One of the main goals of the theory of random matrices is to study distributions concerning the eigenvalues. In recent years, we have witnessed notable progresses on central problems in this area. The PI has been fortunate to participate in some these developments and he would like to propose to continue his research on these topics. In this proposal, he is going to discuss the current state of some of these problems, and propose to study several research problems that would lead to a more complete and deeper understanding of the subject, especially problems related to the universality phenomenon. Beside studying traditional questions on limiting distributions, we will also discuss non-asymptotic aspects of the theory, e.g. problems concerning large deviations. Many of these problems are important in applications in other fields, such as theoretical computer science and data mining.
The theory of random matrices is a rich topic in mathematics. Beside being interesting on their own right, random matrices play fundamental role in various areas such as statistics, mathematical physics, combinatorics, theoretical computer science, etc. A famous example here is the study of physicist Wigner, who used the spectrum of random matrices as a model in nuclear physics, and consequently discovered the fundamental semi-circle law. The main project in this proposal is to study limiting distribution the spectrum of random matrices, at the finest scale, motivated by long standing problems from mathematical physics and probability. We also believe that the methods being developed in the project will be useful for other purposes, such as the study of large (random) networks and large data from random samples.
