These lecture notes are a concise introduction of recent techniques to prove
local spectral universality for a large class of random matrices. The general
strategy is presented following the recent book with H.T. Yau. We extend the
scope of this book by focusing on new techniques developed to deal with
generalizations of Wigner matrices that allow for non-identically distributed
entries and even for correlated entries. This requires to analyze a system of
nonlinear equations, or more generally a nonlinear matrix equation called the
Matrix Dyson Equation (MDE). We demonstrate that stability properties of the
MDE play a central role in random matrix theory. The analysis of MDE is based
upon joint works with J. Alt, O. Ajanki, D. Schr\"oder and T. Kr\"uger that are
supported by the ERC Advanced Grant, RANMAT 338804 of the European Research
Council.
The lecture notes were written for the 27th Annual PCMI Summer Session on
Random Matrices held in 2017. The current edited version will appear in the
IAS/Park City Mathematics Series, Vol. 26.