CAREER: Hyperbolic geometry and knots and links

自20世纪80年代初以来，人们就已经知道，结和连接互补体分解成承认几何结构的碎片，最常见的几何结构是双曲。然而，双曲几何与其他结点和连杆不变量之间的联系仍然没有得到很好的理解。研究者将使用3流形几何和拓扑的最新发展和技术来建立双曲几何和链接不变量之间的联系，重点关注两个领域的问题。首先，她将把双曲几何与量子不变量联系起来，特别是继续她最近的工作，寻找双曲几何与有色琼斯多项式之间的联系。其次，她将根据结点和连杆的图解和拓扑不变量获得双曲量的界，例如边界体积和尖点体积，并找到测地线的同位素类。作为这项工作的教育部分的一部分，她将组织两次关于双曲几何连接的会议，并继续与本科生，研究生和K-12学生一起工作。这个项目涉及三维空间的研究，称为三维流形，其中包括我们的宇宙空间（具有三维空间）。这些空间出现在物理学、力学、微生物学和化学中，因此我们希望更好地理解它们的数学性质。研究3-流形的一种方法是从三维球体上钻出绕着圆的管子，然后以另一种方式重新连接管子。钻管绕圆的空间称为环补，如果只有一个圆，则称为结补。这样，结补和连杆补就是3-流形的组成部分。研究者将研究结补和连杆补的几何，希望在更广泛的3-流形类别的性质上找到新的结果。这个项目包括许多训练学生的规定。大部分研究将在本科生和研究生的协助下进行。此外，作为该项目的一部分，研究者将举办两次研究会议，期间将邀请几位研究生和博士后研究人员介绍他们的相关工作。最后，研究者将在整个学年继续为孩子们举办数学研讨会。

It has been known since the early 1980s that knot and link complements decompose into pieces admitting a geometric structure, with the most common geometry being hyperbolic. However, the connection between hyperbolic geometry and other knot and link invariants is still not well understood. The investigator will use recent developments and techniques in 3-manifold geometry and topology to make connections between hyperbolic geometry and link invariants, focusing on problems in two areas. First, she will relate hyperbolic geometry to quantum invariants, in particular continuing her recent work to find connections between hyperbolic geometry and the colored Jones polynomial. Second, she will obtain bounds on hyperbolic quantities based on diagrammatical and topological invariants of knots and links, for example bounding volume and cusp volume, and finding isotopy classes of geodesics. As part of the educational portion of this work, she will organize two conferences on connections of hyperbolic geometry, and continue her work with undergraduate, graduate, and K-12 students.This project concerns the study of 3-dimensional spaces called 3-manifolds, which include the space of our universe (with three spatial dimensions). These spaces appear in physics, mechanics, microbiology, and chemistry, and so we wish to better understand their mathematical properties. One way to study 3-manifolds is to drill out tubes around circles from a 3-dimensional sphere, and then reattach the tubes in a different manner. The space of drilled tubes about circles is called a link complement, or knot complement if there is just one circle. In this way, knot and link complements are building blocks for 3-manifolds. The investigator will study the geometry of knot and link complements, with the hope of finding new results on the properties of broader classes of 3-manifolds. This project includes many provisions for training students. Much of the research will be carried out with the assistance of undergraduate and graduate students. In addition, as part of this project the investigator will run two research conferences, during which several graduate students and postdoctoral researchers will be invited to present their related work. Finally, the investigator will continue to run mathematical workshops for children throughout the school year.