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

Additive combinatorics and Diophantine problems

Group Analysis and Dynamical Systems
Supervisor
Description

The study of Diophantine equations encompasses a diverse portion of modern number theory.  Recent years have seen spectacular progress on solving linear Diophantine equations in certain sets of interest, such as dense sets or the set of primes.  Much of this progress has been achieved by breaking the problem down into a structure versus randomness dichotomy, using tools from additive combinatorics.  One tackles the structured problem using techniques from classical analytic number theory and dynamical systems, whilst the ‘random' problem is handled using ideas informed by probabilistic combinatorics and Fourier analysis.

The consequences of this rapidly developing theory for non-linear Diophantine equations have yet to be fully explored.  Some possible research topics include (but are not limited to) the following:

Existence of solutions to systems of Diophantine equations in dense sets.  To what extent can Szemerédi’s theorem be generalised to non-linear systems of equations?

Quantitative bounds for sets lacking Diophantine configurations.  Can one obtain good quantitative bounds in the polynomial Szemerédi theorem? What about sets lacking progressions with common difference equal to a prime minus one?

Partition regularity of Diophantine equations.  Can one generalise a Ramsey-theoretic criterion of Rado to systems of degree greater than one?

Higher order Fourier analysis of non-linear equations.  Is it possible to count solutions to hitherto intractable Diophantine equations by developing the Hardy—Littlewood method along the lines of Green and Tao?  What are the obstructions to uniformity for such equations?

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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