Combinatorial iterated integrals and the unipotent Torelli theorem for graphs

The classical Torelli theorem says that a Riemann surface can be recovered from its Jacobian, which is a principally polarized Abelian variety. There is an analogous theorem for graphs, due to Artamkin and Caporaso–Viviani that the 2-isomorphism class of a graph can be recovered from its cycle space, equipped with its cycle pairing. We ask what happens when one considers mildly non-abelian data as in the unipotent Torelli theorem for Riemann surfaces due to Hain and Pulte. This leads us to introducing the analogue of iterated integrals on graphs and encoding them in a particular structure. This structure turns out to recover pointed bridgeless graphs up to isomorphism. We discuss some of the application of this result and connections to Hodge theory, tropical geometry, and number theory. This is joint work with Raymond Cheng.