Rigidity Theory of von Neumann Algebras and its Connections to C*-Algebras, Geometric Group Theory, and Continuous Model Theory

University Of Iowa, Iowa City IA

Investigators

Abstract

Von Neumann algebras are collections of infinite matrices with complex entries and were initially developed to provide a rigorous mathematical framework for quantum mechanics. Early work by pioneers of the field during the 1930s and 1940s revealed that von Neumann algebras are highly complex objects exhibiting remarkably rich structural properties. Over time, their study evolved into an independent and vibrant area of mathematics, spurring the development of powerful mathematical theories and uncovering deep connections with other fields such as group theory, dynamical systems, topology, and more recently, model theory. Beyond mathematics, von Neumann algebras have also provided valuable insights in physics (notably statistical mechanics), biology (DNA molecular structure), engineering (cell phone network design), and computer science (including error-correcting codes, quantum information theory, and quantum computing). These algebras naturally arise from simpler mathematical concepts like groups (symmetries) and their actions on spaces, which are extensively studied in geometric group theory and ergodic theory. The project investigates a number of open problems, with the central goal of developing new methods at the intersection of these disciplines to advance the classification of von Neumann algebras arising from such structures. Additionally, the project offers extensive opportunities for graduate student training and career development. Building on the PI’s prior work, this project will investigate new research avenues in the classification of group and measure space von Neumann algebras, along with their broader applications. The PI will develop innovative techniques that lie at the crossroads of deformation/rigidity theory, group theory, ergodic theory, and continuous model theory to advance several fundamental open problems: (i) identifying new groups and algebraic group structures that are fully determined by the von Neumann and C*-algebraic frameworks, such as property (T) W*- and C*-superrigid groups, including those with infinite centers; (ii) calculating the endomorphism semigroup, the fundamental semigroup, and the Jones index set of property (T) group factors, as well as automorphism groups of reduced group C*-algebras; (iii) constructing new, naturally occurring examples of W*-superrigid actions—an area of mutual interest in orbit equivalence and von Neumann algebras; (iv) investigating tensor product decompositions for factors associated with negatively curved groups and existentially closed factors; and (v) developing new methodologies to distinguish ultrapowers of factors. This work is inherently interdisciplinary, promising significant synergies among geometric group theory, ergodic theory, C*-algebras, model theory, and von Neumann algebras. To foster the professional growth of the PI’s graduate students, the project will continue to support a visiting program that enables students to collaborate with peers at leading institutions, broadening their exposure to diverse research perspectives and expertise. In addition, the PI will continue to promote this field by teaching advanced graduate courses, organizing specialized seminars, and actively disseminating research outcomes through publications, invited talks, and presentations at national and international conferences and research seminars. 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.