REU Site: Investigations in Geometry and Knot Theory

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This program at California State University San Bernardino is an immersive research experience for undergraduate students. In each of three summers, eight participants will learn background material in differential geometry and knot theory and pursue open-ended research questions. Questions under study in differential geometry range widely, from determining efficient methods of expressing curvature, to developing theories regarding the types of curvature that one might possibly encounter, and even geometrically realizing these curvatures. Questions under study in knot theory will emphasize the interplay between topology and geometry. As geometric applications to topology are some of the most subtle and significant discoveries in the last thirty years, these questions are part of an active and vibrant area of mathematical research and are also of interest because of their potential applications. Student participants will be recruited nationally, with emphasis on recruitment from those institutions that have limited research opportunities for their students, and from populations underrepresented in STEM disciplines.The differential geometry component of the project has three main topics. The Riemann curvature tensor is an object that encodes a manifold’s curvature at every point. It is known that this object can be expressed as a combination of other types of curvatures, and students will explore how different curvature tensors could be expressed according to this decomposition. Second, students will investigate novel invariants for curvature tensors. Thirdly, students will seek geometric realizations illustrating these concepts. The knot theory component of this project focuses on hyperbolic links—links whose complements admit a hyperbolic structure. Projects will focus on the class of links called fully augmented links, which have particularly tractable geometric structures but are nevertheless quite useful since they can be used to construct all links via Dehn filling. Both fields are rich with a variety of questions to explore.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.