This series of talks is devoted to modern aspects of random matrix theory.
The eigenvalues of a random matrix form a random measure on the plane, which often converges to a limiting distribution. We would like to introduce some of the most important results in concerning the limiting distribution for different classes of random matrices.
In this talk, we zoom in the local behavior of the nearby eigenvalues. How do they interact and what can we say about the limiting behavior at microscopic scale ?
The key theme is this area is universality: the limiting behavior does not depend too much on the distribution of the entries of the matrix.
We discuss the role of random matrices in data science, with applications concerning basic problems such as matrix completion, clustering, and matrix sparsification.