"Rigidity" in mathematics occurs when objects in a certain class (functions, function algebras, etc.) can be recognized by knowing very little information about them. Regidity results are usually interdisciplinary and can be relevant to many areas of mathematics, with 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 to do with the study of rigidity in so-called von Neumann algebras. These are algebras of infinite matrices in which the product of two elements (A times B, say) may be different from the product in reverse order (B times A), a fact that reflects Heisenberg's Uncertainty Principle in the quantum mechanics of particle physics. They are related to group theory and ergodic theory, since actions of groups on spaces give rise to a class of von Neumann algebras, now known as "factors." Rigidity in this context occurs when the group of transformations can be recognized by merely knowing its associated factor. During the period 2001-2010, the principal investigator developed a series of techniques to study such phenomena (namely, deformation-rigidity theory) to which he recently added a powerful approximation technique. In this project, he intends to combine all these tools to study rigidity in factors and to tackle the two most famous problems in the area: the Connes approximate embedding conjecture and the free group factor problem. The Connes conjecture predicts that factors can be "simulated on a computer" and has an interesting reformulation in quantum information theory. The project should contribute to the cross-pollination of several areas of mathematics (e.g., free probability, random matrices, group theory, logic) and lead to progress in each of them. The principal investigator's research in rigidity theory has already had considerable impact on these areas, with a large number of research articles and Ph.D. theses sprouting directly from his work. 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, with direct and indirect impact in applied mathematics and the aforememtioned areas of computer science.
In this project, the principal investigator intends to further the development of deformation-rigidity theory by incorporating into it a new approximation technology known as incremental patching. Using this array of tools, he will continue to work on the classification of algebras arising from groups and their actions on spaces, as well as on the study of rigidity properties of these objects and the calculation of their symmetries. In particular, the principal investigator will attempt new approaches to two famous problems in two-one factors: the Connes approximate embedding conjecture and the free group factor problem. The problems studied in this project are important to both von Neumann algebra theory and the adjacent areas of group theory, ergodic theory, logic (descriptive set theory), free probability, and subfactor theory. Together with his students and collaborators, the principal investigator will make efforts to broaden the scope of deformation-rigidity theory, strengthening its interactions with all these (and possibly other) areas, an activity that should lead to further surprising results of an interdisciplinary character. Also, the principal investigator intends to continue to disseminate his new techniques through summer programs, mini-courses, textbooks, and expository articles, as well as through conferences that he will conduct and organize.
