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. 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.