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Probability amplitudes in quantum field theory have a rich mathematical structure of infrared singularities.
In the singular limits, amplitudes and loop-integrals simplify. After integration and collinear renormalisation, all singularities cancel for infrared-safe physical cross-sections. However, the presence of singularities renders the integrations in complete amplitudes cumbersome. Notwithstanding the excellent progress which has been achieved in the evaluation of multi-loop amplitudes, a more direct approach is still desired for their evaluation. In this talk, I will present a simple method for the removal of all (overlapping) divergences in momentum space and demonstrate how it works in realistic examples of two-loop integrals of scalar field theories. I will then discuss the generalisation of this approach to gauge theory amplitudes. Finally, I will discuss the prospects for rendering this approach an efficient method for the numerical evaluation of two-loop and higher order amplitudes by direct integration in momentum-space. As a spin-off of this method, I will also discuss the potential of the method to extract analytically the asymptotic behaviour of amplitudes which depend on a small mass.