Spectral geometry: old questions and new answers

Abstract: These are extended notes based on a series of lectures given by the author at the Institut de Mathematiques de Toulouse in June 2025. The goal is to present classical problems as well as the most recent developments in spectral geometry of the Laplace operator in Euclidean domains, subject to Dirichlet, Neumann and Robin boundary conditions. Among the topics covered, there are embedded eigenvalues in unbounded domains, spectral isoperimetric inequalities, nodal-line and hot-spots conjectures, inverse problems of hearing the shape of a drum and geometrically induced eigenvalues and Hardy-type inequalities in curved tubes.