The perspectives: applications and generalisations

Abstract

We have proposed a new, purely algebraic geometric approach to the problem of separation of variables and we have demonstrated that this approach is viable by successfully carrying it out for the simplest non-trivial family of examples – that of spheres. In particular, we elucidated the natural algebro-geometric structure of the parameter space classifying equivalence classes of separation coordinates, which for a long time had only been known as a mere set, and gave a precise description of its topology. In this way we discovered that the theory of Deligne-Mumford-Knudsen moduli spaces and Stasheff polytopes provides the right framework for the classification and construction of all orthogonal separation coordinates on spheres.