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The arithmetic structure of the spectrum of a metric graph

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OOEW04 - Structure and Randomness - a celebration of the mathematics of Timothy Gowers

Endowing a finite combinatorial graph with lengths on its edges defines singular 1-dimensional Riemannian manifolds known as metric graphs. The spectra of their Laplacians have been widely studied.We show that these spectra have a structured linear part described in terms of arithmetic progressions and a nonlinear ”random” part which is highly linearly and even algebraically independent over the rationals.These spectra give rise to exotic crystalline measures and resolve various open problems concerning them. Joint work with Pavel Kurasov.

This talk is part of the Isaac Newton Institute Seminar Series series.

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