Mathematical Sciences: Quantized Analysis

数学科学：量化分析

• 批准号: 9500882
• 负责人: Masamichi Takesaki
• 金额: $ 43.29万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500882&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Quantized; Analysis

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. ***

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。这项工作基于他量化泛函分析的新方法——算子空间理论。波帕计划利用分析和代数的观点，扩大他在快速发展的子因素领域的广泛研究。波帕是极少数一直致力于次因素理论分析方面的研究人员之一。可能有人会说，这个技术上困难的领域迄今为止构成了该主题的最深刻的方面。 Takesaki 与几位同事最近完成了一项中央分类程序，用于算子代数上的离散服从群操作。 他希望在这一成功的基础上再接再厉，回到尚未探索的持续团体行动领域。 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发挥着重要作用。 量子分析由约翰·冯·诺依曼 (John von Neumann) 在 20 世纪 30 年代引入，旨在为量子力学提供数学框架。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“量化”版本。我们现在有了遍历理论的量化版本、阿兰·康内斯的非交换几何、丹·沃伊库列斯库的非交换概率论，以及算子代数、结论、低维拓扑和共形量子场论在大范围内的显着相互作用。部分由沃恩·琼斯（Vaughan Jones）首创。 埃夫罗斯、波帕和竹崎一直积极参与其中一些令人兴奋的数学新领域的前沿工作。 ***