DepartmentSchool of Mathematics and Physics
October, February and April
Applications accepted all year round
Applications are invited for a self-funded, 3 year full-time or 6 year part-time PhD project.
The PhD will be based in the School of Mathematics and Physics and will be supervised by Dr Thomas Kecker.
The work on this project could involve:
- Classifying the types of singularities in the complex plane of solutions of certain complex differential equations
- Computing spaces of initial values of certain classes of differential equations using Mathematica (or similar)
- Exposing the connection between the space of initial values and the underlying symmetries of the equations
This PhD project will offer a unique opportunity to apply cutting-edge techniques originating in algebraic geometry to the study of certain classes of complex differential equations. The PhD project will build on recent work by Dr Thomas Kecker (supervisor) on the singularity structure of solutions of complex differential equations. This work has successfully applied the method of computing the spaces of initial values under continued ‘blowing up’ of ‘base points’ for certain classes of non-linear differential equations, for which one can essentially obtain a complete list of the different types of (movable) singularities that the solutions of the equations can exhibit in the complex plane. It has thus been demonstrated that the space of initial values, originally constructed for the (integrable) Painlevé equations, is a useful concept also for much more general classes of (non-integrable) equations. Building on this work, the PhD project aims at applying and extending this methodology to wider classes of equations and to automate the procedure of computing the space of initial values using computer algebra software, thus providing a useful tool for other researchers to determine the nature of the singularities of solutions of differential equations. It is also known that the geometric properties of the space of initial values, in the case of the Painlevé equations, is reflected in the symmetries of these equation as so-called Bäcklund transformations, a non-linear analogue of raising and lowering operators in linear differential equations. The project will investigate to what extent this also applied to more general classes of equations, connecting the geometric and algebraic properties of an equation. The project will thus involve a unique combination of complex analysis, algebra and geometry to answer fundamental questions in the field of complex differential equations.
Fees and funding
Visit the research subject area page for fees and funding information for this project.
Funding availability: Self-funded PhD students only.
PhD full-time and part-time courses are eligible for the UK Government Doctoral Loan (UK and EU students only).
Some PhD projects may include additional fees – known as bench fees – for equipment and other consumables, and these will be added to your standard tuition fee. Speak to the supervisory team during your interview about any additional fees you may have to pay. Please note, bench fees are not eligible for discounts and are non-refundable.
You'll need a good first degree from an internationally recognised university (minimum upper second class or equivalent, depending on your chosen course) or a Master’s degree in an appropriate subject. In exceptional cases, we may consider equivalent professional experience and/or qualifications. English language proficiency at a minimum of IELTS band 6.5 with no component score below 6.0.
A good working knowledge of either complex analysis, algebraic geometry, the theory of differential equations, or a combination of these, and a genuine interest in applying the methods outlined above to solve fundamental questions in complex differential equations using computer algebra software.
How to apply
When you are ready to apply, please follow the 'Apply now' link on the Mathematics PhD subject area page and select the link for the relevant intake. Make sure you submit a personal statement, proof of your degrees and grades, details of two referees, proof of your English language proficiency and an up-to-date CV. Our ‘How to Apply’ page offers further guidance on the PhD application process.