CAREER: Classification and rigidity for von Neumann algebras

禤浩焯·约阿纳的这一数学研究项目的目的是对由群和群在概率空间上的作用产生的von Neumann代数进行分类。冯·诺依曼代数不包含关于构成它们的群或作用的重要信息。然而，在过去的十年里，波帕的变形/刚性理论导致了刚性结果的爆炸性增长。这些结果表明，群或作用的许多性质可以通过查看它们的von Neumann代数来识别。Ioana将开发新的方法来研究von Neumann代数的刚性。具体的研究方向有三个。首先，Ioana将通过提供可以从von Neumann代数完全重建的群和作用量的新例子来扩展von Neumann代数的超刚性的范围。第二个方向涉及von Neumann代数的Cartan子代数。最后，Ioana将研究群体行动的轨道等价性概念。关于这些主题的几个研究项目将进一步在所有这些方向上发现新的知识。这个数学研究项目是在算子代数的一般领域，它涉及von Neumann代数的分类。这些代数是相互关联的，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)。作为这一数学研究项目的一部分，首席研究员禤浩焯·约阿纳将分别开发本科生和研究生层面的两门新课程。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还将聘请和指导一名博士后学者。Ioana还将组织一次研究讲习班，其目标是让年轻研究人员了解与该项目有关的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。

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.