CAREER: Moduli of curves via topology, geometry, and arithmetic

在几何学的研究中，人们经常通过同时研究所有这些对象来洞察特定几何对象的本质。 例如，人们可以通过研究机器人的所有此类配置的“空间”来找到工厂中机器人的最佳配置，因为后者的几何形状可以洞察哪些特定配置在给定情况下是有问题的（例如当机器人手臂可能卡住或相互干扰时）。 这种几何结构的“空间”称为模空间。调查员的项目是致力于研究的模数空间的表面，一个关键的特点是工作的相互作用的几何和代数的研究模数。其中一些工作将在研究生的协助下进行。调查员将开发一个研究生课题课程在模空间，以刺激学生之间的互动工作在几何和拓扑学和那些工作在代数。研究员还将举办两年一次的研讨会，旨在促进在模量相关领域工作的研究生和本科生的向上专业发展，并刺激这些学科之间跨越障碍的互动。研究员将研究黎曼曲面的模量。 拟议的研究自然分为三个项目福尔斯。 第一个是有关profinite方面的映射类群体的表面。特别是，调查员将进一步发展profinite Teichmueller理论发起的Boggi，也将继续研究中心化的profinite完成的映射类组。 第二个项目涉及曲面丛的几何及其与拓扑和代数的关系。 特别是，调查员将研究一些几何性质之间的区别分组的映射类组在类比与更经典的工作在克莱因集团，也将继续寻找atoroidal表面束的表面。 第三部分是研究双曲三维流形的形变理论。 特别是，研究人员将继续研究瑟斯顿在他的几何化猜想方法中引入的Teichmuller空间之间的某些解析函数，这些解析函数涉及与使几何化有效相关的问题，以及在研究3-流形的一般变形空间时理解某些病理学。

In the study of geometry, one often gains insight into the nature of a particular geometric object by studying all such objects simultaneously. For instance, one might find the optimal configuration of robots in a factory by studying the "space" of all such configurations of robots, as the geometry of the latter can lend insight as to which particular configurations are problematic in a given situation (such as when robotic arms might jam or interfere with each other). Such "spaces" of geometric structures are called moduli spaces. The investigator's project is dedicated to a study of moduli spaces of surfaces, a key feature of the work being the interaction of geometry and algebra in the study of moduli. Some of this work will be conducted with the assistance of graduate students. The investigator will develop a graduate topics course in moduli spaces to stimulate interaction between students working in geometry and topology and those working in algebra. The investigator will also run biennial workshops designed to foster the upward professional development of graduate and undergraduate students working in fields related to moduli as well as to stimulate interaction across barriers between these disciplines.The investigator will study the moduli of Riemann surfaces. The proposed research naturally falls into three projects. The first is concerned with profinite aspects of mapping class groups of surfaces. In particular, the investigator will further develop profinite Teichmueller theory as initiated by Boggi, and will also pursue a study of centralizers in profinite completions of mapping class groups. The second project concerns the geometry of surface bundles and their relation to topology and algebra. In particular, the investigator will study distinctions between a number of geometric properties of subgroups of mapping class groups in analogy with more classical work in Kleinian groups, and will also continue a search for atoroidal surfaces bundles over surfaces. The third portion of the proposed work is dedicated to a study of the deformation theory of hyperbolic 3-manifolds. In particular, the investigator will continue study of certain analytic functions between Teichmuller spaces introduced by Thurston in his approach to his Geometrization conjecture, which come to bear on problems related to making Geometrization effective, as well as toward understanding certain pathology in the study of general deformation spaces of 3-manifolds.