During the last decade, von Neumann algebras of group actions have become a center stage for studying a variety of rigidity phenomena and a playground for various areas of mathematics to interact: operator algebras, group theory (measured, geometric, arithmetic, etc.), ergodic theory, orbit equivalence relations, and descriptive set theory, to name a few. During the period 2001-2010, the principal investigator has developed a series of techniques for studying rigidity in this framework, which is now called deformation/rigidity theory. This led to a large number of striking rigidity results in both the von Neumann algebra and orbit ergodic theory settings, and to the solution of many long-standing problems. The principal investigator's techniques and results naturally entail some exciting new directions of research and problems in all these areas. They also provide new tools for approaching some of the classical (hitherto "intractable") problems in von Neumann algebras, such as: the Connes rigidity conjecture; the structure and classification of free group factors; superrigidity properties of algebras arising from groups and their actions. In this project the principal investigator, with his students and collaborators, will systematically investigate these directions. He intends to deepen his interaction with the areas of group theory, ergodic theory, and descriptive set theory, using operator algebra techniques. This activity should lead to further surprising results and solutions to problems in all these areas. The principal investigator expects the framework of factors to continue to play a crucial role in this interplay between diverse areas of mathematics.
"Rigidity" in mathematics occurs when objects in a certain class (functions, function algebras, etc.) can be recognized without having very much initial information about them. Results of this type are usually interdisciplinary and can be relevant to many areas of mathematics. They can also have interesting applications to computer science, complexity theory, design of computer networks, and the theory of error-correcting codes. The principal investigator's work in recent years has focused on the study of rigidity in the class of objects known as von Neumann algebras. These are algebras of infinite matrices, wherein the outcome of the multiplication of two elements A and B may be different depending on the order in the product (i.e., AB may be different from BA). Theses algebras where introduced by von Neumann in the 1920s in his effort to provide a rigorous approach to quantum mechanics in particle physics. He related the algebras, at the outset, with such areas of mathematics as group theory and ergodic theory by noticing that actions of groups on so-called probability measure spaces give rise to a remarkable class of von Neumann algebras. Rigidity in this context occurs when the group action can be recognized by merely knowing the associated von Neumann algebra. The principal investigator has recently developed a completely new set of techniques for studying such phenomena, creating a framework that is now called deformation/rigidity theory. He has obtained a number of surprising and intrinsically beautiful results that create a bridge from von Neumann algebras to rigidity in other areas of mathematics and lead to deep interdisciplinary activity. The problems that the principal investigator intends to work on over the next three years are increasingly ambitious, having to do with famous unsolved problems about the classification of factors arising from "rigid groups" and "free groups." The projects are important to both von Neumann algebra theory and to the adjacent mathematical areas of group theory, ergodic theory, logic (descriptive set theory), free probability, and subfactor theory. The proposal should further contribute to the cross-pollination of these areas and to substantial progress in each of them. The principal investigator's work in rigidity theory has already had considerable impact in many areas, with a large number of research articles and Ph.D. theses sprouting directly from it. He expects his techniques to have an even broader impact in the future, leading to new developments and solutions to problems in a variety of subjects. He also expects this research to have direct and indirect impact in applied mathematics and in the aforementioned areas of computer science.
