The Topology, Geometry and Arithmetic of Moduli Spaces of Curves

曲线模空间的拓扑、几何与算术

• 批准号: 0103667
• 负责人: Richard Hain
• 金额: $ 10.41万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103667&HistoricalAwards=false
• 关键词: Topology; Geometry; Arithmetic; Moduli; Spaces

DMS-0103667Richard M. HainThe 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 thisstable 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 vanishingresults 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 geometricproperties 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 classgroups 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-0103667 Richard M.这个项目的目标是更好地理解映射类组的结构，然后将这些知识应用于理解整数谱上的动机的问题。主要研究者希望计算Torelli群的较低中心序列的分次子的稳定最高权重分解（用实数张量）作为秩为g的真实的辛群上的模。这应该对那些研究3-流形不变量的人感兴趣。主要研究者计划利用他的知识，这种稳定的分解，研究图像的伽罗瓦集团的有理数适当完成的映射类组。特别是，他（在联合工作与诚松本）希望能够刻画Zeromki封闭的形象伽罗瓦群在组外自同构的相对幂单完成映射类群体的大属。这将有助于加深对霍奇理论和伽罗瓦理论之间联系的理解;特别是，有助于加深对伽罗瓦理论中混合zeta数的作用的理解。首席研究员还计划研究曲线模空间的伪凸性。Looijenga证明了在亏格g曲线的模空间上存在一个真的、非负的（g-2）-伪凸实值函数。海恩在与Looijenga的联合工作中，希望证明他与大卫里德几年前构建的函数就是这样一个函数。这一结果将导致曲线模空间的相干上同调的新的消失结果，以及迪亚兹和Harer关于这些模空间拓扑的几个结果的统一证明.拓扑是研究曲面的几何性质及其推广，这些性质在拉伸（除了撕裂）和其他连续变形下保持不变.几何学是研究曲面的性质及其保持几何性质（如距离和/或角度）的推广。在曲面的拓扑对称性（称为曲面的映射类群）、在曲面上测量角度的所有不同方法的几何学（曲面上的保形结构的模空间）以及曲面作为多项式图形时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数学的许多领域（如数的研究和代数几何），并通过弦理论和共形场论应用于粒子物理学。密码学也有潜在的重要应用。这个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和深刻的联系，特别是那些与数论有关的方面。