U.S.-Polish Collaborative Research on Classical and Noncommutative Gaussian Processes

美国-波兰关于经典和非交换高斯过程的合作研究

• 批准号: 0332062
• 负责人: Wlodzimierz Bryc
• 金额: $ 1.46万
• 依托单位: University of Cincinnati Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-09-01 至 2006-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0332062&HistoricalAwards=false
• 关键词: U.S.; Polish; Collaborative; Research; Classical

This is a US-Polish mathematical research project to explore a new connection between classical probability and non-commutative harmonic analysis. The principal investigator is Dr. Wlodek Bryc from the University of Cincinnati. His Polish collaborator is Dr. Marek Bozejko from Wroclaw University. The need for their collaboration arose from an unexpected occurrence of the same class of processes in their research, indicating an unexplored link between certain models of classical probability and those of deformed quantum theory. The main goal of this collaboration is to advance the understanding of non-commutative processes which have classical versions, and to use classical probability as a guide and motivation in defining appropriate generalizations of the concept of a "non-commutative Brownian motion". The researches will construct new classes of non-commutative processes and will study the classical versions of already known non-commutative processes. To facilitate the exchange of expertise, the researchers will rely on mutual visits, email contacts, internet posting at http://math.uc.edu/free-gauss/, participation in common meetings and workshops. The US participants will attend summer workshops hosted by the Polish affiliates, and likewise, the Polish partners will visit the US for seminars, which will both provide unique training for US graduate students and foster the rapid exchange of ideas.By bringing together leading experts in the U.S. and Central/Eastern Europe, the project will open and advance a new layer of connections between classical probability and non-commutative harmonic analysis. In the past, cross boundary research and exchange of ideas has had great success and was critical for major advances in non-commutative and classical probability. Over the last decade, significant progress in the understanding of free group von Neumann algebras arose from Voiculescu's ingenious introduction of free probability and free convolution. Likewise, the connection between probability and non-commutative harmonic analysis has attained increasing importance, as concepts which originally arose in free probability are nowbecoming powerful tools in classical probability, as in the works of Dembo, Guionnet and Zeitouni. The works of Buchholz, Pisier and Xu on non-commutative martingales have opened new directions of research, and the researchers hope for a similar impact from this project.

这是一个美国-波兰的数学研究项目，旨在探索经典概率和非对易调和分析之间的新联系。首席研究员是来自辛辛那提大学的Wlodek Bryc博士。他的波兰合作者是弗罗茨瓦夫大学的Marek Bozejko博士。需要他们合作的原因是，在他们的研究中，同一类过程出人意料地发生了，这表明经典概率的某些模型与变形量子理论的模型之间存在尚未探索的联系。这一合作的主要目的是促进对具有经典形式的非对易过程的理解，并以经典概率为指导和动机来定义“非对易布朗运动”概念的适当推广。这些研究将构造新的非交换过程类，并将研究已知的非交换过程的经典版本。为了促进专业知识的交流，研究人员将通过相互访问、电子邮件联系、在http://math.uc.edu/free-gauss/，上发布互联网帖子来参与共同的会议和研讨会。美国的参与者将参加由波兰分支机构主办的夏季研讨会，同样，波兰的合作伙伴将访问美国参加研讨会，这些研讨会将为美国研究生提供独特的培训，并促进思想的快速交流。通过将美国和中东欧的顶尖专家聚集在一起，该项目将开启和推进经典概率和非对易调和分析之间的新一层联系。在过去，跨界研究和思想交流取得了巨大成功，对非对易概率和经典概率的重大进展至关重要。在过去的十年里，在理解自由群von Neumann代数方面取得了重大进展，这是由于Vocerescu巧妙地引入了自由概率和自由卷积。同样，概率和非对易调和分析之间的联系也变得越来越重要，因为最初出现在自由概率中的概念现在正在成为经典概率中的强大工具，就像在Dembo、Guion net和Zeitouni的作品中一样。Buchholz、Pisier和Xu关于非对易鞅的工作开辟了研究的新方向，研究人员希望这一项目能产生类似的影响。