RUI: Skeins on Surfaces

RUI：表面上的绞纱

• 批准号: 1510453
• 负责人: Helen Wong
• 金额: $ 16万
• 依托单位: Carleton College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-15 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1510453&HistoricalAwards=false
• 关键词: RUI; Skeins; Surfaces

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.

从主要的角度来看，该研究项目位于几何拓扑与量子物理学之间的广泛跨学科领域。许多激励猜想来自物理学，其数学解决方案将引起理论物理学家的关注。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。从药物的角度来看，这可能很重要，因为某些药物可以设计用于影响特定生物学功能的拓扑特征。 除了其研究目标外，该项目还具有强大的教育部分。 许多提议的问题旨在与本科生进行研究。通过她的研究，教学和其他外展活动，PI打算扩大数学的覆盖范围，例如历史上代表性不足的群体，以及通常不接触过最前沿数学的其他受众。该项目将探索Kauffman Skeinskein代数及其概括在多大程度上可以作为Intermediaries在多种多样的地理上和超级替代层次。 PI将研究Kauffman Kkein代数的代表理论，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Kauffman Skein代数的所有代表，这是该项目将推进的目标。该项目考虑了Kauffman Skein代数及其概括的代数结构（例如，允许表面弧的弧线）。 此外，该项目还包括研究在DNA和蛋白等生物聚合物中可能在拓扑复杂的结构（例如结，链接和非平面图）的问题。