REU Site: Investigations in Geometry and Knot Theory

REU Sites项目“几何和结理论研究”是一个为期8周的项目，招收加州州立大学圣贝纳迪诺分校的8名本科生。该计划旨在进一步加深我们对两个充满活力的数学领域的理解：微分几何和结理论。在每一个夏季，8名参与者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。一组学生将在每个领域工作，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。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。总的来说，这两个领域都有丰富的问题需要探索。在微分几何领域提出的未解决的问题范围很广，从确定表达曲面曲率的有效方法，到发展关于在任何特定曲面上可能遇到的曲率类型的理论。特别强调的是，我们的宇宙本身就是这种表面的一个例子！这个项目的组件有两个主要目标。在任何光滑表面上，黎曼曲率张量是一个对象，它编码表面上每一点的曲率。众所周知，这个物体可以被表示为其他类型曲率的组合，我们的目标是了解不同曲率张量如何根据这种分解来表示的本质。先前的结果表明，这与将表面嵌入到平坦空间的能力之间存在深刻的关系，更好地理解这种关系既旨在区分曲率张量，也有助于阐明嵌入问题。第二个目标是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为一个理论，并举例说明该理论的不同方面。这个项目的结理论部分着重于涉及结不变量的问题。总的来说，结理论在重组DNA和合成化学中有应用；然而，参与者将把注意力集中在与链接的琼斯多项式相关的几个重要的结不变量上。由于琼斯多项式是近三十年来最微妙和最重要的发现之一，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，数学家对这些问题感兴趣，因为他们通常寻求理解宇宙如何运作的结构，但他们也对科学界感兴趣，因为我们的发现的潜在应用。

The REU Sites project "Investigations in Geometry and Knot Theory" is an 8-week program for 8 undergraduates at California State University, San Bernardino. The program 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. A group of students will work in each 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. 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.Overall, both fields are rich with a variety of questions to explore. The unsolved problems presented in the field of Differential Geometry range widely from determining efficient methods of expressing curvature of surfaces, to developing theories regarding the types of curvature that one might possibly encounter on any particular surface. It is of particular emphasis that an example of the type of surface one might consider is our universe itself! This component of the project has two main goals. On any smooth surface, the Riemann Curvature Tensor is an object that encodes the surface's curvature at every point on the surface. It is known that this object can be expressed as a combination of other types of curvatures, and we aim 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 ability to embed the surface into flat space, and a better understanding of this relationship both aims to distinguish between curvature tensors and sheds light on this embedding question. 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 problems involving knot invariants. In general, knot theory has applications in recombinant DNA and synthetic chemistry; however, participants will focus their attention on several significant knot invariants related to the Jones polynomial of a link. As the Jones polynomial is one 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.