RUI: Algebraic Dynamics of Knot Theory

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结和环引起的代数。D元Laurent多项式环上的模被它们的Pontryagin对偶取代，这些对偶是紧交换群。变量的作用变成了可交换的同胚，以及相应的动力学不变量(例如，周期点数、拓扑熵等)。提供了新的拓扑不变量。重点将放在Mahler测度、曲面动力学和扭曲同调上。该项目将加强纽结理论、动力系统和数论之间新出现的联系。尽管在DNA、太阳耀斑和流体流动等一系列科学现象中出现了纽结和联系，但它们的数学研究相当新。在许多数学领域的想法的推动下，纽结理论在过去20年里取得了爆炸性的进展，现在这个主题吸引了物理学家、生物学家、工程师和数学家的广泛兴趣。这个项目将继续研究人员应用另一个数学领域--动力系统的想法，以获得一个新的视角。将使用计算机方法来开发假说。PIS还将撰写一本专著，这是这项跨学科研究的第一次调查，专为研究生和研究人员设计。

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.