Description: Studying the zeroes of a polynomial and the eigenvalues of a matrix is
one of the oldest and most fundamental problem in mathematics (here is a classic problem: how many zeroes of a polynomial is real ?) In this course, we are going to
focus on the case when the coefficients of the polynomial (i.e., the entries of the matrix) are random variables.
The problem goes back to Waring (late 1790s) but a systematic study was started by Littlewood and Offord in the 1940s (for polynomials) and Wigner in the 1950s (for matrices).
For a long time, the two subjects are treated separately, and techniques that are used to study random polynomials usually do not work for random matrices and vice versa.
In this course, we are going to cover the most basic results and methods for both problems, and then focus on a
new approach introduced recently by Tao and the instructor that works for both
subjects equally well. With its help, we are going to discuss many applications, some of which gives solutions to
long standing open problems (such as the one on the number of real zeroes above or Mehta type universality conjecture for correlation functions).
Prerequisites: Basic knowledge in probability, analysis, linear algebra.