I am always interested in recruiting excellent PhD students. Brief outlines of a couple of potential research projects can be found below. If you are interested in working on one of these projects, or in another related area, please get in touch.
Funded PhD Project
Invisible visibility cloaks
In the past decade, there has been an explosion of academic interest in the creation of invisibility cloaks through transformation optics. Although the theoretical framework for such cloaks was established in the 1960’s, the level of scholarly and public interest in invisibility cloaks significantly increased with the publication of three seminal papers in 2006 coupling the theoretical framework with the experimental demonstration of cloaking for electromagnetic waves. In essence, the creation of invisibility cloaks via transformation optics involves transforming space such that a point is mapped to a sphere of finite radius.
Dr DJ Colquitt
Department of Mathematical Sciences
University of Liverpool, UK
Prof Ta-Jen Yen
Department of Materials Science & Engineering
National Tsing Hua University, Taiwan
This project is a part of a 4-year dual PhD programme between National Tsing Hua University (NTHU) in Taiwan and the University of Liverpool in England. It is planned that students will spend time studying in each institution.
Both the University of Liverpool and NTHU have agreed to waive the tuition fees for the duration of the project. Moreover, a stipend of NT$10,000 (approximately £260) per month will be provided to cover part of the living costs.
For academic enquires please contact Dr Daniel Colquitt (firstname.lastname@example.org). For enquires on the application process or to find out more about the Dual programme please contact Miss Hannah Fosh (Hannah.Fosh@liverpool.ac.uk).
In terms of practical applications, cloaking through transformation optics is severely limited by several features. Notably, since the procedure requires mapping a point to a finite region, we obtain infinite wave speeds on the interior boundary of the cloak. In addition, the creation of a “perfect” invisibility cloak relies on the fact that no waves can interact with the region enclosed by the cloak and, whilst this does indeed render any observer inside the cloaked region invisible, it also requires that the observer be rendered blind.
The overall aim of the proposed project is to design and implement an effective invisibility cloak that would permit a hidden observer to interact with the outside world whilst remaining invisible. The project will begin by studying the theoretical foundations of regularised cloaks in order to familiarise yourself with the concepts involved and the state-of-the-art in the field. The initial part of the project will focus on formalising the link between the regularisation parameter ε and the scattering properties; although this has been studied to some extent in a handful of papers it is still not well understood in general. The link between near cloaks and the transmission and scattering of waves has potential impact across a wide range of disciplines. For instance, it has already been shown that invisibility cloaks can affect the behaviour of the interference patterns in the classical Young’s Double Slit experiment; if path information could be extracted through the use of cloaks that there will be substantial impact on the interpretation of this fundamental demonstration of wave-particle duality.
Having established the link between ε and the scattering properties of the cloak, you will investigate the link between ε and the transmission of energy/information through the cloak. Following the analysis of the transmission problem, the student will draw upon the significant knowledge developed in the earlier parts of your Doctoral studies to design an effect “non-blinding” invisibility cloak which effectively cloaks objects whilst allowing them to interact with the environment. Finally, you will fabricate your design.
The project is a collaboration between Applied Mathematics (UoL) and Experimental Physics (NTHU); the supervisory team represents a unique combination of skills and knowledge in Applied Mathematics, Optics, and Experimental Physics. You will receive training in a broad range of highly sort-after areas including, rigorous applied mathematics, wave scattering, numerical simulation, nano-fabrication, and experimental design and measurement. Both teams have expertise in the FEM soft- ware COMSOL, the FDTD software Lumerical, MATLAB, & Mathematica; all are industry standard software and you will receive extensive training in working with these packages. A strong background in Applied Mathematics or Mathematical Physics is required; familiarity with the analysis of Partial Differential Equations in the context of wave propagation is highly desirable. Ideally candidates should have a 1st Class or 2:1 BSc honours degree in Mathematics or closely a related field. An MSc is an advantage but not essential.
Multi-scale mechanical structures for the control of elastic waves
This project involves the mathematical modelling and design of mechanical structures capable of controlling the propagation of surface and bulk waves in elastic solids. The project will employ both numerical and asymptotic analysis to study a combination of continuous and discrete structures in one-, two-, and three-dimensions. The research programme includes scattering, homogenisation, and spectral problems for finite and infinite systems and has a broad range of applications including, filtering of waves, lensing, and cloaking.
Defect states in discrete elastic systems
This project involves the analysis of finite and semi-infinite defects in elastic lattice systems. These defects may be dislocations or variations in inertial properties and, for finite defects, will have an associated spectrum of eigenstates. The focus of the research programme is on the analysis of these eigenstates and the fields in the vicinity of the defect sites; algorithms will be developed to study the solutions in various asymptotic regimes. The research programme will involve both analytical and numerical models. The project may also include the study of edge and interfacial waves in mechanical lattices.
The mathematics of vascular stents - waves and fluid/solid interaction
The project will be carried out within a collaborative framework involving Applied Mathematicians and Medical Practitioners who study fluid flows and deformation of solids and practice the installation of periodically reinforced stents into blood vessels. The successful candidate will have a strong background in the theory and numerical analysis of partial differential equations. The main focus will be made on modelling of dynamic fluid-solid interaction for stents channelling the blood flow, as well as on special regimes leading to stent failure or resonant blockages of the blood flow. The developed multiscale methodology will be applied to cellular interaction problems.
In conjunction with the Liverpool Centre for Mathematics in Healthcare.
Supervisory team: Prof A B Movchan, Dr D Colquitt, Prof N Movchan, Dr R Bearon, Dr A England