Operator algebra seminar

Speaker: Scott Schmieding (Northwestern University)

Title: Strong shift equivalence and the K-theory of endomorphisms

Abstract: For a semiring $R$, shift equivalence (SE) and strong shift equivalence (SSE) are certain equivalence relations on endomorphisms over $R$. SSE over $Z_{+}$ carries deep information about graphs, classifies shifts of finite type, and is still poorly understood. SSE over other semirings appears naturally in other settings. When $R$ is a ring, SSE and SE are often, but not always, the same, the difference depending on K-theoretic information about $R$. We outline connections between SSE and SE over $R$, and algebraic K-theory of endomorphisms over $R$. In particular, we define certain ``higher'' SSE and SE groups, and show there is a long exact sequence relating them, the zeroth part of which records the difference between the ``group completions'' of SSE and SE over $R$. I will pose some questions.