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Jonathan Rohleder: Isoperimetric spectral problems for quantum graphs

Time: Wed 2020-10-07 13.15 - 14.15

Location: Kräftriket, house 5, room 14

Participating: Jonathan Rohleder, Stockholms universitet

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Isoperimetric problems have a long history in spectral theory and mathematical physics. A typical question is to find, under a constraint on the volume, diameter, perimeter or another geometric quantity, the shape which maximizes or minimizes the lowest nontrivial eigenvalue of the Laplacian subject to Dirichlet or Neumann boundary conditions; the famous Faber-Krahn inequality is one such result. In this talk we review corresponding results for Laplacians on metric graphs. In particular, we discuss a recent result on which metric tree graphs maximize higher eigenvalues of the Laplacian with continuity-Kirchhoff vertex conditions under a constraint on the arithmetic mean of the edge lengths. As it turns out, these optimal trees are not balanced with respect to their edge lengths, except for the optimizer of the lowest eigenvalue.