Eugene Wigner's revolutionary vision predicted that the energy levels of large complex quantum systems exhibit a universal behavior. These universal statistics represent a new paradigm of point processes that are characteristically different from the Poisson statistics of independent points. 
A prominent example of Wigner's thesis is the Wigner-Dyson-Gaudin-Mehta conjecture asserting that the spectral statistics of random matrices with independent entries depend only on the symmetry classes but are independent of the distributions of matrix elements.
In this lecture, we will outline the recent solution to this conjecture and demonstrate that the local ergodicity of the Dyson Brownian motion is the intrinsic mechanism behind the universality.
Related topics such as delocalization of eigenvectors for random matrices and Erdős-Renyi graphs will also be discussed.