Random matrices were introduced by E. Wigner to model the excitation spectrum of  large nuclei. The central idea is based on the hypothesis that the local statistics of the excitation spectrum for a large complicated system is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. The random Schrödinger equation was introduced by P. Anderson to model the electron dynamics in a semiconductor. It is believed that in the conducting regime, the dynamics of the electron is given by a diffusion equation despite the fundamental equation is a time-reversible Schrödinger equation. These two models are related via the conjecture that the local spectral properties of the random Schrödinger operator and random matrices are identical in certain limit. In this lecture, we will explain the formulation and relation between these conjectures.  A precise statement of the recent result on the universality of spectral statistics of large random matrices both in the bulk and at the edges will be given.

There is no physical background needed to understand this talk; the essentially backgrounds are basic probability and matrix theory.