REU Site: Investigations in Geometry and Knot Theory

REU 网站：几何和结理论的研究

• 批准号: 2050894
• 负责人: Corey Dunn
• 金额: $ 25.92万
• 依托单位: University Enterprises Corporation at CSUSB
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2050894&HistoricalAwards=false
• 关键词: REU; Site; Investigations; Geometry; Knot

This program at California State University San Bernardino is an immersive research experience for undergraduate students. In each of three summers, eight participants will learn background material in differential geometry and knot theory and pursue open-ended research questions. Questions under study in differential geometry range widely, from determining efficient methods of expressing curvature, to developing theories regarding the types of curvature that one might possibly encounter, and even geometrically realizing these curvatures. Questions under study in knot theory will emphasize the 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 questions are part of an active and vibrant area of mathematical research and are also of interest because of their potential applications. Student participants will be recruited nationally, with emphasis on recruitment from those institutions that have limited research opportunities for their students, and from populations underrepresented in STEM disciplines.The differential geometry component of the project has three main topics. The Riemann curvature tensor is an object that encodes a manifold’s curvature at every point. It is known that this object can be expressed as a combination of other types of curvatures, and students will explore how different curvature tensors could be expressed according to this decomposition. Second, students will investigate novel invariants for curvature tensors. Thirdly, students will seek geometric realizations illustrating these concepts. The knot theory component of this project focuses on hyperbolic links—links whose complements admit a hyperbolic structure. Projects will focus on the class of links called fully augmented links, which have particularly tractable geometric structures but are nevertheless quite useful since they can be used to construct all links via Dehn filling. 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.

加州州立大学圣贝纳迪诺的这个项目是本科生的沉浸式研究体验。在三个夏天的每一个，八名参与者将学习微分几何和纽结理论的背景材料，并追求开放式的研究问题。在微分几何研究的问题范围广泛，从确定有效的方法表示曲率，发展理论的类型曲率，人们可能会遇到的，甚至几何实现这些曲率。纽结理论中的研究问题将强调拓扑学和几何学之间的相互作用。由于几何应用拓扑结构是一些最微妙的和重要的发现在过去的三十年中，这些问题是一个活跃和充满活力的数学研究领域的一部分，也是感兴趣的，因为他们的潜在应用。学生参与者将在全国范围内招募，重点是从学生研究机会有限的机构以及STEM学科代表性不足的人群中招募。该项目的微分几何部分有三个主要主题。黎曼曲率张量是一个对象，它编码了流形在每一点上的曲率。众所周知，该对象可以表示为其他类型曲率的组合，学生将探索如何根据此分解表示不同的曲率张量。其次，学生将研究曲率张量的新不变量。第三，学生将寻求几何实现说明这些概念。这个项目的纽结理论部分集中在双曲链接，其补充承认双曲结构的链接。项目将集中在一类称为完全增强链接的链接上，这些链接具有特别易于处理的几何结构，但仍然非常有用，因为它们可以通过Dehn填充来构建所有链接。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。