Feynman integrals and differential forms on the moduli space of tropical curves

Moduli spaces of graphs/tropical curves appear in various areas of maths and physics, for instance in geometric group theory, algebraic geometry/topology, and perturbative quantum field theory. They provide nice venues for combinatorics, algebra, geometry and topology to interact in interesting and fruitful ways. Quite recently, people have started to think about differential forms and integration on these spaces (they are far from being smooth manifolds). In this talk I will focus on two kinds of forms/integrals, Feynman integrals and "canonical integrals of invariant forms" which were recently constructed by Francis Brown. The former case can be thought of as a "1960's version" of the amplituhedron. The latter case provides a de Rham theory for Kontsevich's commutative graph complex. This relies on the fact that these moduli spaces are geometric models for various graph complexes (which in turn relate to various invariants in low-dimensional topology and group theory). I will explain this geometric viewpoint and introduce some interesting subspaces. One particular example is the "spine" of the moduli space of graphs/tropical curves which is a rational classifying space for Out(F_n), the outer automorphism group of a free group. Here it is then natural to ask if there also exist invariant forms on such subspaces and how to find/construct them. I will discuss a (physics-inspired) geometric approach to this problem and explain how this may shed new light on the overall structure of the homology of Out(F_n).