Mathematical Sciences: Symmetry Groups in Dynamics and Topology

本课题研究位移动力学的对称群 系统和光滑流形。 代数K理论的思想是 适用于每一个领域，并形成一个共同的主题。 研究者将继续研究代数和 一维次对称的同调性质 有限型移位（SFT）、sofic移位和其他动态移位 系统. 这些系统出现在许多情况下，包括 动力学、信息论和统计力学。 他们的 对称性通常称为可逆或可逆细胞对称性 自动机 他还将研究类似的问题， 维马尔可夫位移及其对称性。 经典的亚历山大多项式的一个结在三个- 维空间与代数K理论有着密切的联系 K1组。 最近，又发现了其他的多项式不变量 从沃恩·琼斯的作品开始 大范围内流形的伪合线与伪同胚 的情况下（包括结补）有不变量涉及 代数K理论群K2。 调查员会检查 是否可以定义新的不变量 中纽结补及其它3-流形的同构 与新的纽结多项式相似。 这里的方法的共性是令人惊讶的。 了深刻和 高度发展的代数理论，K理论，将被带到 与对称性问题有关， 外观. 一方面是微妙的拓扑问题 关于Knots 另一方面是微妙的组合， 关于无限符号串的统计问题 的 后一种应用已取得重大成果。

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.