Zeroes of Laplace eigenfunctions

We will discuss geometrical and analytic properties of zero sets of harmonic functions and eigenfunctions of the Laplace operator. For harmonic functions on the plane there is an interesting relation between local length of the zero set and the growth of harmonic functions. The larger the zero set is, the faster the growth of harmonic function should be and vice versa. A curious object is Laplace eigenfunctions on the two dimensional sphere, which are restrictions of homogeneous harmonic polynomials of three variables onto the sphere. They are called spherical harmonics. Zero sets of such functions are unions of smooth curves with equiangular intersections. Topology of zero set could be quite complicated, but the total length of the zero set of any spherical harmonic of degree n is comparable to n. We will discuss recent advances in Yau's and Landis' conjectures.

Based on a joint work with F.Nazarov, N.Nadirashvili and Eu. Malinnikova