The Topology, Geometry and Arithmetic of Moduli Spaces of Curves

Duke University, Durham NC

Investigators

Abstract

DMS-0103667 Richard M. Hain The goal of this project is to better understand the structure of mapping class groups and then to apply this knowledge to the problem of understanding motives over the spectrum of the integers. The Principal Investigator hopes to compute the stable highest weight decomposition of the graded quotients of the lower central series of the Torelli groups (tensored with the reals) as modules over the real symplectic group of rank g. This should be of interest to those studying 3-manifold invariants. The Principal Investigator plans to use his knowledge of this stable decomposition to study the image of the Galois group of the rational numbers on appropriate completions of mapping class groups. In particular, he (in joint work with Makoto Matsumoto) hopes to be able to characterize the Zariski closure of the image of the Galois group in the group of outer automorphisms of the relative unipotent completion of mapping class groups of large genus. This should lead to improved understanding of the connections between Hodge Theory and Galois Theory; in particular, to improved understanding of the role of mixed zeta numbers in Galois theory. The Principal Investigator also plans to study the pseudoconvexity of the moduli spaces of curves. Looijenga has conjectured that there is a proper, non-negative, (g-2)-pseudoconvex real-valued function defined on the moduli space of genus g curves. Hain, in joint work with Looijenga, hopes to prove that the function that he constructed with David Reed several years ago is such a function. This result would lead to new vanishing results for coherent cohomology of moduli spaces of curves as well as unified proofs of several results of Diaz and Harer on the topology of these moduli spaces. 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. 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.