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PhD course: Quantum graphs

First meeting: sal 306, hus 6 , Kräftriket, September 3, 2019.

Time: Tue 2019-09-03 10.15 - 12.00

Location: sal 306, hus 6, Kräftriket

Participating: Pavel Kurasov

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Quantum graphs denote a wide class of systems that can be described by ordinary differential equations on metric graphs. Such models come from physics where they are used to describe systems where the dynamics is confined to a neighborhood of graph-like structures (nanosystems, waveguides, etc.). Studying quantum graphs one may easily see relations between their geometry/topology and spectral properties. I do not know any other example in mathematics where such relation is so straightforward. To study quantum graphs one needs to combine different areas of mathematics as: spectral theory of differential operators, topology, algebraic geometry, number theory, polynomials in several variables, etc. Quantum graphs were introduced at the end of last century making it a relatively young research area with numerous interesting research problems.

This course will give an introduction into the theory of quantum graphs describing their spectral and scattering properties. In particular we are going to discuss how geometric properties of graphs are reflected by the spectrum, how methods from algebraic topology can be used to study number theoretic properties of the eigenvalues. We shall also discuss inverse problems, i.e. how to reconstruct quantum graphs from their spectra.

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