**Speaker:** Oanh Nguyen (Yale University)

**Abstract:** We consider random polynomials of the form $$P_n(x) = \xi_1 p_1(x) + \xi_2 p_2(x) + \dots +\xi_n p_n(x)$$ where \$\xi_1, \dots, \xi_n\$ are independent random variables and \$p_1, \dots, p_n\$ are deterministic polynomials.

Questions about the distribution of the zeros of \$P_n\$ have attracted intensive research for many decades with seminal papers by Kac, Littlewood-Offord, Erdos-Offord, and Tao-Vu, to name a few. In this talk, we will discuss some universality properties of the roots of generalized Kac polynomials and trigonometric random polynomials. As an application, we calculate the average number of real roots and discuss some asymptotic behavior of this number.

The talk is based on some joint works with Yen Do, Hoi Nguyen, and Van Vu.