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

The research of the ERC project RIGIDITY is situated at the crossroads of three areas of mathematics: group theory, functional analysis and ergodic theory. The mathematical concept of a group encodes symmetry. A well known example is the group of all possible transformations of a Rubik's cube. Another one is the group of all distance preserving transformations of the three-dimensional space. This is already a quite complicated group, containing all rotations, translations and mirror symmetries.

Secondly, functional analysis has been developed in the early 20th century in order to provide a solid mathematical framework for particle physics. It turned out that quantum mechanical observables do not behave like functions, but rather like matrices. Furthermore these matrices were infinite in size. Addition of two infinite matrices is easy, but in order to multiply two infinite matrices, the theory of Hilbert spaces is needed. A Hilbert space can be viewed as an infinite-dimensional version of our ambient three-dimensional space. Exactly as with matrices, operators can be added and multiplied so that they can form algebras of operators. A particular class of operator algebras are von Neumann algebras introduced by Murray and von Neumann in the 1930's.

Finally, ergodic theory is the study of area (or length or 'measure') preserving dynamical systems. In ergodic theory, one studies the long time behavior when such a length preserving transformation is iterated. When in average, over many iterations, the dynamical system visits all states an approximately equal number of times, the system is called ergodic.

The above three areas of mathematics - group theory, functional analysis and ergodic theory - meet each other in the theory of von Neumann algebras. The overall theme of the ERC project RIGIDITY is the classification of families of von Neumann algebras. Using Popa's deformation/rigidity theory, we obtained such a classification result for the von Neumann algebras that arise in free probability theory, which is a noncommutative (or quantum) version of usual probability theory. Secondly, we gave the first examples of von Neumann algebras that can be constructed in exactly two ways from ergodic group actions. Finally, we developed a broad and general analysis framework to study the quantum symmetries coming from Jones' theory of subfactors, which are inclusions of von Neumann algebras with a very rich underlying combinatorial structure. This in particular allowed us to introduce new numerical invariants, called L^2-Betti numbers, for these discrete group like structures.

The research results established in the ERC project RIGIDITY, and in earlier research projects of the PI, lead to the award of the Francqui Prize, Belgium's main scientific recognition, to PI Stefaan Vaes in 2015.

Secondly, functional analysis has been developed in the early 20th century in order to provide a solid mathematical framework for particle physics. It turned out that quantum mechanical observables do not behave like functions, but rather like matrices. Furthermore these matrices were infinite in size. Addition of two infinite matrices is easy, but in order to multiply two infinite matrices, the theory of Hilbert spaces is needed. A Hilbert space can be viewed as an infinite-dimensional version of our ambient three-dimensional space. Exactly as with matrices, operators can be added and multiplied so that they can form algebras of operators. A particular class of operator algebras are von Neumann algebras introduced by Murray and von Neumann in the 1930's.

Finally, ergodic theory is the study of area (or length or 'measure') preserving dynamical systems. In ergodic theory, one studies the long time behavior when such a length preserving transformation is iterated. When in average, over many iterations, the dynamical system visits all states an approximately equal number of times, the system is called ergodic.

The above three areas of mathematics - group theory, functional analysis and ergodic theory - meet each other in the theory of von Neumann algebras. The overall theme of the ERC project RIGIDITY is the classification of families of von Neumann algebras. Using Popa's deformation/rigidity theory, we obtained such a classification result for the von Neumann algebras that arise in free probability theory, which is a noncommutative (or quantum) version of usual probability theory. Secondly, we gave the first examples of von Neumann algebras that can be constructed in exactly two ways from ergodic group actions. Finally, we developed a broad and general analysis framework to study the quantum symmetries coming from Jones' theory of subfactors, which are inclusions of von Neumann algebras with a very rich underlying combinatorial structure. This in particular allowed us to introduce new numerical invariants, called L^2-Betti numbers, for these discrete group like structures.

The research results established in the ERC project RIGIDITY, and in earlier research projects of the PI, lead to the award of the Francqui Prize, Belgium's main scientific recognition, to PI Stefaan Vaes in 2015.