Random matrices frequently appear in many different fields — physics,
computer science, applied and pure mathematics. Oftentimes the random
matrix of interest will have non-trivial structure — entries that are
dependent and have potentially different means and variances (e.g.
sparse Wigner matrices, matrices corresponding to adjacencies of random
graphs, sample covariance matrices). However, current understanding of
such complicated random matrices remains lacking. In this talk, I will
discuss recent results concerning the spectrum of sums of independent
random matrices with a.s. bounded operator norms. In particular, I will
demonstrate that under some fairly general conditions, such sums will
exhibit the following universality phenomenon — their spectrum will
lie close to that of a Gaussian random matrix with the same mean and
covariance. No special background in random matrix theory will be
necessary for the audience — basic knowledge of probability and linear
algebra are sufficient.

Joint work with Ramon van Handel https://web.math.princeton.edu/~rvan/tuniv220113.pdf