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

职业：通过拓扑、几何和算术计算曲线模

• 批准号: 1350075
• 负责人: AUTUMN KENT
• 金额: $ 45万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1350075&HistoricalAwards=false
• 关键词: CAREER; Moduli; curves; via; topology

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.

在几何研究中，人们通常通过同时研究所有特定几何对象来深入了解特定几何对象的本质。 例如，人们可以通过研究所有此类机器人配置的“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在给定情况下哪些特定配置存在问题（例如当机械臂可能互相卡住或干扰时）。 这种几何结构的“空间”称为模空间。研究人员的项目致力于研究曲面的模空间，该工作的一个关键特征是模量研究中几何和代数的相互作用。其中一些工作将在研究生的协助下进行。研究人员将开发模空间研究生主题课程，以刺激几何和拓扑学生与代数学生之间的互动。研究人员还将举办两年一次的研讨会，旨在促进模相关领域的研究生和本科生的职业向上发展，并刺激这些学科之间跨越障碍的互动。研究人员将研究黎曼曲面的模。 拟议的研究自然分为三个项目。 第一个涉及表面映射类组的具体方面。特别是，研究者将进一步发展由 Boggi 发起的 profinite Teichmueller 理论，并将继续研究映射类群的 profinite 完成中的中心化器。 第二个项目涉及面丛的几何形状及其与拓扑和代数的关系。 特别是，研究者将研究映射类群的子群的许多几何性质之间的区别，与克莱因群中更经典的工作相类比，并且还将继续搜索曲面上的环面束。 该工作的第三部分致力于研究双曲 3 流形的变形理论。 特别是，研究者将继续研究 Thurston 在其几何化猜想方法中引入的 Teichmuller 空间之间的某些分析函数，这些函数将涉及与使几何化有效相关的问题，以及理解一般研究中的某些病理学。 3-流形的变形空间。