PhD Studentship: Cut and project schemes in hyperbolic space with applications to acoustic metamaterials

The discovery of quasicrystals – honoured with the 2011 Chemistry Nobel prize – has generated and continues to stimulate a tremendous amount of research in the international mathematics and physics communities. Questions concerning the characterisation of order versus disorder, diffraction and spectral theory are extensively studied. Mathematical methods involved range from number theory and geometry, to ergodic theory and harmonic analysis, over to noncommutative geometry. Central to understanding properties of physical quasicrystals and acoustic metamaterials (deliberately engineered composites with bespoke wave-control properties that arise from their structure) is the careful analysis of point sets arising from, for instance, cut and project schemes.

Here, loosely speaking, one takes a straight-line cutting through a square lattice, projects a given window of lattice points onto the straight-line, and considers the resulting point set. This method has recently been adapted by examining what happens when instead of a straight-line one takes a quadratic curve. Using this new construction, Davies, Chaplain, Starkey and Craster, built new acoustic metamaterials with large spectral gaps, yielding high performing sound insulators. This work raised the open question: Can developing new cut and project models, where the lattice is not square or the curve is non-linear, generate better performing graded metamaterials?

Drawing on the rich body of results in the mathematical study of aperiodic order, ergodic theory and hyperbolic geometry, in its first part, this project aims to answer the theoretical aspects of this question, by enhancing the mathematical theory of cut and project schemes in hyperbolic space, and by extracting spectral information of diffraction measures and associated Schrödinger operators arising from the resulting point sets. Building on this, the focus of the second part is to initiate investigations on building and testing new metamaterials based on the mathematical theory developed in the first part of this project.

Funded by: EPSRC Engineering and Physical Sciences Research Council