We will discuss the recent work of Logunov and Malinnikova, for which they were awarded a Clay research award. This work led to the proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecture. Nadirashvili's conjecture roughly states that if \$u\$ is a harmonic function in the unit ball in \$\mathbb{R}^n\$ which is zero at the center of the ball, then the (\$n-1\$)-dimensional Hausdorff measure of the zero set of \$u\$ in the ball is bounded below by a purely dimensional constant. Yau's conjecture concerns the (\$n-1\$)-dimensional Hausdorff measure of the zero set of Laplace eigenfunctions. In particular, Yau conjectured that the zero set of the eigenfunction associated to the eigenvalue \$\lambda\$ should have (\$n-1\$)-dimensional Hausdorff measure comparable to \$\sqrt{\lambda}\$.