REU Site: Investigations in Geometry and Knot Theory

REU网站项目，几何和纽结理论研究，是一个为期8周的项目，在加州州立大学圣贝纳迪诺的8名本科生。参与者将学习两个主题。首先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问题涉及通过将流形作为超曲面嵌入到平坦空间中而获得的各种正则曲率张量之间的相互作用，特别是关于根据代数曲率张量的线性无关性来展示可能遇到的任何曲率的效率。虽然这些问题的范围很广，但这个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，在某些一般情况下，这些结构会分解，但在感兴趣的特定情况下，关于这种分解的性质仍有许多问题。除此之外，还有充分的空间来发现具有指定曲率性质的新流形。第二个研究课题是双曲纽结的研究。双曲几何的主题非常丰富，融合了代数、几何和拓扑技术。此外，这一理论已经足够成熟，足以为数学成熟的本科生提供丰富的问题。该程序的纽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环的辫子理论描述之间的关系。第一个涉及双曲线闭合三个辫子的体积，第二个涉及对双曲节的闭合辫子代表进行分类。学员将被介绍到两个活跃的数学领域，几何和纽结理论，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纽结理论的项目，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会发生重大的数学互动。此外，参与者将在一个丰富的环境中与他们的导师密切合作，完成与其主题相关的背景阅读，就相关材料进行演讲，进行研究，并开始撰写期刊式的论文。随着暑假的进行，学生们将进行自己的文献搜索，进行独立的发现，并从事创造性的数学研究。除了常规的报告和论文作业外，每个学生还将制作一张海报，描述他们的结果，向加州州立大学圣贝纳迪诺分校的校园社区做20分钟的期末报告，并完成一篇关于他们项目的日记风格的论文。因此，参与者将有一个全面和队列的研究经验。该计划将通过积极让本科生参与数学研究并强烈鼓励他们成为数学社区的积极参与者来促进发现。鼓励少数民族服务机构的学生申请。此外，加州州立大学圣贝纳迪诺分校的多样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群体的影响。最后，该方案有一个广泛传播的多方面计划，以增进科学理解。传播途径包括会议报告、提交发表以及在该计划的网站上发布结果。

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.