The Covers, Symmetries, and Combinatorics of Manifolds

几何学和拓扑学研究的是形状。在欧几里得几何中，我们研究圆和矩形等物体。圆是高度对称的；例如，围绕圆心的任何旋转都保持圆不变。矩形虽然在某些方面仍然是对称的，但它的对称性不如圆；它的两边可能有不同的长度。在低维拓扑中，我们研究更复杂的物体或二维、三维和四维空间。这个由美国国家科学基金会资助的项目的中心目标是通过对称性来理解这些空间。拓扑对象在其他领域自然出现，包括生物学、化学、物理学和工程学。有时，了解一个国家的最好方法是通过所谓的“覆盖图”来了解它与另一个国家的关系。第二个项目是分析覆盖地图。研究更复杂的三维或四维空间的一个挑战是，即使使用计算机，人们也不能总是可视化或绘制这些空间。因此，将它们分解成构建模块是有帮助的。其中一个项目就是用所谓的双曲3流形来做这件事。除了数学研究之外，PI还展示了对STEM学科中代表性不足的个人的推广、指导和倡导的强烈奉献。在美国国家科学基金会的旅行资助下，她将继续参与学术界内外旨在促进数学研究和教育的机会。本课题的研究重点是了解低维双曲流形的有限次覆盖空间、对称群和组合学。PI计划在资助期内处理以下项目：(1)使虚拟哈肯定理有效；(2)量化可分性性质，以确定三流形的基本群和闭曲面的映射类群是否为线性；(3)探索无限型曲面及其映射类群，以及这些群对双曲复合体的作用；(4)给出双曲三流形的组合表征。虽然该研究项目主要关注拓扑、几何和几何群论方面的问题，但PI探索的主题与组合学、表示理论、动力学和拓扑量子场论（TQFT）也有很深的联系。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。

Geometry and topology are concerned with the study of shapes. In Euclidean geometry, we study objects such as circles and rectangles. A circle is highly symmetric; for example, any rotation about the center of the circle preserves the circle. A rectangle, though still symmetric in some ways, has less symmetry than the circle; its two sides may have different lengths. In low-dimensional topology we study more complicated objects or spaces of two, three, and four dimensions. A central aim of this National Science Foundation funded project is to understand these spaces through symmetries. Topological objects arise naturally in other fields, including biology, chemistry, physics, and engineering. At times, the best way to understand one is via its relationship with another through what is known as a "covering map". A second project is to analyze covering maps. A challenge in studying more intricate three or four dimensional spaces is that one cannot always visualize or draw these even using a computer. It is therefore helpful to break them into building blocks. One of the projects is to do so with objects called hyperbolic 3-manifolds. In addition to the mathematical research, the PI has demonstrated a strong dedication to outreach, mentoring, and advocating for underrepresented individuals in the STEM disciplines. With the NSF travel funds she will continue to engage in opportunities inside and outside the academia directed at promoting mathematical research and education. The focus of this research project is to understand the finite degree covering spaces, the group of symmetries, and the combinatorics of hyperbolic manifolds in low dimensions. The PI plans to tackle the following projects during the funding period: (1) making effective the Virtually Haken Theorem, (2) quantifying separability properties to determine whether or not the fundamental groups of three-manifolds and the mapping class groups of closed surfaces are linear, (3) exploring infinite-type surfaces, their mapping class groups, and the actions of these groups on hyperbolic complexes, and (4) giving a combinatorial characterization for hyperbolic three-manifolds. Though the research project primarily focuses on questions in topology, geometry, and geometric group theory, the topics explored by the PI have deep connections to combinatorics, representation theory, dynamics, and topological quantum field theory (TQFT) as well.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.