Mathematical Sciences: Symmetry Groups in Dynamics and Topology

这个项目涉及移位动力系统、光滑流形和算子代数的对称群。来自代数K理论的思想适用于每个领域，并形成一个共同的主题，将这些不同的数学结构捆绑在一起。瓦格纳将研究有限型子移位(SFT)、SOFIC和其他移位动力系统的对称性的代数和同调结构。在其他问题中，他将探索的三个SFT核心问题是雾、升力和变速。关于移位系统对称不变量的一些问题对于C*-代数具有非对易形式。三维空间中纽结的经典Alexander多项式与代数K-理论群K1密切相关。最近，从V.Jones的工作开始，已经发现了这些纽结的其他多项式不变量。流形的伪同余和微分同胚在很大范围内(包括纽结补)具有涉及代数K-理论群K2的不变量。瓦格纳将研究是否有可能为纽结的微分同胚定义新的不变量，类似于新的纽结多项式。

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.