LEAPS-MPS: Categorification in Algebra and Topology

About This Grant

Categorification refers to the systematic process of upgrading a mathematical object to one with additional higher-level structure. This idea has proven to be a powerful tool in resolving numerous deep conjectures in mathematics. In this project, the investigator will use categorification techniques to study knot theory. Knot theory aims at understanding the structure of tangled-up loops in space. It has many applications in biology, medicine, chemistry, and physics. This research is expected to create new tools to address challenging open problems in knot theory. Additionally, it will deepen our understanding of the connections between different subfields of mathematics such as algebra and geometry. The investigator will engage students across multiple stages of their educational development (K-12, undergraduate, graduate) to contribute to the research. A key component of the grant activity will be the expansion of the MaPP Challenge, a mathematics puzzle-based scavenger hunt designed by MaPP (Mathematical Puzzle Programs). Data suggests that the MaPP Challenge serves as a useful tool for broadening participation in STEM, since it appeals to students who have not yet developed an appreciation for mathematics. The first goal of the project is to study the geometric, topological, and combinatorial properties of Springer fibers and their generalizations. These algebraic varieties naturally arise in Lie theory and play an important role in Springer’s geometric construction and classification of the representations of Weyl groups. Springer fibers give rise to geometrically defined convolution algebras. The second goal of the project is to use these algebras to construct new homological invariants of knots and links. The algebra underlying Khovanov’s categorification of the Jones polynomial admits a geometric realization in terms of Springer fibers of type A. This project aims to extend this framework by exploring how convolution algebras for Springer fibers of Lie types B, C, and D relate to invariants of knots and links in more general ambient spaces, including manifolds and orbifolds beyond standard Euclidean three-space. 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.