Polylogarithms on higher-genus Riemann surfaces

Multiple polylogarithms and their elliptic counterparts provide powerful tools for the algorithmic evaluation of large classes of integrals on the sphere and the torus which greatly expanded our computational reach for Feynman integrals and string amplitudes. This talk is dedicated to generalizations of multiple polylogarithms to higher-genus Riemann surfaces and the associated function spaces that close under integration on the surface. I will describe two closely related constructions of higher-genus polylogarithms, one via meromorphic integration kernels, the other via modular integration kernels, and highlight their parallels with the Kronecker-Eisenstein kernels of the elliptic case. Recent work has produced families of algebraic and differential relations among these higher-genus kernels which take the same form in their meromorphic and modular formulation. I will comment on possible applications of these results.