If you are interested in joining the group in any capacity, you are encouraged to to reach out to Bryn via email [bryn.davies@warwick.ac.uk]. Informal enquiries of any form are always welcome. It's generally advisable to reach out as early as possible to discuss possible projects and the various options for funding them.
Potential PhD students from overseas (not with "home" fee status) should reach out particularly early, so that we can look into arranging suitable funding.
Current Warwick students are always welcome to join the group for URSS projects, 4th year R-projects and MSc projects. We have a handful of students doing each type of project every year.
Some specific open projects are listed below, however this list will never be exhaustive and we may have other opportunities that are better suited to you, so please get in touch.
Background: PDEs with random coefficients (or, equivalently, materials with random geometries) often lead to localised eigenfuctions (variants of this are known as Anderson localisation). This is important for applications as it corresponds to wave energy being focused or trapped. Various properties of this phenomenon have been studied extensively for the last half a century [PhysToday]. However, until recently, it remained a challenging (and potentially computationally expensive) problem to identify where in a random material waves of certain frequencies would be localised. A breakthrough on this was made with the invention of landscape functions [PNAS], which provide a single easy-to-compute function that estimates where localisation will occur. However, its predictive powers are typically restricted to low frequencies. When we want to study localisation effects in classical (e.g. electromagnetic or acoustic) systems, the localisation typically occurs at higher frequencies [PRB].
Objectives: Develop a landscape function framework specifically tailored to imperfect periodic media. This is the key regime for understanding how imperfections alter the function of metamaterials. We will exploit knowledge of the spectrum of periodic operators and how perturbations create localised eigenfunctions [CiMP] to develop a tool specifically tailored to imperfect periodic systems.
Funding: This project is funded by the Leverhulme Trust through my Research Leadership Award on developing new imperfection-resilient metamaterials. It will be integrated into the Mathematics Centre for Doctoral Training. Candidates will need to apply through the CDT and mention this project (it's also worth reaching out to Bryn via email to let him know that you are applying).
Background: A latent symmetry (sometimes known as a "hidden" symmetry) is the phenomenon whereby systems are not symmetric in any classical sense (i.e. they don't have any reflectional or rotational symmetries) but they still exhibit properties of symmetric systems. These ideas originate in graph theory [PhysA] and have been extended to wave systems [PRL]. This project will extend these ideas to PDE models.
Objectives: Find an appropriate formulation of latent symmetry for continuous PDE models. Use this to advance applications to imaging, sensing and communication problems. Work with external collaborators to develop experimental realisations.
Collaborators: Malte Röntgen and Wenlong Gao (Eastern Institute of Technology, Ningbo); experts in the physics of latent symmetries.
Funding: This project does not have specific funding attached to it, however it would be a suitable project for funding through the Mathematics Centre for Doctoral Training or the Warwick-EIT Collaborative PhD Programme.
Background: Metamaterials are formed by combining many small, locally resonant elements in a repeating pattern. This gives materials with exotic properties such as negative effective masses, negative refractive indices or negative Poisson's ratios. The repeating patterns are often either periodic (in which case Floquet-Bloch theory can be applied) or random (in which case ensemble averages can be used). We have been studying metamaterials based on a class of quasiperiodic patterns, which are non-periodic but have long-range order. Our work so far has focused on developing methods to estimate spectra [PRSA] and using these to identify and develop novel applications [PRL].
Objectives: Continue the search for novel applications of quasiperiodic metamaterials, particularly to multi-dimensional systems. Search for examples of super band gaps [PRSA] in multi-dimensional systems. Develop new mathematical methods to facilitate design, such as new methods for describing their exotic spectra or new superspace homogenisation techniques. Work with external partners to develop experimental demonstrations of new designs and concepts.
Collaborators: Various collaborators, both in the UK and overseas, that will support the development of experimental implementations of our results.
Funding: This project does not have specific funding attached to it, however it would be a suitable project for funding through the Mathematics Centre for Doctoral Training.