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.

几何学和拓扑学都与形状的研究有关。在欧几里德几何中，我们研究诸如圆形和矩形之类的物体。圆是高度对称的;例如，围绕圆心的任何旋转都保持圆。矩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。在低维拓扑学中，我们研究更复杂的物体或二维、三维和四维空间。这个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。拓扑对象自然出现在其他领域，包括生物学，化学，物理学和工程学。 有时，了解一个人的最佳方式是通过它与另一个人的关系，通过所谓的“覆盖图”。第二个项目是分析覆盖图。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，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。其中一个项目是这样做的对象称为双曲三维流形。 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推广，指导和倡导的强烈奉献精神。随着国家科学基金会的旅行资金，她将继续从事机会内外的学术界针对促进数学研究和教育。本研究计画的重点是了解有限度复盖空间、对称群以及低维双曲流形的组合学。计划在资助期内进行以下项目：（1）使虚哈肯定理有效，（2）量化分离性以确定三元流形的基本群和闭曲面的映射类群是否是线性的，（3）探索无限型曲面及其映射类群，以及这些群对双曲复形的作用，（4）给出了双曲三流形的一个组合刻划。虽然该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所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。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。