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REU Site: Investigations in Geometry and Knot Theory

REU 网站：几何和结理论的研究

基本信息

批准号：
1758020
负责人：
Corey Dunn
金额：
$ 27.87万
依托单位：
University Enterprises Corporation at CSUSB
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1758020&HistoricalAwards=false
关键词：
REU Site Investigations Geometry Knot

项目摘要

The REU program "Investigations in Geometry and Knot Theory" aims to further our understanding in two vibrant areas of mathematics: Differential Geometry and Knot Theory. In each of three summers, each of the eight participants will be presented with background material in each of these areas and given the opportunity to pursue open-ended research problems in the field of their choice. The unsolved problems presented in the field of Differential Geometry range widely from determining efficient methods of expressing various sorts of curvature, to developing their own theories regarding the types of curvature that one might possibly encounter and even geometrically realizing these curvatures. The other subject of study is knot theory, or the study of closed, knotted loops in space. Knot and link complements provide excellent examples of three-dimensional manifolds, and the field of three-dimensional topology was revolutionized in the late 1970's and 80's following the discovery of deep connections with geometry. The revolution was spearheaded by the work of William P. Thurston on geometric structures, for which he was awarded the Fields Medal in 1982. The unsolved problems in knot theory will emphasize this interplay between topology and geometry. As geometric applications to topology are some of the most subtle and significant discoveries in the last thirty years, these invariants are part of an active and vibrant area of mathematical research. Overall, these questions are of interest to mathematicians since they generally seek to understand the structure of how the universe works, but they are also of interest to the scientific community because of the potential applications of our findings.The Differential Geometry component of this project has two main goals. On any smooth manifold, the Riemann Curvature Tensor is an object that encodes the surface's curvature at every poin. It is known that this object can be expressed as a combination of other types of curvatures, and the aim is to understand the nature of how different curvature tensors could be expressed according to this decomposition. Previous results present a deep relationship between this and the number of extra dimensions one might need to embed your manifold as a subset of Euclidean space. The second goal aims to collect the known work in a particular class of surfaces and attempt to unify the work in this area into one theory, complete with examples illustrating different aspects of this theory. The Knot Theory component of this project focuses on hyperbolic links-links whose complements admit a hyperbolic structure. The geometric revolution in three-manifold theory demonstrated that "most" links are hyperbolic, and Mostow-Prasad rigidity implies that geometric quantities are topological invariants. Determining and understanding hyperbolic structures on link complements, then, becomes an important problem in knot theory. Knot theory projects will focus on the class of links called fully augmented links-links that have particularly tractable geometric structures, which can be understood by undergraduates from many perspectives. Despite their geometric simplicity, fully augmented links are quite useful since they can be used to construct all links via Dehn filling. Thus projects will further uncover the geometric structure of fully augmented (and related) link complements, and apply this knowledge to links in general via Dehn filling. Overall, both fields are rich with a variety of questions to explore.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.

REU项目“几何和结理论研究”旨在进一步加深我们对两个充满活力的数学领域的理解：微分几何和结理论。在每一个夏季，8名参与者都将获得这些领域的背景材料，并有机会在他们选择的领域进行开放式的研究问题。在微分几何领域中提出的未解决的问题范围很广，从确定表达各种曲率的有效方法，到发展他们自己关于可能遇到的曲率类型的理论，甚至从几何上实现这些曲率。另一个研究主题是结理论，或研究空间中闭合的、打结的环。结和连接互补提供了三维流形的优秀例子，随着与几何的深度联系的发现，三维拓扑学领域在20世纪70年代末和80年代发生了革命性的变化。这场革命由威廉·p·瑟斯顿（William P. Thurston）在几何结构方面的工作引领，他因此在1982年获得了菲尔兹奖（Fields Medal）。结理论中未解决的问题将强调拓扑学和几何学之间的相互作用。由于几何在拓扑学中的应用是近三十年来最微妙和最重要的发现，这些不变量是数学研究中一个活跃和充满活力的领域的一部分。总的来说，数学家对这些问题感兴趣，因为他们通常寻求理解宇宙如何运作的结构，但他们也对科学界感兴趣，因为我们的发现的潜在应用。这个项目的微分几何组件有两个主要目标。在任何光滑流形上，黎曼曲率张量是一个对象，它编码了曲面上每一点的曲率。众所周知，这个对象可以表示为其他类型曲率的组合，目的是了解根据这种分解如何表示不同曲率张量的本质。先前的结果表明，这与将流形嵌入欧几里得空间的子集所需的额外维数之间存在深刻的关系。第二个目标是收集特定类别表面的已知工作，并试图将该领域的工作统一为一个理论，并举例说明该理论的不同方面。这个项目的结理论部分侧重于双曲链接——其补体允许双曲结构的链接。三流形理论中的几何革命证明了“大多数”连杆是双曲的，而Mostow-Prasad刚性意味着几何量是拓扑不变量。因此，确定和理解连杆补上的双曲结构成为结理论中的一个重要问题。结理论项目将集中于一类被称为全增广链的链，这些链具有特别易于处理的几何结构，可以被本科生从许多角度理解。尽管它们的几何简单，但完全增强的链接非常有用，因为它们可以通过Dehn填充来构建所有链接。因此，项目将进一步揭示完全增强（和相关）补链的几何结构，并通过Dehn填充将这些知识应用于一般的链接。总的来说，这两个领域都有丰富的问题需要探索。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。