Integrability in nonlinear systems
PhDs and postgraduate research
Self-funded PhD students only
School of Mathematics and Physics
October and February
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 Sergi Simon.
The work on this project could involve:
- The study of the higher-order variational equations of any given dynamical system, both from the numerical and from the symbolical point of view. This requires some ability in programming which can be either brought as a default by the candidate, or acquired early on.
- The study of the monodromy and Galois groups of any given system, again from a numerical and symbolical point of view.
- A deft use of properties of special functions, linear differential systems and multilinear Algebra.
- The application of results and conjectures to an array of real-life examples; these could be taken from Celestial Mechanics, general Physics and models seen in Finance and Medicine.
Its potential scope is twofold: on one hand, we study the higher-order linearisation of a given system (which in practice entails an infinite matrix, whose structure was clarified by the Proposal author in 2014) and on the other, we study its dual system, whose structure was also found in compact form by the same author.
These two systems are equally important.
The direct system has a relation with the Galois groupoid that we wish to clarify, because it would show light on an infinite-dimensional analogue of well-known results by Ziglin, Morales-Ruiz, Ramis and other authors, and would in fact be the final step in the completion of the algebraic characterisation of integrability of dynamical systems and chaos. Clarifying the Galois groupoid in relation to the Galois groups of the subsystems of the higher-order variational set has long been a goal in the study of integrability, and the 2014 formulation makes this simpler now.
The other system is also important because it encodes necessary conditions on potential first integrals of the given system. This is useful to solve the system itself, or at least to reduce its number of equations.
Whether a PhD thesis will address the direct system (amenable to differential Galois theory as well as knowledge about categories, general Algebra and Complex Analysis in several variables) or the dual system (more susceptible to a study from the point of view of formal calculus as well as the aforementioned Complex Analysis), depends on the applicant's general purview and range of skills and interests. Both systems could theoretically be addressed in the same project, but this is not advised if the time span is set to be three years
Fees and funding
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).
2021/2022 fees (applicable for October 2021 and February 2022 start)
PhD and MPhil
Home/EU/CI full-time students: £4,500 p/a**
Home/EU/CI part-time students: £2,250 p/a**
International full-time students: £17,600 p/a*
International part-time students: £8,800 p/a*
PhD by Publication
External candidates: £4,407*
Members of staff: £1,720
All fees are subject to annual increase. If you are an EU student starting a programme in 2021/22 please visit this page.
*This is the 2020/21 UK Research and Innovation (UKRI) maximum studentship fee; this fee will increase to the 2021/22 UKRI maximum studentship fee when UKRI announces this rate in Spring 2021.
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.
- Computational skills. No previous knowledge of programming is strictly necessary, however the successful candidate should be willing to hone their programming skills for numerical and symbolical computations.
- Basics on Galois theory, Category Theory, Topology, groupoids and complex dynamics-related topics such as monodromy, confluence and Stokes phenomena: no previous knowledge is strictly necessary as long as you can familiarise with these topics early on, or whenever they are needed during the project.
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.
When applying please quote project code: SMAP5271021.