Mathematical Sciences: Symmetry Groups in Dynamics and Topology

数学科学：动力学和拓扑中的对称群

• 批准号: 9102959
• 负责人: John Wagoner
• 金额: $ 9.15万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-07-01 至 1995-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9102959&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Symmetry; Groups; Dynamics

This project concerns the symmetry groups of shift dynamical systems and smooth manifolds. Ideas from algebraic K-theory are applicable to each area and form a common theme between them. The investigator will continue to study the algebraic and homological properties of symmetries of one-dimensional sub- shifts of finite type (SFT's), sofic, and other shift dynamical systems. These systems arise in a number of contexts, including dynamics, information theory, and statistical mechanics. Their symmetries are often called invertible or reversible cellular automata. He will also study similar questions for higher dimensional Markov shifts and their symmetries. The classic Alexander polynomial of a knot in three- dimensional space is closely linked to the algebraic K-theory group K1. Recently, other polynomial invariants have been found for these knots, starting with the work of Vaughan Jones. Pseudo-isotopies and diffeomorphisms of manifolds in a wide range of cases (including knot complements) have invariants involving the algebraic K-theory group K2. The investigator will examine whether it is possible to define new invariants of diffeomorphisms of knot complements and other 3-manifolds in analogy with the new knot polynomials. The commonality of method here is surprising. A deep and highly evolved algebraic theory, K-theory, is to be brought to bear upon symmetry questions which are very far apart in appearance. On the one hand are subtle topological questions about knots. On the other hand are subtle combinatorial and statistical questions about infinite strings of symbols. The latter application has already borne substantial fruit.

本课题研究平移动力系统和光滑流形的对称群。代数k理论的思想适用于每个领域，并在它们之间形成一个共同的主题。研究者将继续研究有限型（SFT's）、sofic和其他位移动力系统的一维子位移对称性的代数和同调性质。这些系统出现在许多环境中，包括动力学、信息论和统计力学。它们的对称性通常被称为可逆或可逆元胞自动机。他还将研究高维马尔可夫位移及其对称性的类似问题。三维空间中结的经典Alexander多项式与代数k理论群K1密切相关。最近，从Vaughan Jones的工作开始，已经发现了这些结的其他多项式不变量。流形的伪同位素和微分同态在广泛的情况下（包括结补）具有涉及代数k理论群K2的不变量。研究者将研究是否有可能用新的结多项式类比来定义结补和其他3流形的微分同态的新不变量。这里的方法的共性令人惊讶。一种深刻而高度发展的代数理论，即k理论，将用于研究表面上相距甚远的对称问题。一方面是关于结的微妙的拓扑问题。另一方面是关于无限符号串的微妙的组合和统计问题。后一种应用已经产生了丰硕的成果。