Department of Mathematics

Prof. Denis Osin recently published a paper in Annals of Mathematics, widely considered the most prestigious journal in pure mathematics. The paper, “Wreath-like products of groups and their von Neumann algebras I: W*-superrigidity” was written with Adrian Ioana (UCSD), Ionut Chifan (U Iowa), and a former Vanderbilt Ph.D. student Bin Sun (Michigan State U).

The topic of the paper is von Neumann algebras. The basis of the Heisenberg uncertainty principle, and more generally quantum mechanics, is that the position and momentum operators do not commute. The theory of von Neumann algebras, developed in the 1940s, provides the mathematical foundation for this principle.  Besides their relation to physics, von Neumann algebras serve as central objects in several areas of contemporary mathematics, including representation theory, dynamical systems, and non-commutative geometry.

Von Neumann algebras are named after John von Neumann, the legendary polymath, famous for his work on the Manhattan project, among many other achievements. Some of the key breakthroughs on von Neumann algebras were achieved by Alain Connes and Vaughan Jones, two Fields medalists and former faculty members at Vanderbilt.

Von Neumann algebras, commonly referred to as non-commutative measure spaces, were originally called “rings of operators”. In their foundational work on the subject, Francis Murray and von Neumann suggested a way to assign a ring of operators (or von Neumann algebra) to every countable group.

Generally, the von Neumann algebra associated with a group tends to retain little to no information about the algebraic structure of the group. The ultimate manifestation of this phenomenon is the famous theorem of Alain Connes asserting that all groups from a broad class (specifically, all amenable groups with infinite conjugacy classes) give rise to the same von Neumann algebra. On the other hand, Connes conjectured in 1980 that the isomorphism class of a group satisfying a certain representation-theoretic property, known as Kazhdan’s property (T), is uniquely determined by its von Neumann algebra.

Despite significant recent progress in understanding von Neumann algebras of groups, Connes’ conjecture remained wide open. Moreover, neither a counterexample nor a single non-trivial example of a group satisfying this conjecture had been known until now. In their paper, Osin, Ioana, Chifan, and Sun proved Connes’ conjecture for a wide class of groups having a particular algebraic structure. Examples of such groups naturally arise in the context of the algebraic analog of Dehn surgery in 3-manifolds, introduced by Osin in 2007.

Prof. Osin and his collaborators intend to apply the technique developed in their first paper to address other open problems in operator algebras. In particular, their second paper “Wreath-like products of groups and their von Neumann algebras II: Outer automorphisms” (available on the arXiv here) makes progress towards another well-known conjecture proposed by Vaughan Jones. This work was supported by a recent NSF Focused Research Group grant, shared by Vanderbilt, UCSD, and the University of Iowa.