  In this thesis, we investigate the low-energy expansion of scattering
amplitudes of closed strings at one-loop level (i.e. at genus one) in a
ten-dimensional Minkowski background using a special class of functions called
modular graph forms. These allow for a systematic evaluation of the low-energy
expansion and satisfy many non-trivial algebraic and differential relations. We
study these relations in detail, leading to basis decompositions for a large
number of modular graph forms which greatly reduce the complexity of the
expansions of the integrals appearing in the amplitude. One of the results of
this thesis is a Mathematica package which automatizes these simplifications.
We use these techniques to compute the leading low-energy orders of the
scattering amplitude of four gluons in the heterotic string at one-loop level.
  Furthermore, we study a generating function which conjecturally contains the
torus integrals of all perturbative closed-string theories. We write this
generating function in terms of iterated integrals of holomorphic Eisenstein
series and use this approach to arrive at a more rigorous characterization of
the space of modular graph forms than was possible before.
  For tree-level string amplitudes, the single-valued map of multiple zeta
values maps open-string amplitudes to closed-string amplitudes. The definition
of a suitable one-loop generalization, a so-called elliptic single-valued map,
is an active area of research and we provide a new perspective on this topic
using our generating function of torus integrals.
  The original version of this thesis, as submitted in June 2020 to the
Humboldt University Berlin, is available under the DOI 10.18452/21829. The
present text contains minor updates compared to this version, reflecting
further developments in the literature, in particular concerning the
construction of an elliptic single-valued map.
