P. Bowers, A. Casella, and L. Ruffoni) where we provide a complete geometric description of a certain slice of the space of projective structures on the thrice punctured sphere.