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

李群中的格是实数内整数的全面推广。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，特别是对双曲流形及其近亲的理解。双曲几何是经典欧几里得几何（零曲率）和球面几何（正曲率）的“负弯曲”对应物，与欧几里得几何和球面几何相比，双曲流形被理解得很少。将双曲流形与李群的离散子群联系起来，提供了对广泛的数学工具的访问，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. Bader， D. Fisher和N. Miller最近的工作，该工作给出了双曲流形的完全几何表征，即“算术”（一个具有巨大意义的族）。但它的定义是纯代数的，与流形的几何关系并不明显)。该项目的总体目标是利用动力学、几何、代数几何和数论的技术来进一步理解双曲流形和与之密切相关的被称为“局部对称空间”的推广。该项目为研究生提供支持，让他们有时间进行研究、合作和参加会议。它还将支持PI继续开发基于探究的学习工具，用于本科教学，并组织暑期学校和研究生专业发展会议等活动。该项目旨在了解局部对称空间的几何和拓扑，特别是实和复杂双曲流形。受到低维拓扑和几何群论的基本问题的启发，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这个领域。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未解决的情况，例如Betti数的（非）琐碎性和与（非）算术性相关的问题。需要研究的特殊问题包括马古利斯超刚性定理的类比，这是基于PI最近与巴德-费雪-米勒的突破，利用代数几何的技术来理解复杂双曲格，并继续/推广PI与D. Toledo的工作，研究复杂双曲格中心扩展的剩余有限性与具有负曲率度量的有趣的光滑投影代数变体的构造之间的联系。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 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.