Random Matrix Program
Large random matrices provide some of the simplest models for large, strongly correlated quantum systems. The statistics of the energy levels of ensembles of such systems are expected to exhibit universality, in the sense that they depend only on the symmetry class of the system. Recent advances have enabled a rigorous understanding of universality in the case of orthogonal, Hermitian, or symplectic matrices with independent entries, resolving a conjecture of Wigner-Dyson-Mehta dating back 50 years. These new developments have exploited techniques from a wide range of mathematical areas in addition to probability, including combinatorics, partial differential equations, and hydrodynamic limits. It is hoped that these new techniques will be useful in the analysis of universal behaviour in matrix ensembles with more complicated structure such as random regular graph models, or 2D matrix ensembles, as well as more physically relevant systems such as band matrices and random Schroedinger-type Hamiltonians. For some of these models, results in the direction of universality have already been obtained.