Hear the shape of a convex drum

Determine whether the Dirichlet Laplace spectrum uniquely determines a bounded convex planar domain in Euclidean space up to Euclidean isometry; equivalently, show that any two bounded convex subsets of R^2 with identical Dirichlet Laplace spectra must be congruent, or construct a counterexample.

Background

The paper surveys 112 years of advances in spectral geometry, from Weyl’s law to modern results on spectral invariants and isospectral non-isometric examples. For planar “drums” (bounded domains in R^2 with Dirichlet boundary conditions), many geometric quantities are known to be spectral invariants (area, perimeter, Euler characteristic, and, for polygonal domains, corner angles).

Despite extensive progress, the central inverse spectral question—whether the spectrum determines the shape—remains unresolved under natural geometric restrictions. In particular, the case of convex planar domains has resisted resolution: although one cannot hear the shape of an arbitrary drum, it is unknown whether convexity suffices for spectral determination. This is explicitly highlighted among the authors’ open questions.