CAREER: Classification and rigidity for von Neumann algebras

Adrian Ioana的这个数学研究项目的目标是对由群和群在概率空间上的作用产生的冯·诺伊曼代数进行分类。冯·诺伊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，Popa的变形/刚性理论导致了刚性结果的爆炸式增长。这些结果表明，群或行为的许多性质可以通过观察它们的冯·诺伊曼代数来识别。Ioana将开发新的方法来研究冯·诺伊曼代数的刚性。将进行三个具体的研究方向。首先，Ioana将通过提供可以完全从其von Neumann代数重构的群和作用的新例子来扩展von Neumann代数的超刚性范围。第二个方向是关于冯诺依曼代数的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·诺伊曼代数的分类。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，并在与该项目研究相关的领域举办研讨会。除了支持研究生，爱荷华大学还将聘请并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与该项目有关的主题的一些最新令人兴奋的发展。最后，爱荷华州将训练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.