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Spectral theory

We study qualitative and quantitative spectral properties of the Laplacian and related operators in different settings such as Euclidean space, Riemannian manifolds and quantum graphs. Our emphasis is on the interplay between analytic and geometric properties and how they reflect themselves in isoperimetric inequalities, the asymptotic behaviour of eigenvalues and properties of eigenfunctions. These include, for instance, the effect of geometric properties on the asymptotic behaviour of extremal sets, dependence of the behaviour in the dimension for global quantities such as the spectral determinant, etc.