  Various methods for the recursive evaluation of scattering amplitudes in
quantum field theory and string theory have been put forward during the last
couple of years. In these proceedings we describe a geometrical framework,
which is believed to be capable of treating many of these recursions in a
unified way. Our recursive framework is based on manipulating iterated
integrals on Riemann surfaces with boundaries. A geometric parameter appears as
variable of a differential equation of KZ or KZB type. The parameter
interpolates between two associated regularized boundary values, which contain
iterated integrals closely related to scattering amplitudes defined on two
different geometries.
