The APDE seminar on Monday, 9/27, will be given by Steve Zelditch (Northwestern University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu)

Title: Riemannian 3-manifolds whose eigenfunctions have just
two nodal domains.

Abstract: In 1977, Hans Lewy published an article constructing a
series  $\phi_N$  of spherical harmonics of degree $N$ whose nodal (zero) sets cut the 2-sphere into just 2 nodal domains. It is an ingenious construction. Recently, Junehyuk Jung and I showed that there is a simple and canonical way to construct an infinite dimensional family of Riemannian metrics on certain 3 manifolds,  all of whose eigenfunctions have this property (except for a certain trivial sequence).  The construction generalizes to all odd  dimensions.  This unexpected behavior of 3D eigenfunctions is heuristically related to numerical computations of nodal sets of 3D spherical harmonics, where only one connected component is visible (the `giant component’). Sarnak conjectured that its genus is maximal among degree N polynomials . Jung and I showed that our result holds for random “equivariant” 3D spherical harmonics and showed that in a certain range the genus has maximal order of growth $N^3$.  The purpose of my talk is to explain these phenomena of 3D nodal sets, which have no analogue for the much more studied 2D case