Topology and motives associated to moduli spaces of curves

Duke University, Durham NC

Investigators

Abstract

The main focus of this project is Hodge and Galois structures on completions of mapping class groups and their relationship to mixed motives over moduli spaces of curves. One case of particular interest is in genus 1, where the PI is studying mixed motives associated to elliptic curves. With Makoto Matsumoto (Hiroshima University) the PI is studying the action of the absolute Galois group on the Malcev completion of the fundamental group of a once-punctured elliptic curve and also on the relative completion of the corresponding mapping class group. With Gregory Pearlstein (Michigan State) the PI is studying variations of mixed Hodge structure over moduli spaces of elliptic curves and their relationship to iterated integrals of modular forms recently defined by Yuri Manin. The Galois and Hodge structures are parallel and the PI expects each approach to illuminate the other. In other projects, the PI and his collaborators are developing general machinery needed to study the case of elliptic motives. On the Galois side, Matsumoto and the PI are proving basic results about Galois actions on completions of arithmetic mapping class groups. On the Hodge side, Pearlstein, Terasoma and the PI are developing fundamental mathematical tools for studying general classes of variations of mixed Hodge structure over complex algebraic varieties. Motives encode deep connections between the theory of whole numbers, integrals of certain algebraic functions and topology. Each of these theories has its own set of symmetries, and all are related through the theory of motives. The PI, together with his collaborators and students, are investigating the interactions of these symmetry groups that are associated to ``elliptic curves'', which are curves defined by cubic polynomials. Although this work is foundational, it has potential applications to cryptography and pseudo random number generation. Indeed, the PI's principal collaborator, Matsumoto (Hiroshima University), is an established expert in these subjects.