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 thatmeet within the theory of von Neumann algebras. The RIGIDITY project aims to classify families ofsuch von Neumann algebras.Each of these three areas of mathematics has an origin in physics. Groups not only describesymmetries of physical systems, but the representation theory of compact groups plays a key rolein the standard model of particle physics. Hilbert space operators - one of the main concepts infunctional analysis - play the role of quantum mechanical observables, while ergodic theoryprovides the necessary tools to describe and understand the long term and global behavior of aphysical system.Whenever a group acts on a measure space, there is an associated crossed product von Neumannalgebra. The most basic question is when two such von Neumann algebras are isomorphic? Towhich extent do they remember the initial data that they were constructed from? In earlier jointwork of Popa and Vaes, we obtained the first families of group actions that can be entirelyrecovered from their ambient von Neumann algebra. In other words, these von Neumannalgebras can be decomposed in exactly one way as a crossed product. The first main resultobtained in the RIGIDITY project was the construction of von Neumann algebras that haveprecisely two and more generally, precisely n, crossed product decompositions.Von Neumann algebras also arise naturally in Voiculescu's free probability theory, anoncommutative or quantum version of classical probability theory. The classification of thesefree 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 freeAraki-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 Neumannalgebras. 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 intricatecombinatorial structure with connections to many areas of mathematics, most notably knottheory. In the RIGIDITY project, we developed several aspects of harmonic analysis for thesegroup-like invariants. Very recently, this led to the first truly quantum instances of Kazhdan'sproperty (T), which is a rigidity property that is for example known for its usage in theconstruction 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 awardof the Francqui Prize to PI Stefaan Vaes and to his election as a member of the Royal Academy ofBelgium.