[gyrovagueGeist]

I love this paper, but something I think is often missed when it comes up is that you CAN hear the shape of many drums if you restrict the shape space, for example with a prior of "what a drum should look like" Zelditch proved spectral uniqueness for convex, fully connected, drums with some symmetry.

[pfortuny]

You are totally right but "convex" is a very strong assumption (so strong that the shape is determined by its harmonics). Very strong.

[DougMerritt]

Aha, I had missed that! Thanks.