REU Site: Investigations in Geometry and Knot Theory

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The REU program "Investigations in Geometry and Knot Theory" 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. The unsolved problems presented in the field of Differential Geometry range widely from determining efficient methods of expressing various sorts of curvature, to developing their own theories regarding the types of curvature that one might possibly encounter and even geometrically realizing these curvatures. The other subject of study is knot theory, or the study of closed, knotted loops in space. Knot and link complements provide excellent examples of three-dimensional manifolds, and the field of three-dimensional topology was revolutionized in the late 1970's and 80's following the discovery of deep connections with geometry. The revolution was spearheaded by the work of William P. Thurston on geometric structures, for which he was awarded the Fields Medal in 1982. The unsolved problems in knot theory will emphasize this 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 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.The Differential Geometry component of this project has two main goals. On any smooth manifold, the Riemann Curvature Tensor is an object that encodes the surface's curvature at every poin. It is known that this object can be expressed as a combination of other types of curvatures, and the aim is 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 number of extra dimensions one might need to embed your manifold as a subset of Euclidean space. 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 hyperbolic links-links whose complements admit a hyperbolic structure. The geometric revolution in three-manifold theory demonstrated that "most" links are hyperbolic, and Mostow-Prasad rigidity implies that geometric quantities are topological invariants. Determining and understanding hyperbolic structures on link complements, then, becomes an important problem in knot theory. Knot theory projects will focus on the class of links called fully augmented links-links that have particularly tractable geometric structures, which can be understood by undergraduates from many perspectives. Despite their geometric simplicity, fully augmented links are quite useful since they can be used to construct all links via Dehn filling. Thus projects will further uncover the geometric structure of fully augmented (and related) link complements, and apply this knowledge to links in general via Dehn filling. Overall, 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.