On the main, this research project lies in the broad interdisciplinary area between geometric topology and quantum physics. Many of the motivating conjectures come from physics, and their mathematical solutions would be of interest to theoretical physicists. A second part of the project concerns the topological characteristics of biopolymers like DNA and proteins. This can be important from a pharmaceutical perspective, as some drugs can be designed to target topological characteristics which affect specific biological functions. Besides its research goals, the project has a strong educational component. Many of the proposed problems are intended for research with undergraduate students. Through her research, teaching and other outreach activities, the PI intends to expand the reach of mathematics, for example to historically under-represented groups and to other audiences not usually exposed to cutting edge mathematics.
This project will explore the extent to which the Kauffman skein algebra and its generalizations can serve as intermediaries between quantum topology and hyperbolic geometry. The PI will study the representation theory of the Kauffman skein algebra, paying particular attention to the representation coming from the Witten-Reshetikhin-Turaev theory. The long-term, overarching goal is to construct and classify all representations of the Kauffman skein algebra, a goal which this project will advance. The project considers the algebraic structure of the Kauffman skein algebra and of its generalizations (e.g., ones that allow arcs on the surface). In addition, the project includes problems investigating which types of topologically complex structures, like knots, links, and non-planar graphs, are possible in biopolymers like DNA and proteins.
