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，调查结理论中的开放性问题。特别是，他们将调查莱默问题。70年前，D.H. Lehmer开始用精确意义上的小多项式构造大素数：马勒测度接近但不同于1的积分多项式。Lehmer只能算出1.17628。这个值是他用一个非凡的10次多项式得到的。然后他问这个价值是否可以提高。尽管许多人尽了最大努力，莱默的问题仍然没有解决。莱默多项式继续出现在令人惊讶的分离领域中。研究者将从拓扑学和几何学的角度来研究莱默问题。他们将研究马勒测度在结和连杆研究中的应用。他们的方法将包括来自交换代数、群论、几何和拓扑的传统方法，以及许多来自动力系统的新技术。计算机方法将用于开发示例。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，吸引了许多领域的科学家的兴趣，包括生物学、化学和物理学。一些吸引人的原因不难看出。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，研究结论中的开放性问题。本研究将利用符号动力系统（信息论的一个数学分支）的技术，为结点和联系提供新的理解。它将促进和加强不同领域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.