CAREER: Hyperbolic geometry and knots and links

职业：双曲几何以及结和链接

• 批准号: 1252687
• 负责人: Jessica Purcell
• 金额: $ 37.32万
• 依托单位: Brigham Young University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-06-01 至 2016-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1252687&HistoricalAwards=false
• 关键词: CAREER; Hyperbolic; geometry; knots; links

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.

自1980年代初以来，人们就已经知道结结和链接的补充，分解为承认几何结构的碎片，最常见的几何形状是双曲线。 但是，双曲线几何形状与其他结和链接不变性之间的连接仍然不太了解。研究人员将在3型manifold几何形状和拓扑中使用最新的发展和技术，以在双曲几何形状和链接不变性之间建立连接，重点关注两个领域的问题。 首先，她将将双曲线几何形状与量子不变性相关联，特别是继续她最近的工作，以找到双曲线几何形状与彩色琼斯多项式之间的联系。 其次，她将基于结和链路的幻想和拓扑不变的额外数量的界限，例如边界体积和尖端体积，以及查找地球化学的同位素类别。 作为这项工作的教育部分的一部分，她将组织两个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。该项目涉及对称为3个manifolds的三维空间的研究，其中包括我们宇宙的空间（包括三个空间上的较高的尺寸）。 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特性。 研究3个序列的一种方法是从3维球体钻出圆圈，然后以不同的方式重新启动管子。 关于圆圈的钻管空间称为链路补充，或者如果只有一个圆，则结节补充。 这样，结和链接补充是3个manifolds的基础。 研究人员将研究结和链接补充的几何形状，希望找到有关更广泛类别的3个manifolds的特性的新结果。 该项目包括许多培训学生的规定。 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，研究人员将举办两个研究会议，在此期间，将邀请几名研究生和博士后研究人员介绍其相关工作。 最后，调查员将在整个学年继续为儿童举办数学研讨会。