The theory of operator algebras originated in the 1930s with the efforts of J. von Neumann and others to develop rigorous mathematical models of quantum mechanical systems. A fundamental precept of this theory is that the time evolution of such a mathematical model is prescribed by a one-parameter group of automorphisms. A more general picture is obtained if one considers evolutions of systems that can move forward but not necessarily backward in time. This notion leads to the study of one-parameter semigroups of endomorphisms, as opposed to (time reversible) one-parameter groups of automorphisms. Price hopes to continue his work with two leaders in the field, Alexis Alevras and Robert T. Powers, on classifying families of semigroups of endomorphisms that act on the operator algebras of all bounded operator on a Hilbert space. Price also plans to continue his study of endomorphisms on the operator algebra called the hyperfinite II_1 factor. The hyperfinite II_1 algebra R is of fundamental importance in the study of operator algebras and is itself related to certain mathematical models of spin systems. Previously he has used number-theoretic ideas to arrive at a complete classification of the family of endomorphisms on this algebra known as binary shifts. His current work in this area is focused on obtaining a complete classification of all endomorphisms on R whose image is maximal in R. This work uses ideas from combinatorics and linear algebra as well as number theory. The linear algebraic and combinatoric aspects of this study are expected to lead to some research problems that are accessible to gifted undergraduate students. Price proposes to continue working with mathematics majors who are interested in honors projects related to this work.
