Abstract: 
In the mid thirties F. J. Murray and J. von Neumann found a natural way to associate a von Neumann algebra, $L^\infty(X)\rtimes \Gamma$ to every measure preserving action of a countable group on a probability space $\Gamma \curvearrowright X$. When $X$ is a singleton this amounts to the so-called group von Neumann algebra, $L(\Gamma)$. The general paradigm of classifying $L^\infty(X)\rtimes \Gamma$ and $L(\Gamma)$ in terms of $\Gamma \curvearrowright X$ and $\Gamma$, respectively, emerged overtime as a natural yet quite challenging theme, as these algebras tend to have very limited “memory” of the initial data. This is best illustrated by Connes’ celebrated result  from mid seventies asserting that all $L^\infty(X)\rtimes \Gamma$ and all $L(\Lambda)$ are isomorphic for all free ergodic actions $\Gamma \curvearrowright X$ with $\Gamma$ amenable and all $\Lambda$  icc amenable groups. Thus in all these cases, besides amenability, the von Neumann algebra has no recollection of the usual dynamical properties of the action (mixing, compactness) or the classical group invariants (torsion, rank, or generators and relations). In the non-amenable case the situation is far more complex and many examples where the von Neumann algebraic structure completely retains various aspects of the initial group/action data have been discovered over the last decade via Popa’s deformation/rigidity theory. In my talk I will survey some of these recent developments, commenting on the methods involved and their importance and also pointing out my contributions to the subject.