Geometry, Topology, and Rank One Lattices

李群中的格是实数内整数的广泛推广。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直接的相似之处，李群中的格位于如此多领域的界面上的方式使它们自19世纪后期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”，它与数论的联系特别强，这个项目的首要目标是利用动力学和几何的技术来理解什么时候一个格是算术的，而且加深我们对算术的几何结果的理解。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，该项目旨在理解局部对称空间的几何和拓扑，特别是实数和复杂双曲流形，灵感来自自Thurston和Gromov的关键工作以来一直主导该领域的低维拓扑和几何群论的基本问题。一方面，了解格罗莫夫-瑟斯顿计划在多大程度上推动了这种更普遍的环境，这是非常有趣的。更重要的是，近四十年来在二维和三维空间中研究的问题与李群的离散子群的经典问题密切相关，而实双曲格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）算术相关的问题。要研究的特殊问题包括算术的几何特征，马古利斯超刚性定理在第一级的类似物，以及使用代数几何的技术来理解复杂的双曲格。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。

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.