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 - 波兰数学研究项目，旨在探索经典概率和非交通谐波分析之间的新联系。主要研究人员是辛辛那提大学的Wlodek Bryc博士。 他的波兰合作者是弗罗克劳大学的Marek Bozejko博士。 他们对他们的协作的需求来自于他们的研究中同一类过程的意外发生，这表明某些经典概率与变形量子理论的某些模型之间存在未探索的联系。这项合作的主要目标是提高对具有经典版本的非共同过程的理解，并使用经典的概率作为指导和动机，以定义“非交通性布朗尼运动”概念的适当概括。 这些研究将构建新的非共同过程类别，并将研究已经知道的非交流过程的经典版本。 为了促进专业知识的交流，研究人员将依靠相互访问，电子邮件联系人，互联网发布，网址为http://math.uc.edu/free-gauss/，参加共同会议和研讨会。 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，波兰合作伙伴也将访问美国参加研讨会，该研讨会都将为美国的研究生提供独特的培训，并促进迅速的想法交流。通过将美国和中欧的主要专家聚集在一起，该项目将开放并提高经典概率和非官能分析之间的新联系。 过去，跨边界研究和思想交流取得了巨大的成功，对于非交通和古典概率的重大进展至关重要。在过去的十年中，对自由群体von Neumann代数的理解取得了重大进展，这是Voiculescu巧妙地引入自由概率和自由卷积的。 同样，概率和非交通谐波分析之间的联系也越来越重要，因为最初在自由概率中出现的概念现在是经典概率的强大工具，就像Dembo，Guionnet和Zeitouni的作品一样。 Buchholz，Pisier和Xu在非交通性的Martingales上的作品开辟了新的研究方向，研究人员希望对该项目产生类似的影响。