In 1920, Haupt classified the homomorphisms from the first homology group of a closed surface to C that arise as the period map of a holomorphic differential for some complex structure on the surface. In this talk, we describe the analogue of Haupt’s theorem for holomorphic differentials on punctured surfaces which extend to meromorphic differentials on the closed surface. We shall see how holomorphic differentials give rise to a translation structure on the punctured surface and describe constructions using translation structures that can be used to prove the analogue of Haupt’s theorem. We will also describe the image of the period map when restricted to any stratum of meromorphic differentials. This is joint work with Gianluca Faraco and Subhojoy Gupta. Speaker(s): Shabarish Chenakkod (University of Michigan)