Separation coordinates, moduli spaces and Stasheff polytopes

Abstract:

We establish a surprising link between two a priori completely unrelated objects:  The space of isometry classes of separation coordinates on an n-dimensional sphere one one hand and the Deligne-Mumford moduli space M_{0,n+2} of stable algebraic curves of genus zero with n+2 marked points on the other hand.  We use the rich combinatorial structure of M_{0,n+2} and the closely related Stasheff polytopes in order to classify the different canonical forms of separation coordinates. Moreover, we infer an explicit construction for separation coordinates and the corresponding quadratic integrals from the mosaic operad on M_{0,n+2}.  The talk ends with an outlook on corresponding results for hyperbolic space.