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.
