Sylvain Chabredier: A journey through the local behaviour of various point processes

Abstract: Finding the roots of a fixed polynomial is an old and hard question, but now for a random polynomial with Gaussian i.i.d. coefficient, can we say something on the statistics on the zeroes? We will see that when the degree N is big, the local behavior of these points can be compared to a statistical physics model of positively charged particles (i.e. a Coulomb gas) and we will explore the theorems that are shown for this Coulomb gas and the associated conjectures for the zeroes of the random polynomial.