The Dirichlet-to-Neumann map records how boundary values of harmonic functions determine their normal derivatives, and is a fundamental object of study in inverse problems and spectral geometry. A natural question is: what geometric conditions are forced on the interior by symmetries or spectral features of this boundary operator? In this talk, we will discuss recent results showing that when the Dirichlet-to-Neumann map commutes with the boundary Laplacian, this strongly constrains the underlying geometry, forcing it to be close to the standard ball. We will then explore stability versions of this statement, and discuss applications in inverse problems. The talk is based on two recent papers: [arXiv:2503.00270] and [arXiv:2510.08822].
