Events

Date: 21 May 2026   Time: 13:00 - 14:00     

Location: MB-503

"Spectral optimization" refers to the study of how shapes affect eigenvalues of self-adjoint operators by trying to maximize or minimize an eigenvalue, or some combination of eigenvalues, under reasonable constraints. This is a well-developed subject for the Laplacian on domains and manifolds, and some other PDEs and ODEs, but only in recent years have such questions been posed on metric graphs. Remarkably, the answers are sometimes quite different from the classical cases.

I'll review some known optimal spectral estimates for metric graphs and will then discuss recent work on optimal ratios of eigenvalues and gaps between eigenvalues. For example, I will prove that the Dirichlet metric trees that produce the largest ratio of the first two eigenvalues are equilateral stars, and will pursue estimates for arbitrary eigenvalue ratios that track with Weyl's distribution law. If time permits I will also describe some optimizers for ratios and gaps for quantum graphs that include potential energies.