Analysis and Dynamical Systems PhD projects

This page provides a (partial) list of specific (and not so specific) PhD projects currently offered around the Analysis and Dynamical Systems topic.

Identifying an interesting, worthwhile and do-able PhD project is not a trivial task since it depends crucially on the interests and abilities of the student (and the supervisor!) The list below therefore contains a mixture of very specific projects and fairly general descriptions of research interests across the School. In either case you should feel free to contact the potential supervisors to find out more if you're interested.

You should also explore the School's research groups' pages, and have a look around the homepages of individual members of staff to find out more about their research interests. You may also contact us for general enquiries.

Title

Fractal geometry and dimension theory

Group Analysis and Dynamical Systems
Supervisor
Description

Roughly speaking, a fractal is an object which displays interesting geometrical features at arbitrarily small scales.  Such objects appear naturally in numerous areas of pure and applied science and fractal geometry is the subject which tries to develop a rigorous mathematical framework for studying them.

This project will focus on, but not necessarily be limited to, the pure side of fractal mathematics. We will set out with an open mind and a flexible programme, but specific topics could include: the geometry of iterated function systems, dimension theory in dynamical systems, or multifractal analysis.

Since we are interested in infinitesimal properties, fractal geometry falls naturally under the umbrella of pure analysis.  However, it has many links to other areas and would be appropriate for students with a background in: geometric measure theory, ergodic theory or dynamical systems, and could be linked when appropriate to: probability theory, number theory, hyperbolic geometry, Fourier analysis, functional analysis, harmonic analysis or aspects of topology.

Title

Self-affine sets: geometry, topology and arithmetic

Group Analysis and Dynamical Systems
Supervisor
Description

Iterated function systems (IFS) are commonly used to produce fractals. While self-similar IFS are well studied, self-affine IFS are still relatively new.

In a recent paper Kevin Hare and I considered a simple family of two-dimensional self-affine sets ($=$ attractors of self-affine IFS) and proved several results on their connectedness, interior points, convex hull and corresponding simultaneous expansions. A great deal of natural questions (simple connectedness, set of uniqueness, dimensions, etc.) remain open - even for this most natural family.

The project is aimed at closing these gaps as well as generalising our results to other 2D families (which are completely classified) as well as higher dimensions.

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