Grant: Discovery Grants Program - Individual

The field of representation theory can be thought of as the study of symmetry and duality in mathematics and physics. Certain groups of symmetries, known as Lie groups, play a vital role in the study of the physical laws that govern our universe. In the quantum world, Lie groups are replaced by quantum groups. Developing a better understanding of these mathematical objects results in a deeper and more sophisticated knowledge of the world around us. While traditional algebra is one-dimensional, monoidal categories can be thought of as a precise way of doing two-dimensional algebra. Together with related concepts in category theory, they provide powerful tools for studying abstract algebra, geometry, topology, and combinatorics. They also provide an intuitive diagrammatic calculus that can be used to simplify technical algebraic arguments and to make apparent the deep connection between different fields in mathematics. One important use of monoidal categories is in the study of a phenomenon known as a categorification, an exciting and rapidly developing field of mathematics. The philosophy of categorification is that many mathematical structures that appear to be fundamental are, in fact, mere shadows of a higher and richer mathematical reality. One of my research objectives is to uncover this hidden higher structure, thus revealing a deeper, more unified mathematical framework. Over the next 5 years, I plan to focus on three areas of categorical representation theory. First, I will develop "odd" analogues of existing results in categorification. These odd versions can be thought of as mirror images, taking place in a world where things behave in an opposite fashion. Second, I will undertake a systematic analysis of the types of dualities and symmetries that are possible in certain monoidal categories. Finally, I will unify and generalize many of the techniques of categorical representation theory, combining several different theories into a single consolidated approach. I expect the outcomes of this research to have significant impact in the fields of representation theory, geometry, topology, combinatorics, and mathematical physics. By involving graduate students and postdoctoral researchers, it will also serve to train new mathematicians in these exciting areas.