Professor Price's project involves mathematical research that is motivated by quantum mechanics. The quantum mechanical point of view is to represent observable quantities by linear operators on Hilbert space rather than by scalars. In some models, the observables give rise to an algebra of operators. Time evolution is given by a coherent rule specifying a transformation of the algebra at each moment. Mathematical manipulations allow one to read off properties of the system being modeled from the way it evolves over time. In the present undertaking, a rather general sort of time evolution, called a semigroup of endomorphisms, is to be studied with the goal of obtaining a scheme of classification. This project is funded under the Research in Undergraduate Institutions activity; there is a component of the research that will be carried out by undergraduate students. Cocycle conjugacy invariants finer than the recently discovered quantized Fredholm index of Arveson and Powers for these semigroups will be investigated. Professor Price will also consider the representation theory of operator algebras generated by pairs of isometric flows on Hilbert space that satisfy a certain commutation relation. The undergraduate component involves work on shift register sequences, a piece of cryptanalytic technology that seems to be related to shift endomorphisms of the hyperfinite factor.
