Hyperbolic Manifolds and Their Embedded Submanifolds

本项目将通过理解有限体积双曲流形的嵌入子流形的结构来研究有限体积双曲流形。该研究结合了多个领域的各个方面，包括几何拓扑、数论、代数几何、组合学和动力学。该项目还将包括参与活动，包括指导和支持学生和早期职业数学家，参加公开讲座和推广活动，以及组织活动和讲习班。本课题包括本科生和研究生的具体研究计划。本项目的研究目标分为三个主要方向。第一个项目是通过构造显式盖继续研究3流形的有效虚性质。该项目将重点关注算术双曲3流形的同余覆盖，其中PI将利用其几何和数论性质之间的丰富联系。第二个课题是研究高维双曲流形中的余维-1嵌入子流形。这个项目的一个特别的焦点是四维双曲流形的情况，以更好地理解双曲4-流形的几何和其他充分研究的4-流形不变量之间的关系。第三个项目是研究3流形中的嵌入曲面和曲面中的简单闭合曲线，通过它们的基本群的表示。该项目还包括旨在扩大学生和初级研究人员参与的更广泛的影响活动。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。

This project will investigate hyperbolic manifolds of finite-volume by understanding the structure of their embedded sub-manifolds. The research combines aspects from multiple fields, including geometric topology, number theory, algebraic geometry, combinatorics, and dynamics. The project will also include participation in activities including mentoring and supporting students and early career mathematicians, participating in public lectures and outreach activities, and organizing events and workshops. This project includes specific research plans for undergraduate and graduate students. The research goals of this project are divided into three main directions. The first project is to continue working in effective virtual properties of 3-manifolds by constructing explicit covers. This project will focus on congruence covers of arithmetic hyperbolic 3-manifolds, where the PI will leverage the rich connection between their geometric and number theoretical properties. The second project is to study codimension-1 embedded sub-manifolds in higher dimensional hyperbolic manifolds. A particular focus of this project is the case of hyperbolic manifolds of dimension 4, to better understand the relationship between the geometry of hyperbolic 4-manifolds and other well-studied 4-manifold invariants. The third project involves the study of embedded surfaces in 3-manifolds and simple closed curves in surfaces through the representations of their fundamental groups. The project also includes broader impact activities aimed at broadening participation among students and junior researchers.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.