Quantifying Rigidity in von Neumann Algebras

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The theory of von Neumann algebras was initiated in the 1930s and 40s by F.J. Murray and John von Neumann as a mathematical framework for quantum mechanics. With recent breakthroughs in quantum computing, the study of von Neumann algebras is poised to yield insights into deep problems in the theory of quantum computation which must be overcome to make quantum computing and quantum cryptography practical, efficient technologies. One goal of this project is to use tools from von Neumann algebras to provide insights into so-called “quantum expanders” which have applications to quantum error correction and quantum cryptography. This is part of the broader goal of the project to investigate quantitative aspects of von Neumann algebras. Other potential applications lie in the theory of random matrices, which are used in diverse applications in many fields from quantum physics to biology and big data. This project will contribute to workforce development by providing research training and mentoring opportunities at the undergraduate and graduate level. The project aims to make progress in several directions around quantifying and developing new invariants for exploring the phenomenon of rigidity in von Neumann algebras. One objective is to further develop the theory and use of cohomological rigidity techniques in Popa’s deformation/rigidity theory based on techniques developed by the PI jointly with collaborators on the existence and uniqueness of maximal rigid subalgebras of deformations. This could lead to progress towards settling two outstanding conjectures in the field, the Peterson-Thom conjecture and absence of Cartan subalgebras for von Neumann algebras of groups having nontrivial first cohomology with coefficients in the left-regular representation. Techniques from continuous model theory will also be explored as potential avenues to these conjectures by attempting to find noncommutative analogs to Anderson and Keisler’s model theoretic approach to stochastic differential equations. A second objective is to develop experimental and quantitative approaches to property Gamma, in part based on the PI’s discovery of malnormal matrices in his work with Mulcahy. The PI will approach these problems using a mix of techniques from ergodic theory, random matrix theory, computability theory, and von Neumann algebras. Results in this direction could lead to new insights at the interface of von Neumann algebras and quantum computing. A third objective is to explore the applications of uniform 2-norms to the classification theory of nuclear C*-algebras based on the PI’s work with Goldbring and Hart on the continuous model theory of correspondences.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.