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
基本信息
- 批准号:2300370
- 负责人:
- 金额:$ 14.64万
- 依托单位:
- 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:
- 关键词:
项目摘要
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.
几何主要研究流形(多维形状)及其固有特性,如体积、曲率和流形上两点之间曲线的长度。在这一领域中,理解给定流形的对称性对研究其其他几何性质起着关键作用。这些对称性被编码在一个称为基本群的代数结构中;该项目旨在研究这一群体与几何之间的联系。具体来说,在双曲流形中,有一个叫做“算术”的特殊类,它往往是最对称的,它的基本群与数论有很强的联系。本项目旨在利用几何学和动力学方面的新技术来研究双曲流形的基本群,试图理解当这样的群是算术的,以及相关流形几何上的算术性(或缺乏算术性)的分支。这个项目更广泛的影响包括与本科生的合作。更具体地说,本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线几何形状,并为探索有限体积实数、复数、四元数和Cayley双曲流形基本群的刚性现象建立一个强大的框架。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,这些技术用于理解由算术流形的胶合子流形构建的流形的测地线几何,以及发展了用于实际双曲空间等长群中的晶格的超刚性风格技术。该项目计划继续开发这些新技术,并着眼于几何应用。具体而言,该项目将解决以下广泛的主题:1)理解低维和高维双曲流形及其测地子流形的构造,2)进一步发展秩1格超刚性结果的一般框架,以及3)尝试使用秩1刚性的最新进展作为理解复杂双曲格的完整性和四元数和Cayley双曲空间的算术的机制。该奖项反映了美国国家科学基金会的法定使命,并通过使用基金会的知识价值和更广泛的影响审查标准进行评估,被认为值得支持。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Nicholas Miller其他文献
A criminal justice comparative: post-9/11 anti-terrorism legislation within legal traditions
刑事司法比较:9/11后法律传统内的反恐立法
- DOI:
- 发表时间:
2010 - 期刊:
- 影响因子:0
- 作者:
Nicholas Miller - 通讯作者:
Nicholas Miller
TCT-160 The Genetic Basis Of Patent Foramen Ovale
- DOI:
10.1016/j.jacc.2014.07.197 - 发表时间:
2014-09-16 - 期刊:
- 影响因子:
- 作者:
Nabil Noureddin;Rubine Gevorgyan;Christopher Low;Nicholas Miller;Peter Debbaneh;Xinmin Li;Jonathan Tobis - 通讯作者:
Jonathan Tobis
Substance Abuse in Oncology
肿瘤学中的药物滥用
- DOI:
- 发表时间:
2014 - 期刊:
- 影响因子:0
- 作者:
S. Passik;Nicholas Miller;Matthew Ruehle;K. Kirsh - 通讯作者:
K. Kirsh
Alliance for a Cavity-Free Future (ACFF) UK Chapter: meeting summary
无龋未来联盟(ACFF)英国分会:会议总结
- DOI:
10.1038/s41415-023-6606-y - 发表时间:
2023-12-15 - 期刊:
- 影响因子:2.300
- 作者:
Avijit Banerjee;Nigel Pitts;Nicholas Miller - 通讯作者:
Nicholas Miller
Definitive LC-MS/MS Drug Monitoring Impacts Substance-use Treatment Planning and Patient Outcomes: A Brief Report
明确的 LC-MS/MS 药物监测影响药物使用治疗计划和患者结果:简要报告
- DOI:
- 发表时间:
2016 - 期刊:
- 影响因子:5.5
- 作者:
Adam Rzetelny;B. Zeller;Nicholas Miller;K. Kirsh;S. Passik - 通讯作者:
S. Passik
Nicholas Miller的其他文献
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{{ truncateString('Nicholas Miller', 18)}}的其他基金
RCN-UBE Incubator: Stem Research on Non-model Genomes Network
RCN-UBE孵化器:非模型基因组网络的干研究
- 批准号:
2120626 - 财政年份:2021
- 资助金额:
$ 14.64万 - 项目类别:
Standard Grant
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:
2005438 - 财政年份:2020
- 资助金额:
$ 14.64万 - 项目类别:
Standard Grant
Agenda Processes and the Theory of Voting
议程流程和投票理论
- 批准号:
8509680 - 财政年份:1985
- 资助金额:
$ 14.64万 - 项目类别:
Standard Grant
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