Mathematical Sciences: Symmetry Groups in Dynamics and Topology

这个项目涉及平移动力系统的对称群、光滑流形和算子代数。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联系在一起。Wagoner将研究有限型（SFT）、sofic和其他移位动力系统的子移位对称的代数和同调结构。在其他问题中，他将探讨SFT的三个核心问题是FOG、LIFT和SHIFT。平移系统对称不变量的一些问题对于C*-代数具有非交换的版本。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V. 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。Wagoner将研究是否有可能用新的结多项式类比来定义结的微分同态的新不变量。

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.