Interview with Cyril Houdayer, invited speaker at ICM2022

I work in the field of operator algebras, which is a branch of functional analysis. More precisely, I’m interested in the study of von Neumann algebras, which are objects that we can define as noncommutative measure spaces. I work on the structure, rigidity and classification of von Neumann algebras arising from free probability theory, ergodic theory of group actions on measure spaces, and geometric group theory. In the last few years, I have been using operator algebras (C*-algebras and von Neumann algebras) to study the theory of unitary representations and the dynamics of the space of positive type functions of lattices in higher rank semisimple Lie groups.

 

After my PhD thesis, I spent two years as a postdoc at the University of California at Los Angeles where I worked under Sorin Popa. Those two years at UCLA were crucial in my mathematical development. I remember a stimulating environment in which I learned a great deal.

 

In 2011, during a CIRM summer school, I took a mini-course with Yves Benoist on lattices in Lie groups. I was deeply impressed by Margulis normal subgroup theorem, which states that any normal subgroup in a higher rank lattice is either finite or has finite index. The proof combines, in a remarkable way, ergodic theory and functional analysis. This result has had a great influence on my recent work.

 

It is a privilege to be able to continue to learn while working with my colleagues and supervising theses. I also like the great flexibility that being a mathematician offers in terms of managing my time. I am very attached to the harmony and balance that I find in it.

 

I will talk about a series of works we began in 2019 with Rémi Boutonnet on noncommutative ergodic theory of higher rank lattices. Our results represent a kind of noncommutative generalisation of Margulis normal subgroup theorem I mentioned earlier. We recently began to notice that our results also allow us to recover the rank of higher rank simple Lie groups in terms of inclusions of von Neumann algebras associated with the lattice. This gives us hope that we will one day solve Alain Connes’ celebrated rigidity conjecture regarding von Neumann algebras of higher rank lattices. In maths, it’s important to dream!