Mathematical Sciences: Quantized Analysis

9500882 Effros Effros将与他以前的博士生z.j合作，继续研究量子群分析的神秘数学世界。阮。这项工作是基于他量化泛函分析的新方法——算子空间理论。Popa计划使用解析和代数的观点，在快速发展的子因子领域扩大他的广泛研究。Popa是少数研究子因子理论分析方面的研究人员之一。也许有人会说，这个技术上困难的领域迄今为止构成了这个主题的最深刻的方面。Takesaki和几位同事最近完成了算子代数上离散可服从群作用的中心分类程序之一。他希望在这一成功的基础上，回归到大部分未被探索的持续群体行动领域。他还计划再次研究他早期的对偶理论，该理论目前在量子群论中起着重要作用。量子化分析是在20世纪30年代由约翰·冯·诺伊曼提出的，目的是为量子力学提供一个数学框架。这一发展对数学领域本身产生了深远的影响，因为它使研究人员能够制定许多截然不同的数学学科的非交换或“量子化”版本。我们现在有了量子化版本的遍历理论，阿兰·康内斯的非交换几何，丹·沃古列斯库的非交换概率论，以及算子代数、结理论、低维拓扑和共形量子场论的显著相互作用，这些在很大程度上是由沃恩·琼斯开创的。Effros， Popa和Takesaki一直积极参与这些令人兴奋的数学新领域的前沿工作。* * *

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. ***