## 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 \(\sqrt{\lambda}\).

## Primary Sources

Nodal Sets of Laplace Eigenfunctions: Estimates of the Hausdorff Measure in Dimensions Two and Three

Nodal Sets of Laplace Eigenfunctions: Polynomial Upper Estimates of the Hausdorff Measure

## Secondary Sources

"Uber Knoten yon Eigenfunktionen des Laplace-Beltrami-Operators", Bruning; 1978

"Nodal sets of eigenfunctions on Riemannian manifolds", Donnelly, Fefferman; 1988

"Nodal Sets of Solutions of Elliptic Equations", Hardt, Simon; 1989

"Nodal Sets of Solutions of Elliptic and Parabolic Equations", Lin; 1991

"Logarithmic Convexity for Supremum Norms of Harmonic Functions", Korevaar, Meyers; 1994