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Spectral Geometry on Graphs and Beyond

Final Report Summary - SPEGYONGRAPHSPLUS (Spectral Geometry on Graphs and Beyond)

This project concerns the extraction of geometric information about graphs from the spectra of the graph’s Schrödinger operator, and from the distribution of zeros of the corresponding eigenfunctions. The spectral geometric point of view shows intrigue links between quantum and combinatorial graphs, which go over towards higher dimensional domains. In this sense our research offers a cross disciplinary perspective - the investigation of spectral geometry covers both quantum and combinatorial graphs. We make connections between quantum graphs which are one dimensional objects and domains which are of higher dimension and further apply graph related methods for real-life composite materials.

During this project, we have made some substantial progress in a few research directions within the field.
From the eigenvalues perspective, we have now a better understanding on construction graphs with a particular set of eigenvalues. This is useful to obtain clusters of graphs which share the same eigenvalues (a.k.a isospectral graphs).
When the graphs in question are infinitely periodic, their eigenvalues "broaden" to form so called spectral bands. We have shown that those spectral bands have some universal features which surprisingly do not change when the topology or geometry of the underlying graph changes.
Even more striking is the observation that the tools used in the study of periodic graphs can be applied after a suitable modification to real-life materials. Specifically, we investigate materials which are formed as periodic layers. The materials can be of different nature (say mechanical or optical), but all of them share the possibility of wave propagation. The frequencies of those waves are arranged similarly to the spectral bands mentioned above. This similarity allowed the adaptation of techniques from the graph regime in order to reveal universal properties in wave frequencies of those real-life materials.

From the eigenfunction perspective, we study zeros of eigenfunctions. These zeros may be physically visualized as the stationary points which are formed while a graph vibrates in a certain natural frequency. The study of those nodal points and nodal domains is a wide research field within spectral theory.
During the course of the project we contribute to this field by adopting the spectral geometric perspective. Namely, we investigate what is the information which those zero points (say their number) encapsulates on the nature of the vibrating graph. We managed to fully solve such a problem for the class of tree graphs, which are graphs not possessing a closed orbit and we continue to investigate it on many other families of graphs.

In the beginning of the project, I joined the mathematics department in the Technion and started a tenure-track assistant professor position. I moved to the Technion after three years of EPSRC postdoctoral fellowship
In the mathematics department of the university of Bristol, UK.
My reintegration was made easier thanks to the obtained Marie-Curie grant.
In addition, I benefited from the vast amount of resources which became available to me upon the move to the Technion. This concerns first of all the human resources; having excellent colleagues and attending high-quality seminars.
The Technion attracts high quality students in all levels. Thanks to this, I was able to mentor a project student (for a semester), two M.Sc. students and a PhD student.