Kontsevich introduced the graph complex $GC_2$ in 1993 and raised the problem of determining its cohomology. This problem is of renewed importance following the recent work of Chan-Galatius-Payne, who related it to the cohomology of the moduli spaces $M_g$ of curves of genus $g$. It is known by Willwacher that the cohomology of $GC_2$ in degree zero is isomorphic to the Grothendieck-Teichmuller Lie algebra $grt$, but in higher degrees, there are infinitely many classes which are mysterious and have no such interpretation.
In this talk, I will define algebraic differential forms on a moduli space of graphs (outer space). Such a form is a map which assigns to every graph an algebraic differential form of fixed degree, satisfying some compatibilities. Using the tropical Torelli map, I will construct an infinite family of such differential forms, which can in turn be integrated over cells. Surprisingly, these integrals are always finite, and therefore one can assign numbers to homology classes in the graph complex. They turn out to be Feynman periods in phi^4 theory, and can be used to detect graph homology classes.
The upshot of all this is a new connection between graph cohomology, Feynman integrals and motivic Galois groups. I will conclude with a conjectural explanation for the higher degree classes in graph cohomology.