II$_1$ factors are non-commutative versions of the function algebra $L^\infty([0,1])$,
the way matrix algebras $M_{n\times n}(\mathbb C)$ are analogue to finite spaces.
They arise as infinite tensor products and ultra products of matrix algebras, but 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 classification results obtained through this technique, showing for instance that 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 $n$.