REU Site: Investigations in Geometry and Knot Theory

加州州立大学圣贝纳迪诺分校的这个项目为本科生提供了身临其境的研究体验。在每三个暑假中，八名参与者将学习微分几何和纽结理论的背景材料，并进行开放式研究问题。在微分几何中研究的问题范围很广，从确定表示曲率的有效方法，到发展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。纽结理论正在研究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年来最微妙和最重大的发现之一，这些问题是数学研究中活跃和充满活力的领域的一部分，也因为它们的潜在应用而引起人们的兴趣。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在STEM学科中代表性不足的人群中招聘。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，它编码流形在每一点的曲率。众所周知，这个对象可以表示为其他类型的曲率的组合，学生将探索如何根据这种分解来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来说明这些概念。本项目的纽结理论部分侧重于双曲链环--其补码采用双曲线结构的链环。项目将重点放在被称为完全增强链接的链接类别上，这种链接具有特别容易处理的几何结构，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。

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.