RUI: Applications of Symbolic and Algebraic Dynamics to Knot Theory

RUI：符号和代数动力学在纽结理论中的应用

• 批准号: 0304971
• 负责人: Daniel Silver
• 金额: --
• 依托单位: University of South Alabama
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0304971&HistoricalAwards=false
• 关键词: RUI; Applications; Symbolic; Algebraic; Dynamics

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.

研究人员将联合收割机结合他们的两个领域的专业知识，拓扑和动力系统，调查悬而未决的问题在结理论。特别是，他们将调查莱默的问题。70年前，D。莱默开始使用精确意义上的小多项式来构造大素数：马勒测度接近但不同于1的整数多项式。Lehmer不能做得比1.17628更好，一个他用10次多项式得到的值。然后他问这个值是否可以提高。尽管许多人尽了最大的努力，莱默的问题仍然悬而未决。Lehmer多项式继续出现在分隔开的领域中。研究者们将从拓扑学和几何学的角度对Lehmer问题进行探讨。他们将研究马勒测度在纽结和链环研究中的应用。他们的方法将包括传统的从交换代数，群论，几何和拓扑学，以及新的技术，许多从动力系统。计算机方法将被用来开发的例子。数学理论的纽结产生于物理理论的世纪。从那时起，这一领域得到了极大的发展，吸引了包括生物学、化学和物理学在内的许多领域的科学家的兴趣。这种吸引力的一些原因并不难看出。例如，DNA、太阳等离子体细丝和流体流动都表现出打结或连接行为。研究人员将联合收割机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悬而未决的问题。这项研究将通过使用信息论的数学分支--符号动力系统中的技术，为节点和链接提供新的理解。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。