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Soon after Mandelstam proposed the double spectral representation of the collision amplitudes, there was for a few years a concerted effort to understand the analytic structure of such amplitudes in the perturbation theory. The aim was firstly to find all the singularities of a scattering matrix element for a given Feynman diagram of a definite order, and then to generalize the investigation to arbitrary orders with the hope of proving the Mandelstam representation in the perturbation theory. The first step was accomplished with relative ease for some simple diagrams. It has yielded a great deal of information on the analytic properties of the associated Feynman integrals. However, the extension of the consideration to more complicated diagrams, let alone general amplitudes of arbitrary order, has met with overwhelming difficulty. It has subsequently become clear that the mathematical technique used (that is, the tracing of singularities) is totally inadequate for an analysis of the general problem. Even for specific and simple integrals the description of the structure of the Riemann surface in the external variables tk is so cumbersome that attempts to communicate the results would invariably lead to confusion and despair. There was the need of a new and more systematic method.
Recently, Fotiadi, Froissart, Lascoux, and Pham proposed that homological methods be used in the study of the analytic structure of Feynman integrals. By formulating the problem in a way that is suitable for the application of some important theorems in algebraic topology, the geometric problem of determining the structure of the Riemann surface associated with a Feynman integral is then converted into an algebraic one. The description of the surface is given in terms of groups and the results can be conveyed in much simpler forms. The homological approach not only provides a systematic language in terms of which the classical results can be described in a highly economical and comprehensive way, but also it seems to be the only feasible approach to the study of the general amplitude