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.

总的来说，该研究项目属于几何拓扑和量子物理学之间广泛的跨学科领域。许多令人兴奋的猜想都来自物理学，它们的数学解决方案将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。从制药的角度来看，这可能很重要，因为一些药物可以针对影响特定生物功能的拓扑特征进行设计。 除了其研究目标外，该项目还具有很强的教育成分。 提出的许多问题是供本科生研究的。通过她的研究、教学和其他外展活动，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通常不接触前沿数学的其他受众开放。该项目将探索考夫曼绞线代数及其推广在多大程度上可以充当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考夫曼绞线代数的所有表示并对其进行分类，这是该项目将推进的目标。该项目考虑了考夫曼绞丝代数的代数结构及其概括（例如，允许表面存在弧的代数）。 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）在 DNA 和蛋白质等生物聚合物中可能存在的问题。