This talk is dedicated to spaces of mathematical functions that close under integration and greatly expanded our computational reach for scattering amplitudes. For instance, multiple polylogarithms complete rational functions under taking primitives and systematize integration on the Riemann sphere. At genus one, integration on a tous became algorithmic through the advent of elliptic polylogarithm. Both types of polylogarithms capture large classes of Feynman integrals relevant to precision computations in particle physics and gravity as well as leading orders in string perturbation theory.
The main results of this talk are explicit constructions of polylogarithms on higher-genus Riemann surfaces inspired by string-theory techniques. The underlying integration kernels that take the role of rational functions on the sphere admit both a non-meromorphic but single-valued description
and a meromorphic but multi-valued alternative. I will conclude with comments on applications to string amplitudes and Feynman integrals.
