RUI: Skeins on Surfaces

本课题主要处于几何拓扑学与量子物理的广泛交叉领域。许多激励性的猜想来自物理学，它们的数学解会引起理论物理学家的兴趣。该项目的第二部分涉及DNA和蛋白质等生物聚合物的拓扑特征。从药学的角度来看，这是很重要的，因为一些药物可以设计成针对影响特定生物功能的拓扑特征。除了研究目标之外，该项目还具有很强的教育成分。许多提出的问题是为本科生研究的。通过她的研究、教学和其他外联活动，PI打算扩大数学的影响范围，例如，向历史上代表性不足的群体和其他通常不接触前沿数学的受众。该项目将探索考夫曼交织代数及其推广在多大程度上可以作为量子拓扑和双曲几何之间的中介。PI将研究Kauffman skein代数的表示理论，特别关注来自Witten-Reshetikhin-Turaev理论的表示。长期的总体目标是构建和分类考夫曼skein代数的所有表示，这是该项目将推进的目标。该项目考虑了考夫曼编织代数的代数结构及其推广（例如，允许在表面上有弧的代数结构）。此外，该项目还包括研究哪些类型的拓扑复杂结构，如结、链接和非平面图形，可能存在于DNA和蛋白质等生物聚合物中。

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.