Location: Ross 70 
Title: Expected number of components of random polynomial lemniscates
Abstract: A lemniscate of a complex polynomial $p$ is a sublevel set of its modulus, i.e., of the form $$\Lambda(t):={z \in \mathbb{C}: |p(z)| < t},$$ for some $t>0.$ The study of lemniscates was pioneered by Erd\H{o}s, Herzog, and Piranian in \cite{metricEHP}, where they asked various questions regarding the number of connected components of a lemniscates (i.e. for $t=1$). In general, the number of components of a unit lemniscate can vary anywhere between 1 and $\deg(p)$. However, for certain random polynomials numerical simulations show a giant component alongside some tiny components. In this talk, we quantify these numerical observations and find their asymptotic behaviour. First, we show that the expected number of connected components of lemniscates whose defining polynomial has $n$ i.i.d. roots chosen uniformly from $\mathbb{D}$, is asymptotically $\gamma \sqrt{n}$, where
$$\gamma=\sqrt{\frac{\zeta(2) -1}{\pi}}.$$
On the other hand, if the i.i.d. roots are chosen uniformly from $\mathbb{S}^1$, we show
that the expected number of connected components is asymptotically $ \frac{n}{2}$. We also describe the phase transition in the number of components of (\Lambda_n(t)) as (t) varies from (0) to (\infty). This talk is based on \cite{SG} and an upcoming paper \cite{GRS} with Atul Shekhar and koushik Ramachandran.