Abstract Boyle Boyle will study various topics involving symbolic dynamics, including the density of periodic points for surjective cellular automata; the inverse spectral problem for nonnegative matrices over the integers and other subrings of the reals; algebraic generalizations of this problem; the classification of reducible shifts of finite type in a functorial setting suitable to the general algebraic analysis of these systems and their relations with C* algebras; topological orbit equivalence; general symbolic dynamical frameworks and residual entropy. To have a crude initial idea of symbolic dynamics, think of studying problems involving very long tapes of zeros and ones. An appropriate idealization is to consider infinite tapes. Such a study is significant when one considers problems of data transmission and storage. It also is one key to the study of how physical systems evolve over time: imagine cutting up a system into (say) two pieces, and associating to a point a sequence of zeros and ones according to its itinerary through the two pieces. It is actually useful to study a system by way of the associated itineraries ... As happens in math, the theory developed for this has turned out to be relevant to superficially unrelated problems in other areas. Two examples are objects of study of this proposal: understanding the algebraic structure of nonnegative matrices, which are basic in many applications of mathematics, and improving a framework for understanding certain relations with the mathematical subject of C*-algebras. Another topic of this proposal involves understanding cellular automata (the computer game of Life is an example) which can be viewed as models for physics and evolution. The final topic, orbit equivalence, involves the study of properties of a dynamical system which do not depend on the way itineraries are ordered.
