Schedule 

13:00 - 14:50 on December 24 (Room 404, Classroom Building 3) 

20:40 - 22:30 on December 25, 26  

Abstract

In this mini-course, we state and prove the local semicircle law for Wigner matrices, which says that the eigenvalue distribution of a Wigner matrix is close to the semicircle distribution, down to spectral scales containing slightly more than one eigenvalue. The proof of the local law has been refined by multiple recent research articles, and we shall present the simplest and most updated version. We then discuss two applications of the local semicircle law, which are the eigenvector delocalization and eigenvalue rigidity. If time permits, we will also talk about the linear statistics of Wigner matrices. 

The preliminaries are elementary analysis, basic probability theory and complex variables. No preknowledge of random matrix theory is needed. Both undergraduate and graduate students are welcome.