The Covers, Symmetries, and Combinatorics of Manifolds

流形的覆盖、对称性和组合学

• 批准号: 1937969
• 负责人: Priyam Patel
• 金额: $ 9.12万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1937969&HistoricalAwards=false
• 关键词: Covers; Symmetries; Combinatorics; Manifolds

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.

几何和拓扑与形状的研究有关。在欧几里得的几何形状中，我们研究圆形和矩形等物体。一个圆形高度对称；例如，圆圈中心的任何旋转都保留了圆。矩形虽然在某些方面仍然对称，但其对称性少于圆。它的两侧可能具有不同的长度。在低维拓扑中，我们研究了两个，三个和四个维度的更复杂的物体或空间。这个国家科学基金会资助的项目的一个核心目的是通过对称性来了解这些空间。拓扑对象自然出现在其他领域，包括生物学，化学，物理和工程。 有时，理解一个人的最佳方法是通过与另一种的关系通过所谓的“覆盖地图”。第二个项目是分析覆盖地图。研究更复杂的三个或四个维空间的挑战是，即使使用计算机，也不能始终可以看到或绘制这些空间。因此，将它们分解成构建基础是有帮助的。其中一个项目是使用称为双曲线3个manifolds的对象。 除了数学研究外，PI还表现出对STEM学科中代表性不足的个人的强烈奉献精神。借助NSF旅行资金，她将继续在学术界内外探讨致力于促进数学研究和教育的机会。该研究项目的重点是了解涵盖空间的有限程度，对称性组以及低维度双曲线歧管的组合。 PI计划在资金期间解决以下项目：（1）使实际上的定理有效，（2）量化可分离性能，以确定三个序列的基本组和封闭表面组的基本组是否是线性的双曲线三个字体的表征。尽管研究项目主要关注拓扑，几何学和几何群体理论的问题，但PI探索的主题与组合学，代表理论，动力学和拓扑量子田地理论（TQFT）以及该奖项都反映了NSF的法定任务，并通过评估基金会的智力效果，并反映了NSF的法定任务，并反映了NSF的法定任务，并反映了基金会的范围和广泛的范围。