Renormalization in dynamical systems
Self-funded PhD students only
School of Mathematics and Physics
Applications accepted all year round
Applications are invited for a self-funded, 3-year full-time or 6-year part-time PhD project, to commence in October 2019 or February 2020. The PhD will be based in the School of Mathematics and Physics and will be supervised by Dr Andrew Burbanks and Prof. Andrew Osbaldestin.
On this project, 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 (doubling) time intervals, leading eventually to chaotic behaviour. Observations of many mathematical models and real physical experiments reveal that some features of this transition are universal; not only are the same qualitative features seen, but also the same quantitative measurements emerge across an enormous number of apparently completely unrelated models and experiments.
The renormalisation approach provides a means to explore and explain this universality. This is done by examining the properties of a renormalisation operator. We can think of this operator as a simplifying transformation that simplifies a complicated system, while preserving features of interest.
More complicated examples occur when two or more physical systems are coupled together. The simplest type of coupled system is one in which two systems have their own intrinsic dynamics, but one system drives, or forces, the behaviour of the other. Such systems have their own universal features, along the lines described above for a single system. A corresponding renormalisation analysis can therefore 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. Although analytical proofs for such results are possible in certain cases, historically such results have been extremely difficult to come by. A number of problems remain open after several decades. Instead, a number of these questions have been settled via rigorous computer-assisted proofs. We will take this approach, to give existence proofs that are constructive.
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).
Home/EU full-time students: £4,327 p/a*
Home/EU part-time students: £2,164 p/a*
International full-time students: £15,900 p/a*
International part-time students: £7,950 p/a*
*Fees are subject to annual increase
By Publication Fees 2019/2020
Members of staff: £1,610 p/a*
External candidates: £4,327 p/a*
*Fees are subject to annual increase
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 Civil Engineering or related area. 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.
How to apply
We’d encourage you to contact Dr Andrew Burbanks (email@example.com) to discuss your interest before you apply, quoting the project code MPHY4470219.
When you're ready to apply, you can use our online application form and select ‘Mathematics and Physics’ as the subject area. 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 also offers further guidance on the PhD application process.