Bulk universality for Wigner matrices

In this talk, I am going to consider ensembles of NxN Wigner matrices whose entries are, up to the symmetry constraints, independent and identically distributed. Assuming sub-exponential decay of the matrix entries, I will present a proof of the universality of the local eigenvalue statistic; in the limit of large N we show that the all eigenvalue correlation functions are independent of the particular distribution of the entries and are determined only by the symmetry of the matrix (hermitian, real symmetric, or symplectic).