Hodge Theory, Galois Theory and the Topology of Moduli Spaces

霍奇理论、伽罗瓦理论和模空间拓扑

• 批准号: 0405440
• 负责人: Richard Hain
• 金额: --
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405440&HistoricalAwards=false
• 关键词: Hodge; Theory; Galois; Topology; Moduli

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.

DMS-0405440Richard M.Hain本项目的目标是应用Hodge理论、Galois理论和表示论的方法来研究曲线和交换簇的模空间的几何和拓扑，并利用几何和拓扑通过绝对Galois群(即代数数的Galois群)在映射类群的完备化上的作用来研究绝对Galois群。具体地说，Hain有三个主要项目：(1)解决超椭圆曲线的模空间的拓扑中的某些基本问题，这些问题是在研究超椭圆曲线的基本群上的Galois作用时出现的；(2)解决稳定的n点曲线的Deligne-Mumford模空间上的泛雅可比交理论中的某些问题，这些问题出现在辛几何和物理学中；(3)研究绝对Galois群的适当完备化在数域上定义的基本曲线群的PRO-UNPOLY和PROCEL完备化上的作用。第三个问题是与广岛大学松本诚的合作项目的一部分，该项目的目标是确定这一行为是否忠诚，这是动机理论中的一个基本问题。映射类群及其上同调在每个项目中都起着核心作用。拓扑学是研究曲面的几何性质及其在拉伸(简称撕裂)和其他连续变形下保持不变的推广。几何学是研究曲面的那些属性及其保持几何属性(如距离和/或角度)的推广。曲面的拓扑对称性(称为曲面的映射类群)、曲面上所有不同测角方法的几何(曲面上保角结构的模空间)以及曲面作为多项式函数的图形时的算术性质之间有着深刻的联系。曲面上共形结构的类群和模空间的映射问题出现在许多数学领域(如数论和代数几何)，并通过弦理论和共形场理论在粒子物理中得到应用。密码学也有潜在的重要应用。这项提议的目的是进一步探索和理解曲面理论的这些拓扑、几何和算术方面之间的复杂而深刻的联系，特别是那些与数论有关的方面。