# The Work of Logunov and Malinnikova in Relation to Yau's Conjecture (Fall 2019)

## COURSE DESCRIPTION

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 which is approximately $$\sqrt{\lambda}$$.