Algebra, Geometry and Physics: From Feynman graphs to moduli spaces

We present a program which is designed to take combinatorial categorical data and produce geometric spaces
and classify universal algebraic operations. Specifically when applied to categories of graphs as founded by Borisov-Manin thought of as Feynman graphs, this program on one hand produces cubical and simplicial complexes corresponding to moduli spaces of Riemann surfaces and their Kontsevich-Penner compactifications.
On the other hand, algebraically, it gives rise to the notions of modular operads as put forth by Getzler and Kapranov, as well as operations related to 2d open and closed topological field theories and Chas-Sullivan string topology.