Many key classes of C*-algebras, and of closely related abstract algebras, arise as representations of dynamical or combinatorial objects. For such algebras, a natural question is: “What collection of (C*)-algebraic information is needed to ‘remember’ the dynamics?” For example, a rotation algebra remembers the defining rotation up to a sign; but you need a Cuntz-Krieger algebra \emph{and} its canonical Cartan subalgebra to remember flow-equivalence class of the associated subshift. I will discuss the problem of recovering a given dynamical or combinatorial object from an associated (C*)-algebra, and outline what is known and some of what is not.
