Working Group 2 focuses on spectral geometry in continuous and non-smooth settings, addressing fundamental questions on how geometric structure influences spectral behavior. Its overarching aim is to understand and control eigenvalues, eigenfunctions, and spectral stability for a wide class of operators arising in mathematics and physics.

A key strand of research within WG2 concerns spectral optimization problems, including the existence, characterization, and stability of optimal geometries for eigenvalue functionals. Closely related is the study of nodal sets and nodal domains of eigenfunctions, which encode fine geometric and topological information about the underlying space. Particular emphasis is placed on non-local and pseudo-differential operators, such as Dirichlet-to-Neumann maps, as well as Laplacians on non-smooth spaces.

WG2 also addresses modern challenges related to spectral instability and pseudospectra, especially for non-self-adjoint operators. These questions are motivated by applications ranging from wave propagation to mathematical physics, including the analysis of quasi-normal modes in black hole models. Through these investigations, WG2 contributes advanced theoretical insights that naturally connect to random geometry, materials science, and complex networks studied elsewhere in the Action.