## Final Report Summary - RIGIDITY (Rigidity and classification of von Neumann algebras)

(A version of this text appeared in the ERC brochure "Mathematics - Spotlight on ERC projects -

2018".)

Group theory, functional analysis and ergodic theory are three distinct areas of mathematics that

meet within the theory of von Neumann algebras. The RIGIDITY project aims to classify families of

such von Neumann algebras.

Each of these three areas of mathematics has an origin in physics. Groups not only describe

symmetries of physical systems, but the representation theory of compact groups plays a key role

in the standard model of particle physics. Hilbert space operators - one of the main concepts in

functional analysis - play the role of quantum mechanical observables, while ergodic theory

provides the necessary tools to describe and understand the long term and global behavior of a

physical system.

Whenever a group acts on a measure space, there is an associated crossed product von Neumann

algebra. The most basic question is when two such von Neumann algebras are isomorphic? To

which extent do they remember the initial data that they were constructed from? In earlier joint

work of Popa and Vaes, we obtained the first families of group actions that can be entirely

recovered from their ambient von Neumann algebra. In other words, these von Neumann

algebras can be decomposed in exactly one way as a crossed product. The first main result

obtained in the RIGIDITY project was the construction of von Neumann algebras that have

precisely two and more generally, precisely n, crossed product decompositions.

Von Neumann algebras also arise naturally in Voiculescu's free probability theory, a

noncommutative or quantum version of classical probability theory. The classification of these

free Araki-Woods von Neumann algebras is wide open and only very partial results were known.

In the RIGIDITY project, we established definitive classification theorems for large families of free

Araki-Woods von Neumann algebras, in terms of their defining spectral measure.

A third focus of the RIGIDITY project has been on the quantum symmetries of von Neumann

algebras. These arise in the form of inclusions of von Neumann algebras, known as Jones'

subfactors. To every such subfactor is associated a group-like invariant, which is a very intricate

combinatorial structure with connections to many areas of mathematics, most notably knot

theory. In the RIGIDITY project, we developed several aspects of harmonic analysis for these

group-like invariants. Very recently, this led to the first truly quantum instances of Kazhdan's

property (T), which is a rigidity property that is for example known for its usage in the

construction of expander graphs and thus widely used in theoretical aspects of computer science.

The research achievements of the RIGIDITY project were broadly recognized and led to the award

of the Francqui Prize to PI Stefaan Vaes and to his election as a member of the Royal Academy of

Belgium.

2018".)

Group theory, functional analysis and ergodic theory are three distinct areas of mathematics that

meet within the theory of von Neumann algebras. The RIGIDITY project aims to classify families of

such von Neumann algebras.

Each of these three areas of mathematics has an origin in physics. Groups not only describe

symmetries of physical systems, but the representation theory of compact groups plays a key role

in the standard model of particle physics. Hilbert space operators - one of the main concepts in

functional analysis - play the role of quantum mechanical observables, while ergodic theory

provides the necessary tools to describe and understand the long term and global behavior of a

physical system.

Whenever a group acts on a measure space, there is an associated crossed product von Neumann

algebra. The most basic question is when two such von Neumann algebras are isomorphic? To

which extent do they remember the initial data that they were constructed from? In earlier joint

work of Popa and Vaes, we obtained the first families of group actions that can be entirely

recovered from their ambient von Neumann algebra. In other words, these von Neumann

algebras can be decomposed in exactly one way as a crossed product. The first main result

obtained in the RIGIDITY project was the construction of von Neumann algebras that have

precisely two and more generally, precisely n, crossed product decompositions.

Von Neumann algebras also arise naturally in Voiculescu's free probability theory, a

noncommutative or quantum version of classical probability theory. The classification of these

free Araki-Woods von Neumann algebras is wide open and only very partial results were known.

In the RIGIDITY project, we established definitive classification theorems for large families of free

Araki-Woods von Neumann algebras, in terms of their defining spectral measure.

A third focus of the RIGIDITY project has been on the quantum symmetries of von Neumann

algebras. These arise in the form of inclusions of von Neumann algebras, known as Jones'

subfactors. To every such subfactor is associated a group-like invariant, which is a very intricate

combinatorial structure with connections to many areas of mathematics, most notably knot

theory. In the RIGIDITY project, we developed several aspects of harmonic analysis for these

group-like invariants. Very recently, this led to the first truly quantum instances of Kazhdan's

property (T), which is a rigidity property that is for example known for its usage in the

construction of expander graphs and thus widely used in theoretical aspects of computer science.

The research achievements of the RIGIDITY project were broadly recognized and led to the award

of the Francqui Prize to PI Stefaan Vaes and to his election as a member of the Royal Academy of

Belgium.