Appears in collection : Algebraic Structures in Perturbative Quantum Field Theory: a conference in honour of Dirk Kreimer's 60th birthday

I will report on joint work with Dirk on his vision of exploring quantum fields in outer space. Our expeditions have so far uncovered an exciting wonderland of algebraic, geometric, and topological relations in a magical galaxy, populated by Feynman graphs. Its inhabitants seem to play a mysterious game of hide-and-seek, hopfing around singularities of various kinds, all ruled by the mighty King Cutkosky.