| Abstract: |
| I will give a quick overview of the problem of understanding flat bands (eigenvalues) of periodic graphs. I will then focus on maximal abelian covers, which form an important class of periodic graphs, and for which we obtained a complete characterization of the flat bands in terms of the combinatorics of the base graph. Coupling this criterion with a sophisticated analysis, we proved a conjecture of Higuchi and Nomura from 2009, stating that maximal abelian covers of regular graphs have no eigenvalues. I will also mention relations between these spectral atoms and those of universal covers, which are a lot easier to analyze as there are much fewer such covers (a single one in case of a regular graph of a fixed degree, in contrast to a whole bunch of covers in case of maximal abelian ones).
Based on joint work with Wenbo Li, Michael Magee and Joe Thomas. |
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