FRG: Collaborative Research: von Neumann Algebras Associated to Groups Acting on Hyperbolic Spaces

University Of California-San Diego, La Jolla CA

Investigators

Abstract

The study of von Neumann algebras was initiated in the 1930s and 1940s by F. Murray and J. von Neumann as a mathematical tool to understand particle physics. Subsequently, it became an independent discipline that has stimulated the development of powerful mathematical theories and bringing valuable insight to physics (statistical mechanics), biology (DNA structure), and engineering (cell phone network design). Von Neumann algebras are highly interdisciplinary in nature as they arise canonically from simpler mathematical structures, such as symmetries and actions, often present in many areas of mathematics. Over time their study remained closely connected with various topics in dynamical systems, measured group theory, and more recently geometric group theory. This project investigates several major open problems inspired by the rich interaction between operator algebras and the aforementioned fields. This research project explores new horizons in the classification of group von Neumann algebras. The first objective of the project is to advance Connes' rigidity conjecture, a major wide-open problem predicting that ICC property (T) groups are completely recognizable from their von Neumann algebras (W*-superrigid). The PIs proposed several natural constructions of W*-superrigid property (T) groups based on new developments in geometric group theory and deformation/rigidity theory. The second objective of the project revolves around the study of prime II1 factors. The main focus is to understand the relationship between various manifestations of negative curvature in group theory and primeness aspects of the corresponding group factor. The results arising from this project are expected to reveal significant cross-pollination between, geometric group theory, ergodic theory, random walks, C*-algebras, and von Neumann algebras. The PIs intend to organize a series of workshops aimed at stimulating collaboration between experts in these fields. To promote the career development of graduate students the proposal also involves a student exchange program between the participating institutions aimed at exposing students to different expertise and research environments. 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.