Hyperbolic Manifolds and Their Moduli Spaces

双曲流形及其模空间

• 批准号: 1904130
• 负责人: AUTUMN KENT
• 金额: $ 33.2万
• 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: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2022-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1904130&HistoricalAwards=false
• 关键词: Hyperbolic; Manifolds; Their; Moduli; Spaces

In mathematics one is often faced with the problem of understanding a given geometric object. Surprisingly often, it is advantageous to deform the geometry of this object in order to better understand it. That is, one may understand something geometric by understanding how one may change it. The prototypical example of this is Riemann's moduli space of curves, which is a space each of whose points represents a two-dimensional object called a Riemann surface. This space captures all of the ways one may deform these surfaces. The proposed research will explore the geometry of this space, its analogs in spaces of deformations of certain three-dimensional spaces called hyperbolic manifolds, and their interrelationships. It also contains suitable sub-projects for graduate students.The project falls into two parts: the deformation theory of hyperbolic three-manifolds; and the study of geometric and algebraic properties of subgroups of mapping class groups of surfaces, their associated surface bundles, and profinite completions. The first is principally concerned with understanding a certain function called the skinning map, which was discovered and studied by Thurston in his work on the geometrization of three-manifolds, and which measures the effects of deformations of hyperbolic three-manifolds. A better understanding of this function sheds light on the geometry of three-manifolds and its relation to topology. In particular, the PI will continue an ongoing project with collaborators Bromberg and Minsky to establish effective bounds on the diameter of this map and its relation to the notion of renormalized volume. The second part is concerned with the study of the geometry of subgroups of mapping class groups and their relation to the geometry of four-manifolds in collaboration with Leininger. The second part will also continue the PI's work in profinite aspects of mapping class groups, with a view toward Ivanov's congruence subgroup problem.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在数学中，人们经常面临理解给定几何对象的问题。 令人惊讶的是，为了更好地理解它，使该对象的几何形状变形是有利的。 也就是说，人们可以通过理解如何改变几何学来理解它。 典型的例子是黎曼曲线模空间，该空间的每个点都代表一个称为黎曼曲面的二维对象。 这个空间捕捉了人们可能使这些表面变形的所有方式。 拟议的研究将探索该空间的几何形状、其在某些三维空间（称为双曲流形）的变形空间中的类似物以及它们的相互关系。它还包含适合研究生的子项目。该项目分为两部分：双曲三流形的变形理论；以及曲面映射类群的子群的几何和代数性质、其相关的曲面丛和有限完成的研究。第一个主要涉及理解称为蒙皮图的特定函数，该函数是瑟斯顿在其三流形几何化工作中发现和研究的，用于测量双曲三流形变形的影响。 更好地理解该函数有助于了解三流形的几何结构及其与拓扑的关系。 特别是，PI 将继续与合作者 Bromberg 和 Minsky 进行一个正在进行的项目，以建立该地图直径的有效界限及其与重整化体积概念的关系。 第二部分是与 Leininger 合作研究映射类群的子群的几何形状及其与四流形几何的关系。 第二部分还将继续 PI 在映射类组的具体方面的工作，着眼于伊万诺夫的同余子组问题。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查进行评估，被认为值得支持标准。