REU Site: Investigations in Geometry and Knot Theory

REU计划“几何和纽结理论的研究”旨在进一步我们在数学的两个充满活力的领域的理解：微分几何和纽结理论。 在三个夏天的每一个，八个参与者中的每一个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。 在微分几何领域提出的未解决的问题范围很广，从确定有效的方法来表达各种曲率，发展自己的理论，关于类型的曲率，人们可能会遇到，甚至几何实现这些曲率。 另一个研究课题是纽结理论，或者说是对空间中封闭的、打结的环的研究。 纽结和链补提供了三维流形的很好的例子，并且三维拓扑学领域在20世纪70年代末和80年代随着与几何的深层联系的发现而发生了革命性的变化。 威廉·瑟斯顿（William P. Thurston）在几何结构方面的工作是这场革命的先驱，他因此在1982年获得菲尔兹奖。 纽结理论中未解决的问题将强调拓扑学和几何学之间的这种相互作用。 由于几何应用拓扑是一些最微妙的和重要的发现在过去的三十年里，这些不变量是一个活跃和充满活力的数学研究领域的一部分。 总的来说，这些问题是数学家感兴趣的，因为他们通常寻求了解宇宙如何运作的结构，但他们也感兴趣的科学界，因为我们的研究结果的潜在应用。微分几何部分这个项目有两个主要目标。 在任何光滑流形上，黎曼曲率张量是一个对象，它编码了曲面在每一点的曲率。 众所周知，这个对象可以表示为其他类型的曲率的组合，目的是理解如何根据这种分解来表达不同的曲率张量的性质。 以前的结果提出了一个深刻的关系，这和额外的维度的数量可能需要嵌入你的流形作为一个子集的欧几里德空间。第二个目标旨在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到一个理论中，并通过示例说明该理论的不同方面。 这个项目的纽结理论部分集中在双曲链接，链接的补充承认双曲结构。 在三流形理论中的几何革命证明了“大多数”环节是双曲的，而Mostow-Prasad刚性意味着几何量是拓扑不变量。 确定和理解链补上的双曲结构，成为纽结理论中的一个重要问题。纽结理论项目将集中在一类链接称为完全增广链接链接，具有特别易处理的几何结构，这可以从许多角度理解的本科生。 尽管它们的几何简单，完全增广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广（和相关）链接补的几何结构，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各种问题需要探索。这个奖项反映了NSF的法定使命，并被认为是值得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的支持。

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.