RUI: Algebraic Dynamics of Knot Theory

RUI：结理论的代数动力学

• 批准号: 0706798
• 负责人: Daniel Silver
• 金额: $ 22.76万
• 依托单位: University of South Alabama
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-15 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706798&HistoricalAwards=false
• 关键词: RUI; Algebraic; Dynamics; Knot; Theory

The PIs will continue their program to understand the algebra arising from knots and links by using new techniques of algebraic dynamics. Modules over Laurent polynomial rings in d variables are replaced by their Pontryagin duals, which are compact abelian groups. Actions by the variables become commuting homeomorphisms, and corresponding dynamical invariants (e.g., numbers of periodic points, topological entropy, etc.) provide new topologicial invariants.The focus will be on Mahler measure, surface dynamics and twisted homology. The project will strengthen newly emerging connections between knot theory, dynamical systems and number theory.Although knots and links arise in a host of scientific phenomena such as DNA, solar flares and fluid flows, their mathematical study is fairly new. Fueled by ideas from many areas of mathematics, progress in knot theory has exploded during the past two decades, and the subject now attracts wide interest from physicists, biologists and engineers as well as mathematicians. This project will continue the investigators' program of applying ideas from another mathematical field, dynamical systems, in order to gain a new perspective. Computer methods will be used to develop hypotheses. The PIs will also write a monograph, the first survey of this interdisciplinary study, designed for graduate students and researchers.

PI将继续他们的计划，通过使用代数动力学的新技术来理解由节点和链接产生的代数。 d元Laurent多项式环上的模被其Pontryagin群所取代，它们是紧交换群。变量的作用成为交换同胚，以及相应的动力学不变量（例如，周期点的数量、拓扑熵等）提供新的拓扑不变量。重点将放在马勒措施，表面动力学和扭曲的同源。该项目将加强纽结理论、动力系统和数论之间的新兴联系。尽管纽结和链接出现在DNA、太阳耀斑和流体流动等许多科学现象中，但它们的数学研究还是相当新的。在许多数学领域的思想的推动下，纽结理论在过去的二十年中取得了爆炸性的进展，这个主题现在吸引了物理学家，生物学家和工程师以及数学家的广泛兴趣。该项目将继续研究人员的应用程序从另一个数学领域，动力系统的想法，以获得一个新的视角。 计算机方法将被用来发展假设。PI还将撰写专着，这是为研究生和研究人员设计的跨学科研究的第一次调查。