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.

几何广泛关注于研究流形（多维形状）及其内在属性，例如体积、曲率和流形上两点之间的曲线长度。在这一领域，理解给定流形的对称性在研究其其他几何性质中起着关键作用。这些对称性被编码在称为基本群的代数结构中。该项目旨在研究该群与几何之间的联系。具体来说，在双曲流形中，有一个称为“算术”的特殊类别，它往往是最对称的，并且其基本群与数论有很强的联系。该项目旨在利用几何和动力学方面的新技术来研究双曲流形的基本群，试图了解这样的群何时是算术性的，以及算术性（或缺乏算术性）对相关流形几何的影响。该项目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——更好地理解双曲流形的分类及其测地几何，并建立一个强大的框架来探索基本群的刚性现象有限体积实数、复数、四元数和凯莱双曲流形。 首席研究员最近取得了一系列进展，促进了几何、群论和动力学技术的发展，以理解通过粘合算术流形的子流形构建的流形的测地几何，以及开发网格中的超刚性风格技术。实双曲空间的等距群。该项目计划继续开发这些新技术，着眼于几何应用。具体来说，该项目将解决以下广泛主题：1）了解低维和高维双曲流形及其测地子流形的构造，2）进一步开发 1 阶晶格的超刚性结果的通用框架，以及 3）尝试利用 1 阶刚性方面的最新进展作为一种机制来理解复杂双曲格子的完整性以及四元数和凯莱双曲空间的算术性。该奖项反映了 NSF 的法定使命，并被视为值得通过使用基金会的智力优点和更广泛的影响审查标准进行评估来支持。