Hyperbolic Manifolds, Geodesic Submanifolds, and Rigidity for Rank-1 Lattices

双曲流形、测地线子流形和 1 阶晶格的刚度

• 批准号: 2300370
• 负责人: Nicholas Miller
• 金额: $ 14.64万
• 依托单位: University of Oklahoma Norman Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-11-15 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2300370&HistoricalAwards=false
• 关键词: Hyperbolic; Manifolds; Geodesic; Submanifolds; Rigidity

Geometry is broadly focused on studying manifolds (multi-dimensional shapes) and their intrinsic properties, such as volume, curvature, and lengths of curves between two points on the manifold. In this field, understanding symmetries of a given manifold plays a key role in studying its other geometric properties. These symmetries are encoded in an algebraic construction called the fundamental group; this project aims at studying the connections between this group and geometry. Specifically, among hyperbolic manifolds there is a special class called "arithmetic" that tend to be the most symmetric and whose fundamental group has strong connections to number theory. This project aims to use new techniques in geometry and dynamics to study the fundamental group of hyperbolic manifolds in an attempt to understand when such a group is arithmetic and the ramifications of arithmeticity (or lack thereof) on the geometry of the associated manifold. Broader impacts of this project include work with undergraduates.More specifically, the overarching goal of this research project is twofold -- to better understand the classification of hyperbolic manifolds and their geodesic geometry and to build a robust framework for exploring rigidity phenomenon for fundamental groups of finite-volume real, complex, quaternionic, and Cayley hyperbolic manifolds. The principal investigator has recently made a series of advances that facilitate the development of geometric, group theoretic, and dynamical techniques for understanding the geodesic geometry of manifolds built by gluing submanifolds of arithmetic manifolds, as well as the development of superrigidity style techniques for lattices in the isometry group of real hyperbolic space. This project plans to continue to develop these new techniques with an eye toward geometric applications. Specifically, the project will address the following broad themes: 1) understanding constructions of both low- and high-dimensional hyperbolic manifolds and their geodesic submanifolds, 2) further developing a general framework for superrigidity results for rank-1 lattices, and 3) attempting to use recent advances in rank-1 rigidity as a mechanism to understand integrality of complex hyperbolic lattices and arithmeticity of quaternionic and Cayley hyperbolic spaces.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.

几何形状广泛地集中在研究歧管（多维形状）及其内在特性，例如歧管上两个点之间的曲线的体积，曲率和曲线长度。在这个领域，理解给定流形的对称性在研究其其他几何特性中起着关键作用。这些对称性是在称为基本组的代数结构中编码的；该项目旨在研究该组与几何形状之间的联系。具体而言，在双曲线歧管中，有一个称为“算术”的特殊类别，倾向于最对称，其基本群体与数字理论具有牢固的联系。该项目旨在使用几何技术和动力学中的新技术来研究双曲线歧管的基本组，以便了解何时在相关歧管的几何学上，算术（或缺乏）算术的后果（或缺乏）。该项目的更广泛的影响包括与本科生的合作。更具体地说，该研究项目的总体目标是双重的 - 更好地了解双曲线歧管及其地质几何形状的分类，并建立一个强大的框架，以探索有限，复杂，复杂，复杂，Quaternionic和Cayley的基本群体的刚性现象，以探索刚性的僵化框架。 首席研究者最近取得了一系列进步，以促进几何，群体理论和动力学技术的发展，以理解歧管歧管的粘贴子序列的歧管的地理几何形状，以及在实际超质量较高级超过的超级量化群中的超级繁殖风格技术的开发。该项目计划继续开发这些新技术，以注重几何应用。具体而言，该项目将解决以下广泛的主题：1）了解低维和高维超歧管的结构及其地球次符号，2）进一步为等级-1晶格的超级疏松结果开发了一个一般框架，以及3）试图在等级1中使用Quarties grountity and quarties grountity Queltition y Ingroultity Qualtery y Ingroltity Quystry Explybolic Expyerbolic Lattices and Ariith aircolic Latterics的机制，该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准评估值得支持。