CAREER: Classification and rigidity for von Neumann algebras

职业：冯诺依曼代数的分类和刚性

• 批准号: 1253402
• 负责人: Adrian Ioana
• 金额: $ 55万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1253402&HistoricalAwards=false
• 关键词: CAREER; Classification; rigidity; von; Neumann

The goal of this mathematics research project by Adrian Ioana is the classification of von Neumann algebras arising from groups and actions of groups on probability spaces. Von Neumann algebras do not contain significant information about the group or action that they were constructed from. Nevertheless, in the past decade, Popa's deformation/rigidity theory has led to an explosion of rigidity results. These show that many properties of groups or actions can be recognized by looking at their von Neumann algebras. Ioana will develop new methods to study rigidity for von Neumann algebras. Three specific directions of research will be pursued. Firstly, Ioana will expand the scope of super-rigidity for von Neumann algebras by providing new examples of groups and actions that can be entirely reconstructed from their von Neumann algebras. A second direction concerns Cartan subalgebras of von Neumann algebras. Finally, Ioana will investigate the notion of orbit equivalence for group actions. Several research projects on these topics will further the discovery of new knowledge in all of these directions.This mathematics research project is in the general area of operator algebras, and it deals with the classifications of von Neumann algebras. These algebras are connected and have direct applications to a number of areas of science, such as quantum computing, biology (structure of DNA) and engineering (cell phone design). As part of this mathematics research project, the principal investigator Adrian Ioana will develop two new courses at the undergraduate and graduate level, respectively. Ioana is currently mentoring several graduate students and running a seminar in areas close to the research-related content of this project. In addition to supporting graduate students, Ioana will hire and mentor one postdoctoral scholar. Ioana will also organize a research workshop, whose goal is to acquaint young researchers with some of the latest exciting developments on the topics pertaining this project. Finally, Ioana will train high school students and college students to participate in mathematics competitions.

这个数学研究项目的目标是由阿德里安Ioana是分类冯诺依曼代数所产生的群体和行动的群体的概率空间。冯·诺伊曼代数不包含有关构造它们的群或作用的重要信息。然而，在过去的十年中，Popa的变形/刚度理论导致了刚度结果的爆炸。这些表明，许多性质的群体或行动可以认识到，通过看他们的冯诺依曼代数。Ioana将开发新的方法来研究冯诺依曼代数的刚性。将开展三个具体的研究方向。首先，Ioana将通过提供可以完全从其冯诺依曼代数重建的群和作用的新例子来扩展冯诺依曼代数的超刚性范围。第二个方向涉及冯诺依曼代数的Cartan子代数。 最后，Ioana将研究群作用的轨道等价性概念。关于这些主题的几个研究项目将进一步发现所有这些方向的新知识。这个数学研究项目是在算子代数的一般领域，它涉及冯诺依曼代数的分类。这些代数是相互联系的，并直接应用于许多科学领域，如量子计算，生物学（DNA结构）和工程学（手机设计）。 作为这个数学研究项目的一部分，首席研究员阿德里安·约阿纳（Adrian Ioana）将分别在本科和研究生阶段开发两门新课程。 Ioana目前正在指导几名研究生，并在接近该项目研究相关内容的领域举办研讨会。 除了支持研究生，Ioana将聘请和指导一名博士后学者。 Ioana还将组织一个研究研讨会，其目标是让年轻的研究人员了解有关该项目主题的一些最新令人兴奋的发展。 最后，Ioana将培训高中生和大学生参加数学竞赛。