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

这是一个美国和波兰的数学研究项目，旨在探索经典概率和非交换调和分析之间的新联系。主要研究者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。 他们合作的需要来自于他们研究中同一类过程的意外发生，表明经典概率的某些模型与变形量子理论之间存在未探索的联系。这项合作的主要目标是推进非交换过程的理解，其中有经典的版本，并使用经典概率作为指导和动机，在定义适当的推广的概念的“非交换布朗运动”。 这些研究将构造新的非对易过程类，并将研究已知非对易过程的经典版本。 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，在http：//math.uc.edu/free-gauss/上发布互联网，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举办的夏季研讨会，同样，波兰合作伙伴将访问美国参加研讨会，这将为美国研究生提供独特的培训，并促进思想的快速交流。通过汇集美国和中欧/东欧的领先专家，该项目将开启和推进经典概率和非交换调和分析之间的新一层联系。 在过去，跨边界的研究和思想交流取得了巨大的成功，并在非交换和经典概率的重大进展的关键。在过去的十年中，对自由群冯诺依曼代数的理解取得了重大进展，这源于Voiculescu巧妙地引入了自由概率和自由卷积。 同样，概率和非交换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已经成为经典概率中的有力工具，如Dembo，Guionnet和Zeitouni的作品。布赫霍尔茨，皮西尔和徐非交换鞅的工作开辟了新的研究方向，研究人员希望从这个项目产生类似的影响。

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.