REU Site: Investigations in Geometry and Knot Theory

REU 网站：几何和结理论的研究

• 批准号: 1156608
• 负责人: Corey Dunn
• 金额: $ 28.43万
• 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: 2012
• 资助国家: 美国
• 起止时间: 2012-04-01 至 2015-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1156608&HistoricalAwards=false
• 关键词: REU; Site; Investigations; Geometry; Knot

The REU Sites project, Investigations in Geometry and Knot Theory, is an 8-week program for 8 undergraduates at California State University, San Bernardino. There are two topics which the participants will study. The first is an investigation of the algebraic properties of the Riemann curvature tensor on a smooth manifold. The possible questions we intend to pose relate to the interaction between the sorts of canonical curvature tensors one obtains from an embedding of a manifold into flat space as a hypersurface, in particular, the efficiency one has in exhibiting any curvature one might encounter in terms of the linear independence of algebraic curvature tensors. While these questions are broad in scope, another main area of study in this realm will be the decomposability of the algebraic structures that each tangent space of a smooth manifold is equipped with: the tangent space itself, the metric, and the curvature tensor. While it has been shown there are certain general circumstances when these structures decompose, there are many questions about the nature of this decomposition in specific instances that is of interest. Beyond that, there is ample room for the discovery of new manifolds with prescribed curvature properties. The second topic of investigation is a study of hyperbolic knots. The subject of hyperbolic geometry is very rich, incorporating algebraic, geometric and topological techniques. Moreover, the theory is developed enough to offer a wealth of problems accessible to mathematically mature undergraduates. There are two classes of questions that will be investigated in the knot theory portion of the program, both of which pursue the relationship between hyperbolic geometry and braid theoretic descriptions of links. The first involves volumes of hyperbolic closed three braids, and the second involves classifying closed braid representatives of hyperbolic knots. Participants will be introduced to two vibrant areas of mathematics, geometry and knot theory, and will be actively engaged in significant research experiences.Experienced faculty advisors design projects in Geometry and Knot theory that introduce participants to significant mathematics while exploring creative and original concepts in their respective fields. A group of students will work in each relevant field, with significant mathematical interaction occurring between students working in the same field. Moreover, participants will work closely with their mentors in an enriching environment to complete background reading related to their topic, give presentations on relevant material, conduct research, and begin writing a journal-style paper. As the summer progresses, students will perform their own literature searches, make independent discoveries and engage in creative mathematical research. In addition to regular presentations and paper assignments, each student will create a poster describing their results, give a twenty-minute final presentation to the campus community at California State University, San Bernardino, and complete a journal-style paper about their project. Thus participants will have a comprehensive and cohort research experience. The program will advance discovery through actively engaging undergraduate students in mathematical research and strongly encouraging them to become active participants in the mathematical community. Students from minority-serving institutions are encouraged to apply. Further, California State University, San Bernardino's diverse student population attends events sponsored by the program, broadening the impact it has on underrepresented groups. Finally, the program has a multifaceted plan for broad dissemination in order to enhance scientific understanding. Avenues for dissemination include conference presentations, submission for publication, and posting results on the program's web site.

REU Sites项目，几何和结理论研究，是一个为期8周的项目，由加州州立大学圣贝纳迪诺分校的8名本科生参加。参与者将学习两个主题。第一部分是研究光滑流形上黎曼曲率张量的代数性质。我们打算提出的可能的问题与从流形嵌入到平面空间作为超曲面中获得的各种正则曲率张量之间的相互作用有关，特别是，根据代数曲率张量的线性无关性，人们在展示任何曲率时所具有的效率。虽然这些问题的范围很广，但该领域的另一个主要研究领域将是光滑流形的每个切线空间所配备的代数结构的可分解性：切线空间本身、度量和曲率张量。虽然已经表明存在这些结构分解的某些一般情况，但是在令人感兴趣的特定实例中，关于这种分解的性质存在许多问题。除此之外，有足够的空间来发现具有规定曲率特性的新流形。研究的第二个主题是双曲结的研究。双曲几何的主题是非常丰富的，结合了代数，几何和拓扑技术。此外，该理论的发展足以为数学成熟的本科生提供丰富的问题。在程序的结理论部分将研究两类问题，这两类问题都追求双曲几何和链路的编织理论描述之间的关系。第一个涉及双曲闭合的三个辫子卷，第二个涉及对双曲结的闭合辫子代表进行分类。参与者将被介绍到两个充满活力的数学领域，几何和结理论，并将积极参与重要的研究经验。经验丰富的教师顾问设计几何和结理论的项目，向参与者介绍重要的数学，同时探索各自领域的创造性和原创性概念。一组学生将在每个相关领域工作，在同一领域工作的学生之间发生重要的数学互动。此外，参与者将在丰富的环境中与导师密切合作，完成与主题相关的背景阅读，就相关材料进行演讲，进行研究，并开始撰写期刊风格的论文。随着暑假的进行，学生们将进行自己的文献搜索，独立发现并从事创造性的数学研究。除了定期的报告和论文作业外，每个学生还要制作一张海报来描述他们的结果，在加州州立大学圣贝纳迪诺分校的校园社区做一个20分钟的期末报告，并完成一篇关于他们项目的期刊式论文。因此，参与者将有一个全面的和队列的研究经验。该计划将通过积极参与数学研究的本科生，并强烈鼓励他们成为数学社区的积极参与者，从而促进发现。鼓励来自少数民族院校的学生申请。此外，加州州立大学圣贝纳迪诺分校的多元化学生群体参加了该项目赞助的活动，扩大了该项目对代表性不足的群体的影响。最后，该方案有一个多方面的计划，以广泛传播，以提高科学认识。传播途径包括会议演讲、提交出版物和在项目网站上发布结果。