Braids and Contact Geometry

首席研究员（PI）计划应用几何辫理论研究接触几何中的各种问题。特别是，她引入了一个新的概念——开卷叶化，它起源于编织理论。她打算应用开卷叶理来研究三维流形的接触结构，并寻找新的紧度和过扭度标准。此外，PI继续研究辫指数（经典结不变量）与各种几何量（如wrthes、HOMFLY多项式、接触几何中的自连接数和Khovanov-Rozansky同调）之间的关系。PI将通过向她在爱荷华大学的在读和潜在研究生提供建议，并通过她的研讨会和讲座与本科生和研究生进行互动，来传播她的研究成果。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。发展计算机图形技术，将打开的书的叶子可视化，将为学生提供一个很好的现代数学入门。PI将在几个低维拓扑的项目中工作。低维拓扑学广泛应用于包括生物学和医学在内的各种科学学科，例如蛋白质折叠机制和DNA结构在这些学科中起着重要作用。拟议项目的结果可能会促进PI与数学以及其他科学学科的研究人员之间的交流和合作。

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.