Quasicrystals and networks PhD student project

Announced on www.math.su.se

Funded by the Swedish Research Council (VR)

Interesting mathematical results are often obtained when ideas from completely di˙erent of

c www.sigma-aldrich.com

areas of mathematics come together and interact in a non-trivial way and fertilizing each other. One such example is a recent construction of non-trivial crystalline measures generalising classical periodic measures. The corresponding object is often referred to as a Fourier quasicrystal. Such measures were studied for decades by specialists in Fourier analysis (J.-P. Kahane, Y. Meyer, A. Olevskii, N. Lev, etc.) proving spectacular results, but it was still unclear how to construct explicit non-trivial examples.

It appeared that precisely such measures describe spectral properties of Laplacians on metric graphs, also known as quantum graphs studied by P. Exner, U. Smilansky, P. Kuchment, G. Berko-laiko, etc. In contrast to discrete graphs, quantum graphs also take into account distances between the nodes, turning them into geometric objects. The proof of the crystalline character of their spec-tral measures is based on scrupulous investigation of the relation between the topology/geometry of quantum graphs and their spectrum (P.Kurasov and P. Sarnak).

This discovery opened a small Pandora’s box in Fourier analysis with several alternative con-structions of crystalline measures appearing in just few months. The impact on the spectral theory of quantum graphs was also tremendous proving fne arithmetic properties of their spectra, and thus providing a rather unique example of a quantum system where the additive structure of the spectrum is fully understood.

The connection between di˙erential operators on singular manifolds (a metric graph can be seen as a singular one-dimensional manifold) and exotic measures in Fourier analysis has not been fully explored. The aim of the this PhD project is to go further constructing crystalline measures in several dimensions clarifying their connections to physical and chemical quasicrystals (Penrose tiling, Dan Shechtman’s quasicrystals (Nobel Prize 2011)).

Mathematically, the project lies on the border between Fourier analysis, discrete mathematics, spectral theory, and number theory, and therefore knowledge in some of these areas is a requirement, whilst certain acquaintance with quantum physics is desirable.

The successful candidate will join the active research environment of one of the leading mathe-matical institutions with established collaborations all over the world (including Bonn, Princeton, etc) and actively participate in two international Networks

• Research and Training Network "QGRAPH" (http://sta˙.math.su.se/kurasov/QGRAPH/); • COST Action CA18232 "Mathematical models for interacting dynamics .networks"on

Supervisor: Pavel Kurasov, Stockholm university. http://sta˙.math.su.se/kurasov/

If you are interested, please send an e-mail to Pavel Kurasov at kurasov@math.su.se

Deadline for application: April 23, 2021

c Dan Shechtman