Abstract Price Endomorphisms on von Neumann algebras are examples of noncommutative dynamical systems which may proceed forward, but not backward, in time. The analysis of the cocycle conjugacy classes of discrete endomorphisms is related to the Jones theory of index for subfactors and has numerous connections to the classification theory of Connes of the outer conjugacy classes of automorphisms on the tracial hyperfinite factor. An important family of endomorphisms is the family of Powers shifts, which appear naturally in the construction of shifts on Jones tunnels of subfactors and as restrictions of endomorphisms with finite relative commutant index. Using a connection between Powers shifts and linear shift register systems, Price will attempt to complete his classification of the rational Powers shifts. Price has also computed the entropy of all rational Powers shifts and expects to work with E. Stormer on the problem of determining the range of values of the Connes-Stormer entropy among the family of all Powers shifts. In addition, a number of combinatorial problems have arisen from the interplay between the study of Powers shifts and shift register systems, and as part of this project Price will work with some undergraduate students to answer some related questions about the ranks of Toeplitz and Hankel matrices over finite fields. This research project represents an attempt to exploit connections which Price has shown to exist between two fields of mathematics, one of which is relatively abstract and one of which is comparatively concrete. Since the 1930's operator algebras have been used as mathematical constructs to model the behavior of quantum mechanical systems. The field of shift register systems, on the other hand, is a branch of cryptology, and is one of the means by which secure messages are encoded and transmitted. Using some number-theoretic aspects of linear shift register systems, and with the participation of undergraduate research assistants, Price will con tinue his program of classifying families of mappings known as Powers shifts. Powers shifts are mappings which may be viewed as determining the forward time evolution of a particular algebra of physical observables. Although this algebra has been used in modeling quantum mechanical systems, it has also been studied for its applications to knot theory and for its potential applications towards the understanding of the structure of DNA.
