The Topology, Geometry and Arithmetic of Moduli Spaces of Curves

DMS-0103667 Richard M.Hain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到理解整数谱上的动机的问题上。主要研究者希望计算Torelli群(实数张量)的下中心级数的分次商的稳定的最高权分解作为g阶实辛群上的模。这对研究3-流形不变量的人应该是感兴趣的。首席调查者计划利用他关于这种稳定分解的知识来研究有理数的伽罗瓦群在映射类群的适当完备化上的映象。特别地，他(与Makoto MatSumoto合作)希望能够刻画大亏格映射类群的相对么正完备的外自同构群中Galois群的像的Zariski闭包。这将导致对Hodge理论和Galois理论之间的联系的更好的理解，特别是对混合Zeta数在Galois理论中的作用的更好的理解。首席研究者还计划研究曲线的模空间的伪凸性。Looijenga猜想在亏格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。Hain在与Looijenga的合作中，希望证明他几年前与David Reed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凝聚上同调的新的零化结果，以及Diaz和Heller关于这些模空间的拓扑的几个结果的统一证明。拓扑学是研究曲面的那些几何性质及其在拉伸(短于撕裂)和其他连续变形下保持不变的广义。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)和曲面的算术性质之间有着深刻的联系。关于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。

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.