Geodesic Submanifolds, Rigidity, and Other New Phenomena in Rank 1

李群中的格是真实的数中整数的一个全面推广。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连接的理解，特别是双曲流形及其近亲。双曲几何是经典欧几里德几何（零曲率）和球面几何（正曲率）的“负弯曲”对应物，而双曲流形与它们的欧几里德和球面对应物相比知之甚少。将双曲流形与李群的离散子群联系起来提供了一个广泛的数学工具，包括代数和动力学，这些工具最近取得了重大成果，包括PI与U的最近工作。Bader，D. Fisher和N.米勒，使一个完全的几何特征，其中双曲流形是“算术”（一个家庭的巨大意义，但其中的定义是纯粹的代数，并没有明显绑在几何的流形）。这个项目的首要目标是使用动力学，几何，代数几何和数论的技术来进一步理解双曲流形和称为“局部对称空间”的密切相关的推广。该项目为研究生提供支持，使他们有时间进行研究，合作和参加会议。它还将支持PI继续为本科教学开发基于探究的学习工具，并组织暑期学校和会议等活动，重点关注研究生专业发展。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，灵感来自于低维拓扑和几何群论的基本问题，这些问题自瑟斯顿和格罗莫夫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，Betti数的（非）平凡性及与（非）算术性有关的问题要研究的具体问题包括类似物的马古利斯超刚性定理在秩一，建设PI的最近的突破与Bader-Fisher-米勒，理解复杂的双曲格使用技术从代数几何和继续/推广PI的工作与D。Toledo关于复双曲格的中心扩展的剩余有限性和有趣的新光滑投影代数簇的结构之间的连接承认度量的负曲率。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Lattices in Lie groups are a 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, in particular for hyperbolic manifolds and their close relatives. Hyperbolic geometry is the "negatively curved" counterpart to classical Euclidean geometry (zero curvature) and spherical geometry (positive curvature), and hyperbolic manifolds are poorly understood compared with their Euclidean and spherical counterparts. Relating hyperbolic manifolds to discrete subgroups of Lie groups provides access to a wide spectrum of mathematical tools, including algebra and dynamics, that have borne significant recent fruit, including recent work of the PI with U. Bader, D. Fisher, and N. Miller that gives a completely geometric characterization of which hyperbolic manifolds that are "arithmetic" (a family of immense significance, but one for which the definition is purely algebraic and not obviously tied to the geometry of the manifold). The overarching goal of this project is to use techniques from dynamics, geometry, algebraic geometry, and number theory to further our understanding of hyperbolic manifolds and closely-related generalizations called "locally symmetric spaces". The project provides support for graduate students allowing them time for research, collaboration, and travel to conferences. It will also support the PI's continued development of Inquiry-Based Learning tools for undergraduate teaching and organizing events like summer schools and conferences focused on graduate student professional developmentThis 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., (non)triviality of Betti numbers and problems related to (non)arithmeticity. Particular problems to be studied include analogues of the Margulis superrigidity theorem in rank one, building on the PI's recent breakthroughs with Bader-Fisher-Miller, and understanding complex hyperbolic lattices using techniques from algebraic geometry and continuing/generalizing the PI's work with D. Toledo on connections between residual finiteness of central extensions of complex hyperbolic lattices and constructions of interesting new smooth projective algebraic varieties admitting metrics of negative curvature.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.