Recently, there has been considerable amount of work in the area of random polynomials, where the "randomness" comes from the selection of coefficients of the polynomial. However, if the "randomness" comes from the choice of zeros, can we infer something about the critical points of the polynomial? More formally, Robin Pemantle and Igor Rivin conjectured that given any distribution $\mu$ on the complex plane, if we select points $z_1, z_2, ..., z_n$ using $\mu$, then the empirical distribution of the critical points of the polynomial $p_n(z) = (z-z_1)(z-z_2)...(z-z_n)$ converges weakly to $\mu$ as $n$ goes to $\infty$. Pemantle and Rivin, in their paper, proved the conjecture for a special class of measures, called finite 1-energy measures.

The speaker proved the result for another special case: measures on the unit disc, while, more recently Zakhar Kabluchko proved the conjecture in full generality. In this talk, we shall go over these results and their proofs.