On the nodal lines of random and deterministic Laplace eigenfunctions

Abstract: In the present survey we present some of the recent results concerning the geometry of nodal lines of random Gaussian eigenfunctions (in case of spectral degeneracies) or wavepackets and related issues. The most fundamental example, where the spectral degeneracy allows us to consider random eigenfunctions (i.e. endow the eigenspace with Gaussian probability measure), is the sphere, and the corresponding eigenspaces are the spaces of spherical harmonics; this model is the primary focus of the present survey. The list of results presented is, by no means, complete.