Alexander Betts

Iterated integrals, Green's functions, and fundamental groups

Chen's theory of iterated integration provides a way of making sense of, for example, double-integrals along paths in smooth manifolds, and it turns out that many functions of number-theoretic interest -- theta functions, Arakelov--Green functions, admissible metrics -- admit descriptions as double-integrals in a natural way. In this talk, I will survey some of these descriptions, before showing how to reinterpret these results in the language of Hodge theory and Hain's higher Albanese maps. Time permitting, I may also explain how this Hodge-theoretic description of admissible metrics fits into the wider context of non-abelian Kummer maps for realisations of motivic fundamental groups, and how this provides an analogue over non-archimedean fields.