The central focus of Loring's work will be relations on operators and the associated universal C*-algebras. These will be studied in connection with K-theory, perturbation problems, lifting problems and representation theory. Gilfeather will work on extending recent results on Hochshild cohomology of an algebra taking values in that algebra. In particular, he will investigate completely bounded cohomology for non-selfadjoint algebras. A related area involves automorphisms motivated by various constructions from algebraic topology. The general area of mathematics of this project has its basis in the theory of algebras of Hilbert space operators. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA, and they are of fundamental importance in noncommutative geometry, which is becoming increasingly important in physics.
