Random matrix theory I

This is the first general lecture of a mini-course that gives a brief introduction to the random matrix theory. In the first part we show the main examples of random matrices and discuss some typical problems that appear in the theory. In addition, we illustrate some standard methods of random matrix theory, in particular, the random polynomials techniques, and prove the well-known Wigner semicircle law. The second part of the mini-course will be devoted to the universality of the local regime of the random matrix theory and especially, to the supersymmetry approach to these questions.