The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue
statistics of large real and complex Hermitian matrices with independent,
identically distributed entries are universal in a sense that they depend only
on the symmetry class of the matrix and otherwise are independent of the
details of the distribution. We present the recent solution to this
half-century old conjecture. We explain how stochastic tools, such as the Dyson
Brownian motion, and PDE ideas, such as De Giorgi-Nash-Moser regularity theory,
were combined in the solution.
We also show related results for log-gases that represent a universal model
for strongly correlated systems. Finally, in the spirit of Wigner's original
vision, we discuss the extensions of these universality results to more
realistic physical systems such as random band matrices.Comment: Proceedings of ICM 201
