One of the main goals of the theory of random matrices is to establish the limiting distributions of the eigenvalues.
In the 1950s, Wigner proved his famous semi-cirle law (subsequently extended by Anord, Pastur and others),
which established the global distribution of the eigenvalues of random Hermitian matrices.
In the last fifty years or so, the focus of the theory has been on the local distributions,
such as the distribution of the gaps between consecutive eigenvalues, the k-point correlations,
the local fluctuation of a particular eigenvalue, or the distribution of the least singular value.
Many of these problems have connections to other fields of mathematics, such as combinatorics,
number theory, statistics and numerical linear algebra.
Most of the local statistics can be computed explicitly for random matrices with gaussian entries (GUE or GOE),
thanks to Ginibre's formulae of the joint density of eigenvalues.
It has been conjectured that these results can be extended to other models of random matrices.
This is generally known as the Universality phenomenon, with several specific conjectures
posed by Wigner, Dyson, Mehta etc.
In this talk, we would like to discuss recent progresses concerning the Universality phenomenon,
focusing on a recent result (obtained jointly with T. Tao), which asserts that all
local statistics of eigenvalues of a random matrix are determined by the first four moments of the entries.
This (combining with results of Johansson, Erdos-Ramirez-Schlein-Yau and many others)
provides the answer to several old problems.