A short while ago I was blogging about the classical isospectrality problem which goes back to Mark Kac’ famous question: Can one hear the shape of a drum? Some basic information about it can be found in that blog post of mine: Can one hear orientability?
The original question of Mark Kac was answered negatively by Gordon-Webb-Wolpert, in the following way: they constructed non-isometric planar domains with both the same Dirichlet and Neumann spectrum. However, these domains do not have a smooth boundary, they have corners:
Kac’ original question, if only considered for planar domains with smooth boundary, is still open. One can wonder now if there is any connection between the question for planar domains with corners and for domains with smooth boundary. The answer to this was recently provided by Nursultanov-Rowlett-Sher: https://arxiv.org/abs/2012.03366. They proved that no planar domain with corners can be isospectral to a planar domain with smooth boundary.