RUI: Applications of Symbolic and Algebraic Dynamics to Knot Theory

研究人员将结合他们的两个专业领域--拓扑学和动力系统--来研究纽结理论中的未决问题。特别是，他们将调查莱默的问题。七十年前，D.H.莱默开始使用精确意义上的小的多项式来构造大的素数：马勒度量接近但不同于1的整多项式。莱默的计算结果不会超过1.17628……，他用一个了不起的10次多项式得到了这个值。然后他问这个值是否还能提高。尽管许多人尽了最大努力，莱默的问题仍然悬而未决。莱默多项式继续出现在令人惊讶的分开的田野中。研究人员将从拓扑学和几何学的角度来攻击Lehmer的问题。他们将研究马勒度量在节点和链环研究中的应用。他们的方法将包括交换代数、群论、几何学和拓扑学的传统方法，以及许多来自动力系统的新技术。将使用计算机方法来开发实例。纽结的数学理论起源于19世纪的物理理论。从那时起，这一领域得到了极大的扩展，吸引了包括生物、化学和物理在内的许多领域的科学家的兴趣。吸引人的一些原因并不难看出。例如，DNA、太阳等离子体细丝和流体都表现出打结或连接的行为。研究人员将结合他们的两个专业领域--拓扑学和动力系统--来研究纽结理论中的未决问题。这项拟议的研究将通过使用符号动力学系统的技术来提供对节点和链接的新理解，符号动力学系统是信息论的一个数学分支。它将促进和加强不同领域的研究人员之间的互动。本科生和研究生将参与该项目。

The investigators will combine their two areas of expertise,topology and dynamical systems, to investigate open questionsin knot theory. In particular, they will investigate Lehmer'sQuestion. Seventy years ago D.H. Lehmer began constructing large primenumbers using polynomials that are small in a precise sense:integral polynomials with Mahler measure close to butdifferent from 1. Lehmer could do no better than 1.17628...,a value that he achieved with a remarkable polynomial ofdegree 10. He then asked if that value could improved.Lehmer's Question remains open despite the best efforts ofmany. Lehmer's polynomial continues to appear in surprisinglyseparated fields. The investigators will attack Lehmer's Question from theperspective of topology and geometry. They will investigateapplications of Mahler measure to the study of knots andlinks. Their methods will include traditional ones fromcommutative algebra, group theory, geometry and topology, aswell as new techniques, many from dynamical systems. Computermethods will be used to develop examples.The mathematical theory of knots arose fromphysical theories of the nineteenth century. Since then,the field has expanded greatly, attracting the interestsof scientists in many fields, including biology, chemistryand physics. Some reasons for the attraction are not hard tosee. DNA, solar plasma filaments and fluid flow, for example,all exhibit knotting or linking behaviour. The investigatorswill combine their two areas of expertise, topology anddynamical systems, to investigate open questions in knottheory. The proposed research will provide new understandingof knots and links by using techniques from symbolicdynamical systems, a mathematical branch of informationtheory. It will promote and strengthen interaction betweenresearchers in different fields. Undergraduate and graduatestudents will participate in the project.