A sequence of tuples of random matrices $(X_1^N, \dots, X_d^N)$ converges strongly to a tuple of operators $ (x_1, . . . , x_d)$ in a $C^*$-algebra if for any noncommutative polynomial $P$, $\|P(X_1^N, \dots, X_d^N)\|$ converges (say, in probability) to $\|P(x_1, . . . , x_d)\|$ as $N$ goes to infinity. This phenomenon plays a central role in breakthrough results  in operator algebras, as well as in the construction of: expander graphs, hyperbolic surfaces with nearly optimal spectral gaps and minimal surfaces. Given its far-reaching implications, it is no surprise that strong convergence is notoriously difficult to prove and has generally required delicate problem-specific methods.
In this talk I will discuss recent joint work with Chi-Fang Chen, Joel Tropp and Ramon van Handel, where we introduce a new flexible and elementary technique for proving strong convergence. This technique can be applied to random matrix models that have a lot of symmetry, for example, random permutation matrices, classical Gaussian and unitary matrices (i.e. GOE, GUE, GSE, $O(N)$, $U(N)$, and $Sp(N)$), and some others, constructed via representations of the symmetric and unitary group, for which other methods seem to break. In all of these models, the technique yields the sharpest quantitative results known so far.