Project code



School of Mathematics and Physics

Start dates

February and October

Closing date

Applications accepted all year round

Applications are invited for a 3 year PhD to commence in October or February.

The PhD will be based in the Faculty of Technology, and will be supervised by Dr Andrew Burbanks and Professor Andrew Osbaldestin.

The work on this project could involve:

  • Investigating universal behaviour in nonlinear dynamical systems and estimating the corresponding universal constants
  • Proving the existence of fixed-points of renormalisation operators via rigorous computer-assisted proofs
  • Bounding the spectrum of the derivative of renormalisation operators at fixed points, in order to prove universality

We will prove the existence of objects, known as renormalisation fixed points, that are crucial to understanding how physical systems undergo a transition from predictable to chaotic behaviour. Our approach is extremely powerful: results gained via renormalisation techniques are often universal – they apply to an enormous range of physical, biological, meteorological, ecological, chemical, and mathematical systems.

A classic example is provided by period doubling. In this scenario, a system undergoes cyclic behaviour that repeats over ever-longer time intervals leading eventually to chaos. Observations of mathematical models and physical experiments reveal that features of this transition are universal; the same qualitative features and quantitative measurements emerge across an enormous number of apparently unrelated models and experiments.

Renormalisation provides a means to explore and explain this universality by examining the properties of a renormalisation operator that we can think of as a simplifying transformation that preserves phenomena of interest.

More complicated examples occur when two systems each have their own intrinsic dynamics, but one system drives, or forces, the behaviour of the other. Such systems have their own universal features. A corresponding renormalisation analysis can reveal and explain the behaviour of a wide variety of physical and mathematical systems that have this structure.

Our central challenges are to prove that fixed points of renormalisation operators exist, to gain rigorous bounds on their properties, and to use this information to deduce new results about broad classes of systems. Historically, analytical proofs of such results have been extremely difficult to come by; a number of problems remain open after several decades. Instead, some of these questions have been settled via rigorous computer-assisted proofs. We will take this approach, which leads to existence proofs that are constructive, yielding rigorous bounds on the objects concerned, that may then be used in further analysis.

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).

Bench fees

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.

Entry requirements

You'll need a good first degree from an internationally recognised university 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.

The project would suit candidates with a good background in Mathematics, Applied Mathematics, or Mathematics and Computation, with an interest in using a combination of mathematical and computer-assisted techniques.

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.