Abstract: von Neumann algebras are non-commutative versions of the function algebra $L^\infty([0,1])$, the same way matrix algebras $M_{n\times n}(\mathbb C)$ are analogue to finite spaces. A particularly important class, called II$_1$ {\it factors}, arise as infinite tensor products and ultra products of matrix algebras, and also from groups $\Gamma$ and their measure preserving ergodic actions on probability spaces $\Gamma \curvearrowright X$. A key analysis tool to study II$_1$ factors is {\it deformation-rigidity theory}, which exploits the tension between ``soft'' and ``rigid'' parts of the algebra to unravel its building data. This fits within the fundamental dichotomy {\it structure versus randomness}, which appeared in many areas of mathematics in recent years. I will present several rigidity results obtained through this technique, showing for instance that II$_1$ factors arising from Bernoulli actions of property (T) groups $\Gamma \curvearrowright X$ ``remember'' both the group and the action, and that free ergodic actions of the free groups $\Bbb F_n$ remember the rank $n$.