REU Site: Investigations in Geometry and Knot Theory

REU课程“几何和纽结理论的研究”旨在加深我们对数学两个活跃领域的理解：微分几何和纽结理论。在三个暑假中的每一个暑假，八名参与者每人都将收到这些领域的背景材料，并有机会在他们选择的领域内进行开放式研究问题。微分几何领域中尚未解决的问题广泛，从确定表示各种曲率的有效方法，到发展自己的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。另一个研究课题是纽结理论，也就是研究空间中闭合的打结环。纽结和链接补码提供了三维流形的极好例子，随着与几何的深层联系的发现，三维拓扑学领域在20世纪70年代末的S和80年代的S发生了革命性的变化。这场革命是由威廉·P·瑟斯顿在几何结构方面的工作带头进行的，他因此于1982年被授予菲尔兹奖章。纽结理论中尚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过去30年来最微妙和最重要的发现之一，这些不变量是数学研究中活跃和充满活力的领域的一部分。总体而言，数学家对这些问题很感兴趣，因为他们通常试图了解宇宙如何运行的结构，但科学界也对它们感兴趣，因为我们的发现具有潜在的应用价值。这个项目的微分几何部分有两个主要目标。在任何光滑流形上，黎曼曲率张量是一个对象，它编码曲面在每个点的曲率。众所周知，这个对象可以表示为其他类型的曲率的组合，目的是了解根据这种分解可以如何表示不同的曲率张量的本质。以前的结果表明，这与将流形嵌入为欧几里得空间的子集可能需要的额外维度的数量之间存在深刻的关系。第二个目标是收集一类特定曲面的已知工作，并试图将这一领域的工作统一到一个理论中，并用实例说明该理论的不同方面。本项目的纽结理论部分侧重于双曲线链接--其补语采用双曲线结构的链接。三维流形理论中的几何革命表明，“大多数”环是双曲的，Mostow-Prasad刚性意味着几何量是拓扑不变量。因此，确定和理解连结补语上的双曲结构就成为纽结理论中的一个重要问题。纽结理论项目将专注于一类被称为完全增强链接的链接-具有特别容易处理的几何结构的链接，可以被本科生从多个角度理解。尽管它们的几何结构简单，但完全增强的链接是非常有用的，因为它们可以通过Dehn填充来构造所有的链接。因此，项目将进一步揭示完全增强的(和相关的)链接补语的几何结构，并通过Dehn填充将这些知识应用于一般的链接。总体而言，这两个领域都有各种各样的问题需要探索。这一奖项反映了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。

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.