Professor Rieffel's project involves a synthesis of ideas from differential geometry and the theory of operator algebras. He will study from several viewpoints the process of deforming geometric structures by passing to an algebra of functions or to a Lie algebra and then smoothly changing the multiplication. Specific subjects of investigation include quantum groups, deformation quantization, and Yang-Mills theory. Fundamental to Rieffel's work is the notion that certain algebras of operators can profitably be viewed as noncommutative analogs of familiar geometric objects. One might for instance begin with the two-torus, the surface of an inner tube. All of the information about the shape of this object is contained in the algebra of complex valued functions on it. Two such functions are the ones that measure phase as one goes around the torus the long, respectively the short, way around. These functions generate the algebra of continuous functions on the torus in a certain universal way. Replace them by two (unitary) operators, a species of generalized functions, that instead of commuting when multiplied, commute only up to a phase. As the phase varies, the operator algebras generated by these projectively commuting unitaries form a family of noncommutative tori, creating a setting in which geometric ideas can be formulated and studied without reference to a geometric object in the usual sense.
