Braids and Contact Geometry

编织物和接触几何形状

• 批准号: 1206770
• 负责人: Keiko Kawamuro
• 金额: $ 13.45万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1206770&HistoricalAwards=false
• 关键词: Braids; Contact; Geometry

The principal Investigator (PI) plans to apply the theory of geometric braids to study a variety of problems in contact geometry. Especially, she introduces a new notion, open book foliation, that has its origin in braid theory. She intends to apply the open book foliation to investigate contact structures for three dimensional manifolds and find new tightness and overtwistedness criteria. In addition, the PI continues to study the relationship between the braid index, a classical knot invariant, and various geometric quantities such as writhes, HOMFLY polynomials, the self-linking number in contact geometry, and Khovanov-Rozansky homology. The PI will disseminate the results of her research by advising her current and potential graduate students at the University of Iowa, and by interacting with both undergraduate and graduate students through her seminars and lectures. The projects afford undergraduate and graduate students the opportunity to study contact geometry from both a theoretical and practical point of view. Developing computer graphic techniques with a view towards visualizing the open book foliations will provide students with an excellent introduction to a modern mathematicsThe PI will work on several projects in low-dimensional topology. Low dimensional topology is broadly applicable in a variety of scientific disciplines including biology and medicine where, for example, protein folding mechanisms and DNA structure play a seminal role. Results of the proposed projects will likely stimulate communication and collaboration between the PI and researchers within mathematics as well as other scientific disciplines.

主要研究者（PI）计划应用几何编织理论研究接触几何中的各种问题。特别是，她介绍了一个新的概念，打开书叶理，它起源于辫子理论。她打算应用开卷叶理来研究三维流形的接触结构，并找到新的紧性和过扭性准则。 此外，PI继续研究辫子指数，一个经典的结不变量，和各种几何量之间的关系，如writes，HOMFLY多项式，接触几何中的自链接数，和Khovanov-Rozansky同调。PI将通过为爱荷华州大学现有和潜在的研究生提供建议，并通过研讨会和讲座与本科生和研究生进行互动，传播她的研究成果。这些项目为本科生和研究生提供了从理论和实践角度研究接触几何的机会。开发计算机图形技术，以期可视化的开放式图书叶理将为学生提供一个很好的介绍现代marticsThe PI将在低维拓扑结构的几个项目。低维拓扑广泛应用于各种科学学科，包括生物学和医学，例如，蛋白质折叠机制和DNA结构起着开创性的作用。拟议项目的结果可能会刺激PI和数学以及其他科学学科的研究人员之间的沟通和合作。