Quantum Graphs and Their Applications

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The project addresses an emerging field of quantum graphs and their applications to modeling quasi-one-dimensional complex physical systems. In this project, quantum graphs are graphs considered as one-dimensional singular varieties equipped with differential or pseudo-differential "Hamiltonians." The main problems to be considered are modeling wave propagation in thin systems by graph models and studying properties of these models of relevance for many areas of application, ranging from chemistry, to optics, to nanotechnology.Motivation for this study comes from various areas of engineering, sciences, and mathematics. Among them are materials science (nanotechnology, in particular circuits containing quantum wires, and novel optical materials, so-called photonic crystals) and high-performance computing (optical and quantum computing). The project aims to develop the subject of quantum graph theory, which, in spite of its wide applicability, is still in early stages of development. The project will advance knowledge in this important area of mathematics with applications to several active branches of science and engineering.