In this lecture, I focus on ergodic theory of group actions. We consider the translation action of a discrete group G on the product space {0,1}^G equipped with the product of probability measures \mu_g on {0,1}. When all \mu_g are equal, these are the classical Bernoulli actions, which are probability measure preserving. When the \mu_g are distinct, non measure preserving actions of Krieger type III may appear. I will explain an intricate connection to L^2-cohomology. In particular, I will show that a group G admits a Bernoulli action of type III_1 if and only if G has nonzero first L^2-cohomology. I will also explain why the group of integers does not admit a Bernoulli action of type II_\infty and why type III_\lambda only arises when G has more than one end. This is joint work with J. Wahl, and with M. Björklund and Z. Kosloff.

Stefaan Vaes is a full professor and the head of the Analysis section at KU Leuven (Belgium). His research focuses on operator algebras and their connections to group theory and ergodic theory. He was an invited speaker at the International Congress of Mathematicians in 2010 and at the European Congress of Mathematics in 2016. In 2015, he was awarded the Francqui Prize, which is the highest scientific distinction in Belgium. In the spring of 2017, he was a Rothschild Fellow at the Newton Institute in Cambridge. Stefaan Vaes is a member of the Royal Academy of Belgium (KVAB). He is one of the editors-in-chief of Journal of Functional Analysis.

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