Shifts of finite type and $C^*$-algebras

The presentation is based on joint work with Adam Dor-On and Efren Ruiz.
To each directed graph, one can associate a dynamical system with discrete time, consisting of bi-infinite paths, where the evolution mapping is defined by a shift. Such a dynamical system is called a shift of finite type and is a central object of study in symbolic dynamics.
One of the main open questions in this area is the classification of shifts of finite type up to conjugacy and eventual conjugacy. In a foundational work from 1973, Williams reduced this problem to the classification of incidence matrices of corresponding graphs up to shift (SE) and strong shift (SSE) equivalence. Williams presented a reasonable classification of matrices up to shift equivalence and hypothesized that SE and SSE coincide. Nearly 20 years later, this was refuted by Kim and Roush through a counterexample.
This classification problem is closely related to the $C^*$-algebras of graphs. It turns out that two graphs with SSE incidence matrices have stably isomorphic $C^*$-algebras. Furthermore, if we equip the $C^*$-algebras of graphs with two additional structures: a commutative diagonal subalgebra and a gauge action of the circle, then we obtain a complete invariant of strong shift equivalence. In our work with A. Dor-On and E. Ruiz, it was shown that stable equivariant homotopic equivalence of $C^*$-algebras is equivalent to shift equivalence of graphs.
In the presentation, I will discuss these constructions and results in more detail and explain how the perspective of $C^*$-algebras can help resolve open questions in symbolic dynamics.