Von Neumann Algebras: Rigidity, Applications to Measurable Dynamics, and Model Theory

Purdue University, West Lafayette IN

Investigators

Abstract

The theory of von Neumann algebras was initially developed in the 1930s and 1940s by Francis J. Murray and John von Neumann as a vast generalization of matrix algebra necessary to capture the mathematical complexities of quantum mechanics by encoding the symmetries of the quantum mechanical system in roughly the same way that matrix algebra encodes the symmetries of classical mechanics and general relativity. Over the intervening decades it has been realized that von Neumann algebras represent the symmetries of many objects studied in the physical and biological sciences, leading to applications in diverse fields such as the theory of statistical mechanics, the structure of DNA molecules, the theory of error correcting codes, and the theory of quantum information theory and quantum computing. The building blocks of these symmetries are called "factors." The main goal of the theory of von Neumann algebras is to provide an effective classification of factors to understand how symmetries arise in these various settings. One approach is through so-called rigidity, where it is shown that a small, known group of symmetries actually describes the entire, vast set of symmetries of a factor. Another direction addresses the computability of the symmetries of a factor via the ability to simulate the factor in a computer by matrix algebras, what is known as the Connes embedding problem. The last decade has seen a rapid growth in the understanding of the structure of factors, due to the development of deformation-rigidity theory that was initiated by Sorin Popa in the early 2000s. These results and approaches have additionally had a remarkable impact on the fields of measurable dynamics of groups, descriptive set theory of Polish group actions, probability theory, as well as on geometric group theory and coarse geometry. Working to develop and expand this theory, the principal investigator's research focuses on the connections between operator algebras, geometric and measurable group theory, Lie groups, and ergodic theory, specifically through the use of geometric deformations that were first investigated in work by the principal investigator and Ionut Chifan. The principal investigator is also interested in the application of logic and model-theoretic techniques to operator algebras. The project aims to achieve the following broad objectives: (1) to continue developing a perspective from which tools from Lie groups and geometric group theory can be used to classify the structure of and to demonstrate new examples of rigidity phenomena in the context of group and measure-measure space von Neumann algebras; (2) to find applications of these techniques to measurable group theory, probability theory, and the classification of C*-algebras; (3) to further explore and develop connections between continuous model theory and von Neumann algebras, C*-algebras, and the theory of operator systems with applications to the Connes embedding problem. The principal investigator will continue to teach advanced courses and organize seminars in operator algebras, working closely with graduate students to develop and foster talent and discovery. He will also continue to broadly and actively disseminate his research through publications, seminar and colloquium talks, and conferences and workshops, at both the national and international level.