Nodal Geometry and Yau's Conjecture

Following Chapter 4 in Levitin--Mangoubi--Polterovich’s book, we will investigate the nodal geometry of Laplace eigenfunctions on bounded domains and Riemannian manifolds. We will prove Courant’s nodal domain theorem, and then show that nodal sets are dense on the wavelength scale. Following a discussion of Yau’s conjecture, time permitting, we will discuss other interesting known estimates.