Special topics on Introduction to Random Matrices offered in Spring 2016 by Ionel Popescu.
Random matrices started in physics and statistics and nowadays it has become an independent field.
I plan on covering the basic results in random matrices, like for instance the convergence of the distributions of eigenvalues and spend some time on the most important ensemble, which is GUE.
(Gaussian unitary ensemble). The GUE is the richest ensemble because there are closed formulae for important objects.
Many results for the general ensembles obey the same behavior. We will discuss the limiting behavior of the distribution of eigenvalues, its fluctuations and then expose the audience a little bit to free probability which is in some sense the study of joint random matrices. If time permits, I will also get into the intricate business of behavior of top eigenvalue, local semicircular laws, tridiagonal models and estimation of covariance matrices.