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.
