Hodge Theory, Galois Theory and the Topology of Moduli Spaces

本课题的目标是应用Hodge理论、伽罗瓦理论和表示理论的方法研究曲线和阿贝变的模空间的几何和拓扑，并利用几何和拓扑通过绝对伽罗瓦群（即代数数的伽罗瓦群）对映射类群补全的作用来研究伽罗瓦群。具体而言，Hain有三个主要项目：(1)解决了在研究超椭圆曲线基本群上的伽罗瓦作用时出现的超椭圆曲线模空间拓扑中的一些基本问题；(2)解决了辛几何和物理中稳定n点曲线的delign - mumford模空间上的泛雅可比矩阵的交点理论中的若干问题；(3)研究绝对伽罗瓦群的适当补全对数域上曲线基本群的亲幂偶和亲良补全的作用。第三个问题是与广岛大学的松本诚（Makoto Matsumoto）合作项目的一部分，该项目的目标是确定这种行为是否忠实，这是动机理论中的一个基本问题。映射类组及其上同调在每个项目中都起着中心作用。拓扑学是研究在拉伸（撕裂除外）和其他连续变形下保持不变的表面的几何性质及其概化。几何是研究表面的那些性质及其保持几何性质（如距离和/或角度）的概括。一个曲面的拓扑对称性（称为曲面的映射类群）、在这样一个曲面上测量角度的所有不同方法的几何形状（曲面上共形结构的模空间）和当被视为多项式函数的图时，曲面的算术性质之间存在着深刻的联系。关于曲面上共形结构的映射类群和模空间的问题出现在许多数学领域（如数学和代数几何的研究），并通过弦理论和共形场论应用于粒子物理。密码学也有潜在的重要应用。本提案的目标是进一步探索和理解曲面理论的拓扑、几何和算术方面，特别是与数论有关的那些方面之间复杂而深刻的联系。

DMS-0405440Richard M. HainThe 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.