The Covers, Symmetries, and Combinatorics of Manifolds

几何学和拓扑学是研究形状的学科。在欧几里得几何学中，我们研究对象，如圆和矩形。圆是高度对称的；例如，围绕圆中心的任何旋转都将保留该圆。长方形虽然在某些方面仍然是对称的，但它的对称性不如圆形；它的两边可能有不同的长度。在低维拓扑中，我们研究二维、三维和四维的更复杂的对象或空间。这个由国家科学基金会资助的项目的一个中心目标是通过对称性来理解这些空间。拓扑物在生物、化学、物理和工程等其他领域中自然产生。有时，最好的理解方式是通过所谓的“覆盖地图”来了解它与另一个人的关系。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，将它们分解为构建块是有帮助的。其中一个项目是用被称为双曲3-流形的对象来做到这一点。除了数学研究外，国际和平协会还表现出强烈的奉献精神，致力于在STEM学科中为代表性不足的个人提供外联、指导和倡导。利用美国国家科学基金会的旅行基金，她将继续参与学术界内外的机会，以促进数学研究和教育。这项研究的重点是了解低维双曲流形的有限度覆盖空间、对称群和组合学。PI计划在资助期间处理以下项目：(1)使虚拟Haken定理有效；(2)量化可分性以确定三维流形的基本群和闭曲面的映射类群是否是线性的；(3)探索无限类型曲面、它们的映射类群以及这些群在双曲复形上的作用；(4)给出双曲三维流形的一个组合刻画。虽然这项研究项目主要集中在拓扑学、几何学和几何群论的问题上，但PI探索的主题与组合学、表示论、动力学和拓扑量子场论(TQFT)也有很深的联系。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.