Recent Advances in the Spectral Geometry of Domains and Approaches with Computer-Assisted Proofs

What can we determine about the shape of a drum from its sound?"—This inverse problem has given rise to spectral geometry and has attracted researchers for over 110 years. The first half of the talk explains recent advances in shape optimization problems for domains with respect to eigenvalues of the Laplacian and the inverse problem known as "hearing the shape of a drum," presented in an accessible manner for experts from other disciplines. The second half introduces verified computation methods for eigenvalues, eigenfunctions, and shape derivatives. As applications, it presents newly established computer-assisted proofs for the minimization problem of eigenvalues with non-homogeneous Neumann boundary conditions, and the conjecture on the simplicity of the second Dirichlet eigenvalues for non-equilateral triangles.