Scattering amplitude of one-loop closed strings

One of the best ways to study a theory of particles (or strings) is to collide the components and measure the probability of different scenarios. The mathematical objects describing these probabilities are scattering amplitudes. In this talk, we use the genus expansion of string theory and study the amplitude of one-loop closed strings. They are integrals over the moduli space of the punctured torus. We see how "Modular Graph Forms" (MGFs) arise from the perturbative solution of the integrals over the configuration space of punctures. The main focus of the talk will be the iterated integral representation of MGFs, their number-theoretical properties, and their integral over the moduli space of the torus.