Lectures on Random Matrices

Lecture notes for a short course on the school for complex systems in Sao Jose, Brazil. The goal of these lecture is to expose the student to the main concepts and tools of random matrices. This short course consists of a few lectures to students of various backgrounds. Therefore, I have chosen to include many elementary exam-ples throughout the text. I tried to combine heuristic arguments and to communicate the main ideas. There are beautiful connections between many branches of mathe-matics, and the theory is aesthetically very pleasing. I tried to capture some of these connections and to highlight the main points, however, often I chose to sacrifice the precision of the statements. I also don’t present the proofs, but I will offer the ref-erence. I’m in debt with Alexei Veneziani and Daniel Maia for his critical reading of the text and for the exercises