REU Site: Investigations in Geometry and Knot Theory

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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.