Mathematical Sciences: Symmetry Groups in Dynamics and Topology

本课题研究位移动力学的对称群 系统、光滑流形和算子代数。 想法从 代数K-理论适用于每个领域，并形成一个共同的 将这些不同的数学结构联系在一起的主题。 瓦戈纳将研究代数和同调结构的 有限型、sofic和其他子移位的对称性 移动动力系统 除其他问题外，三个中心 他将探索的SFT的问题是FOG、LIFT和PACK。 关于移位对称不变量的几个问题 系统具有C*-代数的非交换版本。 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V·琼斯的作品开始 伪 在很宽的范围内的流形的合线和同胚 例（包括纽结补）具有涉及 代数K-理论群K2. 瓦戈纳将检查它是否是 可以定义新的不变量的纽结的同构， 与新的纽结多项式相似。

This project concerns the symmetry groups of shift dynamical systems, smooth manifolds, and operator algebras. Ideas from algebraic K-theory are applicable to each area and form a common theme tying together these disparate mathematical constructs. Wagoner will study algebraic and homological structure of symmetries of sub-shifts of finite type (SFT), sofic, and other shift dynamical systems. Among other problems, three central questions for SFT's he will explore are FOG, LIFT, and SHIFT. Some of the problems concerning invariants of symmetries of shift systems have non-commutative versions for C*-algebras. 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 V. 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. Wagoner will examine whether it is possible to define new invariants for diffeomorphisms of knots in analogy with the new knot polynomials.