Title: Extended modularity and deformation of Riemann surfaces Date: 26/02/2021 at 10:30 Abstract: I will discuss modular-type functions arising from the classical theory of uniformizing differential equations (deformation of accessory parameters) and the deformation of Riemann surfaces. While the holomorphic solutions of the uniformizing differential equation are modular forms, these new functions can be described in terms of iterated integrals of cusp forms, quasimodular forms, and Rankin-Cohen brackets. Moreover, they appear as components of vector-valued modular forms associated to extensions of symmetric tensor representations of Fuchsian groups.