Abstract:
The inverse spectral problem for a bounded planar domain $\Omega \subseteq \mathbb{R}^2$ is to ascertain whether any other domain $\Omega_0 \subseteq \mathbb{R}^2$ with same spectrum (namely, the Laplacian spectrum with Dirichlet boundary conditions) is the image of $\Omega$ under a rigid motion of the plane. The methods used to understand this problem draw on diverse areas: for example, PDE, group theory, number theory, and probability. In this talk, we will provide an introduction to the inverse spectral problem and tools that arise in the case of domains with polygonal boundary. In order to give a picture of this, we will show an elegant proof from Grieser and Maronna of how triangles are determined spectrally.