Graphs, curves, and their moduli spaces (Part 1 of 2)

I will give a gentle introduction to the moduli space of graphs and its fine moduli space cousin known as Outer Space. This moduli space of graphs has many applications to various branches of mathematical physics, algebraic geometry, and geometric group theory. It is a natural object to consider while studying Feynman amplitudes in parametric space, and it can be seen as the configuration space of one-dimensional quantum gravity. I will explain how this moduli space of graphs recently became the largest provider of information on the homology of the moduli space of curves of genus g and how associated graph complexes can be used to shed light on the 'dark-matter problems' of these moduli space's cohomology.