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Spaces of initial values for Hamiltonian systems

   School of Mathematics and Physics

   Tuesday, January 31, 2023  Competition Funded PhD Project (Students Worldwide)

About the Project

Applications are invited for a three year full-time (or up to six year part-time) PhD to commence in October 2023.

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:

  • Mathematical analysis of differential equations in the complex plane, in particular Hamiltonian systems and their singularities
  • Using methods from algebraic geometry to resolve points of indeterminacy in certain augmented phase spaces of these equations
  • Developing code for computer algebra systems such as Mathematica to perform the relevant calculations and automate the process of regularising the systems of equations

Project description

This research project aims to make advances on the singularity structure of certain ordinary differential equations in the complex plane (in particular Hamiltonian systems which are motivated from physical applications) via their so-called spaces of initial values. These are augmented phase spaces of the equations that can be computed using certain methods from algebraic geometry. Computing such spaces usually involves lengthy calculations that in most cases require a computer algebra system to be performed. Spaces of initial values where originally invented by K. Okamoto for the Painlevé equations, a set of certain non-linear second-order ordinary differential equations that play an important role in mathematical physics, in particular in the theory of nonlinear phenomena and solitons, as they are obtained as symmetry reductions from nonlinear evolution equations / soliton equations. 

The project aim is to construct spaces of initial values for more general equations than the Painlevé equations, such as equations which exhibit a more complicated singularitiy structure in the complex plane (the only movable singularities of the Painlevé equations are poles, apart from 1 or 2 other, fixed singularities), using so-called blow-ups, a method from algebraic geometry that can resolve certain points of the space where the equations are ill-defined. One particular goal is to automate the process of obtaining these spaces as much as possible using computer algebra. This will deliver an important software tool that can be applied to wider classes of equations and have applications in many areas where such equations are studied, such as fluid dynamics, nonlinear optics, biomathematics and many more.

This PhD project builds on recent progress made by the supervisor in this area and the successful candidate will be integrated into a wider research collaboration (with Warsaw University, as well as potential postdocs from third-party funding) in which this research is being performed.  

General admissions criteria

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.

Specific candidate requirements

A strong background in mathematics, in particular (Complex) Analysis, Differential Equations and Geometry. Motivation to work on pure as well as computational mathematical problems, independently and under the guidance of the supervisory team. Willingness to travel to conferences and present research.

How to Apply

We encourage you to contact Dr Thomas Kecker () to discuss your interest before you apply, quoting the project code.

When you are ready to apply, please follow the 'Apply now' link on the https://www.port.ac.uk/study/courses/postgraduate-research/pgr-mathematics?utm_campaign=pg2023-uop-pgr-bursaries&utm_medium=paid-3rd-party-website&utm_source=findaphd&utm_content=course-pgr-mathematics" rel="nofollow" target="_blank">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:SMAP7430423.

Funding Notes

Candidates applying for this project may be eligible to compete for one of a small number of bursaries available; these may provide tuition fee discounts and/or a stipend. This project is also available to students who are self-funded or hold (or likely to hold) scholarships from other organisations or governments.

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