The goal of this project is to apply the methods of Hodge theory, Galois theory and representation theory to study the geometry and topology of moduli spaces of curves and abelian varieties, and to use geometry and topology to study the absolute Galois group (i.e., the Galois group of the algebraic numbers) via its action on completions of mapping class groups. Specifically, Hain has three main projects: (1) resolving certain fundamental questions in the topology of moduli spaces of hyperelliptic curves that are arise in the study of Galois actions on fundamental groups of hyperelliptic curves; (2) resolving certain problems in the intersection theory of the universal jacobian over the Deligne-Mumford moduli spaces of stable, n-pointed curves, which arise in symplectic geometry and physics; (3) studying the action of an appropriate completion of the absolute Galois group on pro-unipotent and pro-ell completions of fundamental groups of curves defined over number fields. The third problem is part of a joint project with Makoto Matsumoto of Hiroshima University whose goal is to determine whether this action is faithful, a fundamental question in the theory of motives. Mapping class groups and their cohomology play a central role in each of the projects.
Topology is the study of those geometrical properties of surfaces and their generalizations that remain unchanged under stretching (short of tearing) and other continuous deformations. Geometry is the study of those properties of surfaces and their generalizations that preserve geometric properties such as distances and/or angles. There is a profound connection between the topological symmetries of a surface (called the mapping class group of the surface), the geometry of all of the different ways of measuring angles on such a surface (the moduli space of conformal structures on the surface) and the arithmetical properties of the surface when viewed as the graph of a polynomial function. Questions about mapping class groups and moduli spaces of conformal structures on surfaces arise in many areas of mathematics (such as the study of numbers, and algebraic geometry), and have applications to particle physics through string theory and conformal field theory. There are also potential significant applications to cryptography. The goal of this proposal is to further explore and understand the intricate and deep connections between these topological, geometrical and arithmetical aspects of surface theory, especially those aspects with connections to number theory.
