Mathematical Sciences: Quantized Analysis

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9500882 Effros Effros will continue to investigate the mysterious mathematical world of quantum group analysis, in collaboration with his former Ph. D student Z.-J. Ruan. This work is based on his new approach to quantizing functional analysis - the theory of operator spaces. Popa plans to broaden his wide-ranging studies in the rapidly evolving field of subfactors, using both analytic and algebraic points of view. Popa is one of very few researchers who has been working on theanalytical aspects of subfactor theory. It may be argued that this technically difficult area has thus far constituted the deepest aspect of the subject. Takesaki, together with several coworkers, has very recently completed one of the central classification programs for discrete amenable group actions on operator algebras. He would like to build on this success by returning to the largely unexplored area of continuous group actions. He also plans to work once again on his earlier theory of duality, which is currently playing a major role in quantum group theory. Quantized analysis was introduced in the 1930's by John von Neumann in order to provide a mathematical framework for quantum mechanics. This development has had a profound effect on the field of mathematics itself, since it has enabled researchers to formulate non-commutative or "quantized" versions of many quite distinct mathematical disciplines. We now have quantized versions of ergodic theory, the non-commutative geometry of Alain Connes, the non-commutative probability theory of Dan Voiculescu, and the remarkable interactions of operator algebras, knot theory, low dimensional topology, and conformal quantum field theory in large part pioneered by Vaughan Jones. Effros, Popa and Takesaki have been actively involved in working on the frontier of some of these exciting new areas of mathematics. ***