Student Arithmetic Geometry Seminar: Iterated Integrals and Unipotent Connections

Seminar | August 30 | 4:10-5 p.m. | 891 Evans Hall

 David Corwin, UC Berkeley

 Department of Mathematics

Line integrals of differential one-forms define functions on the path space, and these functions are homotopy invariant if the form is closed. They thus define functions on the fundamental group, which in fact factor through its abelianization, the first homology. Iterated integrals are a way of defining integrals that detect non-abelian quotients of the fundamental group. We will explain how these work and then explain how they may be understood in terms of unipotent objects in the category of vector bundles with connection. We will discuss some specific examples on algebraic curves and explain Deligne's canonical extension of unipotent connections in this context. In particular, we can see how Deligne's canonical extension helps one explicitly understand the the Hodge filtration for the first de Rham cohomology of a curve.