Iterated integrals III

The path integral of a closed differential form on a manifold only sees the abelianization of the fundamental group. At the end of the 60s, Chen made the remarkable observation that higher length iterated integrals can detect elements which are trivial in homology. More precisely, they describe the pro-unipotent completion of the fundamental group. This has a number of interesting consequences –including the fact that this completion carries a mixed Hodge structure– and is a key ingredient in the study of multiple zeta values. In this series of lectures, I will present an overview of the theory. An important topic will be a theorem of Beilinson which expresses each step in the pro-unipotent completion as a relative homology group. If time permits, I will discuss some new developments at the end.