权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Geometry, Topology, and Rank One Lattices

几何、拓扑和一阶晶格

基本信息

批准号：
1906088
负责人：
Matthew Stover
金额：
$ 20.63万
依托单位：
Temple University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-15 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906088&HistoricalAwards=false
关键词：
Geometry Topology Rank Lattices

项目摘要

Lattices in Lie groups are a wide-sweeping generalization of the integers inside the real numbers. The deep connections between the integers and a broad range of mathematical areas from geometry to number theory have direct analogues for lattices in Lie groups, and the way in which lattices in Lie groups sit at the interface of so many fields has made them of fundamental importance since the late 19th century. The primary motivation for this project is to deepen our understanding of these connections. There is a special class of lattices that are called "arithmetic", where the connections to number theory are particularly strong, and this an overarching goal of this project is to use techniques from dynamics and geometry to understand when a lattice is arithmetic, and moreover deepen our understanding of the geometric consequences of arithmeticity. The project provides summer support for graduate students allowing them time for research and collaboration.More specifically, this project aims to understand the geometry and topology of locally symmetric spaces, particularly real and complex hyperbolic manifolds, inspired by the fundamental problems in low-dimensional topology and geometric group theory that have dominated the fields since the pivotal work of Thurston and Gromov. On the one hand, it is of significant interest to learn the extent to which the Gromov-Thurston program pushes into this more general setting. More importantly, the questions studied in dimensions 2 and 3 for the last forty years are closely related to classical problems about discrete subgroups of Lie groups, and real and complex hyperbolic lattices are precisely the cases where many of the basic questions remain open, e.g., Betti numbers and problems related to (non)arithmeticity. Particular problems to be studied include geometric characterizations of arithmeticity, analogues of the Margulis superrigidity theorem in rank one, and understanding complex hyperbolic lattices using techniques from algebraic geometry.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.

李群中的格是真实的数中整数的广泛推广。整数与从几何到数论的广泛数学领域之间的深刻联系与李群中的格有着直接的相似之处，李群中的格处于如此多的领域的界面，这使得它们自世纪以来具有根本的重要性。这个项目的主要动机是加深我们对这些联系的理解。有一类特殊的格被称为“算术”，与数论的联系特别强，这个项目的首要目标是使用动力学和几何学的技术来理解格何时是算术的，并加深我们对算术性的几何后果的理解。该项目为研究生提供暑期支持，让他们有时间进行研究和合作。更具体地说，该项目旨在了解局部对称空间的几何和拓扑，特别是真实的和复双曲流形，灵感来自低维拓扑和几何群论的基本问题，这些问题自Thurston和Gromov的关键工作以来一直主导着该领域。一方面，了解Gromov-Thurston程序在多大程度上推动了这一更一般的设置是非常有趣的。更重要的是，在过去的四十年中，在2维和3维中研究的问题与李群的离散子群的经典问题密切相关，而真实的和复双曲格正是许多基本问题仍然开放的情况，例如，贝蒂数和与（非）算术性有关的问题。特别是要研究的问题包括算术性的几何特征，类似物的马古利斯超刚性定理在秩一，并理解复杂的双曲格使用技术从代数几何。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。