Funded PhD opportunities

Geodesics on surfaces and manifolds: caustics and integrability.

  • Application end date: 11th February 2018
  • Funding Availability: Funded PhD project (EU/UK/International)
  • Department: Mathematics
  • PhD Supervisor: Dr Thomas Waters, Professor Andrew Osbaldestin

Project code: MPHY4280218

Project description

If we picture a surface such as a sphere or ellipsoid, then in order to understand the geometry of the surface we need some notion of straight lines. Geodesics are a special class of curves in surfaces which play the role of straight lines. They are defined by taking the properties of straight lines and generalising them. One option is to imagine a particle moving across the surface free from any forces; its path would be a geodesic. It is for this reason that geodesics play such a central role in, for example, General Relativity.

Now suppose we choose a point in the surface and consider a spray of geodesics emanating from that point. Due to the curvature of the surface these geodesics may focus along a curve, much like light reflecting off a cup of coffee will focus along a curve on the surface of the liquid. The focussing line is known as a caustic, and the caustic of geodesics is called the conjugate locus.

The equations which describe the geodesics of a surface can be surprisingly complicated, indeed there are simple surfaces for which the geodesic equations cannot be solved. The question of whether we can or cannot solve the equations is known as integrability, and one consequence of non-integrability is the chaotic behaviour of geodesic curves.

This project will make a detailed study of the caustics of geodesics on certain classes of surfaces, a topic of which surprisingly little is known and much remains to be discovered. We will also consider the integrability of the geodesic equations on certain surfaces and seek new surfaces which are integrable. Finally we will combine these two notions and examine what effect non-integrability has on the caustic of geodesics.

The background theory of this project is Differential/Riemannian Geometry on one hand and Lagrangian/Hamiltonian dynamics on the other. We will also use techniques from bifurcation/singularity theory and perturbation methods, and at every stage we will use software to visualise and numerically experiment with this rich and complex topic. The techniques and methods of this project have found application in various fields such as seismology, optimal control theory and crystallographic optics.

Supervisor profiles

Dr Thomas Waters

Professor Andrew Osbaldestin

Admissions criteria

You’ll need a good first degree from an internationally recognised university (depending upon chosen course, minimum second class or equivalent) or a Master’s degree in an appropriate subject. Exceptionally, equivalent professional experience and/or qualifications will be considered. English language proficiency at a minimum of IELTS band 6.5 with no component score below 6.0.  

Enquiries

Informal enquiries are encouraged and can be made to Dr. Thomas Waters at thomas.waters@port.ac.uk (02392 846383) or Professor Andrew Osbaldestin at andrew.osbaldestin@port.ac.uk (02392 846366).

For administrative and admissions enquiries please contact tech-enquiries@port.ac.uk.

How to Apply

We welcome applications from highly motivated prospective students who are committed to develop outstanding research outcomes. You can apply online at www.port.ac.uk/applyonline. You are required to create an account which gives you the flexibility to save the form, log out and return to it at any time convenient to you.

A link to the online application form and comprehensive guidance notes can be found at www.port.ac.uk/pgapply.

Applications should include:

- Full CV including personal details, qualifications, educational history and, where applicable, any employment or other experience relevant to the application

- Contact details for two referees able to comment on your academic performance

- Research proposal of 1,000 words outlining the main features of a research design you would propose to meet the stated objectives, identifying the challenges this project might present and discussing how the work will build on or challenge existing research in the above field.

- Proof of English language proficiency (for EU students)

When applying, please quote project code: MPHY4280218

Interview date: TBC

Start date: October 2018.

Funding notes

The fully-funded, full-time three-year studentship provides a stipend that is in line with that offered by Research Councils UK of £14,553 per annum.

The above applies for Home/EU students only.

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