Iva Halacheva

Sponsor: NSF

Categorical centers, cactus actions, and diagram algebras

This project delves into several research directions within representation theory, which is the mathematical framework for studying objects through their symmetries and the operations which preserve them. Such operations can carry a classical, or even more intriguingly a quantum algebraic structure. Originally appearing in physical models within statistical mechanics and quantum integrable systems, quantum groups and the theory surrounding them are now a thriving source of uncovering new mathematical principles. This project will develop a richer understanding of this theory by building a common ground for combining algebraic, combinatorial, and higher-structural categorical techniques for the study of quantum groups and associated diagram algebras. This will lead to a more unified approach and provide connections between several areas of mathematics, as well as potential physical applications. The project will involve the participation of undergraduate students and create opportunities for discussion and collaboration among early-career researchers.