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Texas A&M University

Events for 11/19/2020 from all calendars

Mathematical Physics and Harmonic Analysis Seminar

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Time: 10:00AM - 11:00AM

Location: Zoom

Speaker: James Kennedy, University of Lisbon

Title: Spectral partitions of metric graphs


We introduce a theory of partitions of metric graphs via spectral-type functionals, inspired by the theory of spectral minimal partitions of domains but also with a view to understanding how to detect "clusters" in metric graphs.

The goal is to associate with any given partition a spectral energy built around eigenvalues of differential operators like the Laplacian, and then minimize (or maximize) this energy over all admissible partitions. Since metric graphs are essentially one-dimensional manifolds with singularities (the vertices), the range of well-posed problems is much greater than on domains. We first sketch a general existence theory for optimizers of such partition functionals, and discuss a number of natural functionals and optimization problems.

We also illustrate how changing the functionals and the classes of partitions under consideration -- for example, imposing Dirichlet versus standard conditions at the cut vertices or considering min-max versus max-min type functionals -- may lead to qualitatively different optimal partitions which seek out different features of the graph.

Finally, we show how for many problems the optimal energies behave very similarly to the eigenvalues of the Laplacian (with Dirichlet or standard vertex conditions), in terms of Weyl asymptotics, upper and lower bounds, and interlacing inequalities.

This is based on joint works with Matthias Hofmann, Pavel Kurasov, Corentin Léna, Delio Mugnolo and Marvin Plümer.