Operator Spaces and Locally Compact Quantum Groups

在过去的几年里，他的研究重点是算子空间的局部理论以及算子空间在抽象调和分析和局部紧量子群中的应用。在这份提案中，他计划继续在这些方向上进行研究，并提出了以下研究项目。(1)研究算子空间的一些局部性质。(2)研究了局部紧量子群的若干问题。(3)研究Kadison-Singer问题和Bozejko和Speicher引入的q-变形因子的相关问题。算符空间理论是现代分析中近年来发展起来的一个新的研究领域。PI和其他一些数学家一起对这一领域做出了重大贡献。本文所提出的课题包含了算子空间中的一些重要问题，以及它们在算子代数、抽象调和分析和局部紧量子群中的应用。这些问题的进展将对这些领域产生重大影响。它还将对量子信息论、非交换几何、几何群论等相关领域产生重要影响。本提案中的项目也为PI的博士生提供了很好的资源和研究问题。PI将继续组织关于伊利诺伊大学最新研究进展的学习研讨会。这将鼓励和吸引他的学生参与这些研究项目。PI还将继续组织沃巴什研讨会和迷你会议，这为他的学生，博士后和访客提供了一个非常独特的机会，定期与一些领域的顶尖专家会面和交流思想。

RuanDuring the last few years, the PI has focused his research on the local theory of operator spaces and the applications of operator spaces to abstract harmonic analysis and locally compact quantum groups. In this proposal, he plans to continue his research in these directions and he proposes the following research projects. (1) Investigate some local properties of operator spaces. (2) Investigate some problems on locally compact quantum groups.(3) Investigate some problems related to Kadison-Singer problem and the q-deformation factors introduced by Bozejko and Speicher.The theory of operator space theory is a recently developed new research area in modern analysis. The PI, together with some other mathematicians, has made significant contributions to this area. The projects proposed in this proposal contain some important questions in operator spaces, and their applications to operator algebras, abstract harmonic analysis, and locally compact quantum groups. The progress on these problems will have significant impact to these areas. It will also have important impact to some other related areas such as quantum information theory, noncommutative geometry, and geometric group theory. Projects in this proposal also provide excellent resources and research problems for PI's Ph.D students. The PI will continue to organize learning seminars on recent research progress at the University of Illinois. This will encourage and attract his students to get involved in these research projects.The PI will also continue to organize Wabash Seminar and Miniconference, which has provided a very unique opportunity for his students, postdocs, and visitors to regularly meet and exchange ideas with some leading experts in the fields.