Gaultier Lambert: Eigenvalues of Haar distributed matrices

Large random matrices are probabilistic models for hamiltonians of physical systems with high complexity, therefore the main question is what can we say about the distribution of the eigenvalues? This topic has brought a lot of interest over the last 50 years and many other connections with mathematical physics, but also with number theory or geometry have been found. I will focus on the so-called Circular Unitary Ensemble (CUE) which consists of matrices drawn at random from the Haar measure on  the Unitary Group. I will start by motivating some of the basic results of Random Matrix Theory in the simple case of the CUE. Then I will discuss the Strong Szegö theorem and the method of Diaconis and Shahshahani to analyze the eigenvalue distribution. I do not intend to dwell on any technical details from the proof but rather to point out connections between questions related to the Circular Ensemble and other areas of mathematics such as the orthogonal Polynomials and Representation theory.