Dynamics, operator algebra, and groups: from one to many dimensions

This proposal is in pure mathematics and aims to broaden the myriad connections between topological dynamics, operator algebra, and group theory.PART A deepens connections between one-dimensional symbolic dynamics, C*-algebras, and topological full groups.

A central problem is the Conjugacy Problem (when are two shifts of finite type the same?), and this part aims to reduce the question to an isomorphism problem for groups.

PART B develops a theory of ‘noncommutative dynamical system’ and inspired by symbolic dynamics I will address the Shift Equivalence Problem in this context. This is done in the formalism of C*-correspondences which are a flexible notion of morphism between C*-algebras. PART C connects multidimensional symbolic dynamics to C*-algebras.

This framework models many examples from higher-rank directed graphs, tiling theory, and statistical mechanics. It is marred by many properties that are formally undecidable and a lack of computable invariants. In particular, computing entropy is difficult.

I propose to associate a C*-algebra to such systems and explore invariants such a (equivariant) K-theory, KMS structure, and topological groupoids (and ensuing topological full groups) to gain deeper insight into such systems.

This approach seems to not have been explored so far.The explicit aim of the project is thus to develop and explore tools from operator algebra and groups and apply them to gain insight into symbolic dynamical systems.