CAREER: Hyperbolic geometry and knots and links

人们从20世纪80年代初就知道，纽结和链接补语会分解成碎片，形成几何结构，其中最常见的几何图形是双曲线。然而，双曲几何与其他纽结和链接不变量之间的联系仍然没有被很好地理解。研究人员将利用三维流形几何和拓扑学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，她将把双曲几何和量子不变量联系起来，特别是继续她最近的工作，寻找双曲几何和有色琼斯多项式之间的联系。其次，她将根据纽结和链环的图解和拓扑不变量，例如边界体积和尖点体积，以及寻找测地线的等规类，获得双曲线量的界限。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联系的会议，并继续她与本科生、研究生和K-12学生的工作。这个项目涉及被称为三维流形的三维空间的研究，其中包括我们宇宙的空间(具有三个空间维度)。这些空间出现在物理、力学、微生物学和化学中，因此我们希望更好地了解它们的数学性质。研究三维流形的一种方法是在三维球体的圆形周围钻出管，然后以不同的方式重新连接管。围绕圆钻出的管子的空间称为链节补，如果只有一个圆，则称为结补。在这种情况下，纽结和链补是3-流形的构建块。研究人员将研究纽结和链补的几何，希望在更广泛的3-流形类的性质上找到新的结果。这个项目包括了许多培训学生的条款。大部分研究将在本科生和研究生的帮助下进行。此外，作为该项目的一部分，研究人员将举办两次研究会议，期间将邀请几名研究生和博士后研究人员介绍他们的相关工作。最后，调查员将在整个学年继续为儿童举办数学讲习班。

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.