Title: The Growth of Harmonic Functions and their Nodal Volume
 
Abstract: In this talk, I will discuss the relation between the growth of harmonic functions and their nodal volume, that is volume of their zero set. One way to quantify the growth of a harmonic function $u$ in a ball $B=B(0,1)\subset \mathbb{R}^n$ is via the \textit{doubling index} $N$, defined by
        $$ \sup_{B(0, 1)}|u| = 2^N \sup_{B(0,\frac{1}{2})}|u|.$$
Since (as we will see) the doubling index gives a notion of ``degree" for harmonic functions, it would be reasonable to expect that the nodal volume of $u$ in $B$ is proportional to $N$. I will present a joint work with A. Logunov and Lakshmi Priya, where we prove the almost sharp lower bound
        $$ \mathcal{H}^{n-1}(\{u=0\}\cap B(0,2))\gtrsim_{\varepsilon}N^{1-\varepsilon},$$
provided that $u(0)=0$.