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

Scenery flow and fine structure of fractals

Group Analysis and Dynamical Systems
Supervisor
Description

After the recent influential works of Furstenberg and Hochman-Shmerkin techniques based on magnifying measures (and taking their tangent measures) have been essential in the study of arithmetic and geometric features of sets and measures in new settings. For example, these approaches work well with notions that involve questions on the entropy or dimensions of a measure, projections and distance sets, or features related to equidistribution. The key ideas are based on the dynamics or stochastics of the process of magnification and applying classical tools from ergodic theory and Markov chains. I have been recently working on developing these techniques with new arithmetic and geometric applications in mind. The project would try to attempt develop these in the setting of nonconformal dynamics, in particular self-affine fractals such as Baranski carpets and also fractals arising from nonsmooth dynamics such as quasiregular geometry.

Title

Additive combinatorics in dynamics and spectral theory

Group Analysis and Dynamical Systems
Supervisor
Description

The high-frequency asymptotics of Fourier coefficients of functions and measures describe their local structure. For example, they can be used to yield geometric or arithmetic features of the object under study such as on dimension, curvature, equidistribution or combinatoric structure. In my recent work I have been working on finding conditions based on ergodic theory, dynamical systems and stochastics which yield efficient estimates for Fourier transforms of dynamically or randomly constructed objects. Furthermore, we aim to use these estimates to obtain new arithmetic/geometric applications for them. Recent revolutions on applications of additive combinatorics to dynamics by Bourgain, Dyatlov, Hochman, Shmerkin et al. have presented many interesting problems that we will attempt to now solve in the context of thermodynamical formalism. Specific projects include establishing connections between nonlinearity and Fourier transforms, and Fractal Uncertainty Principle for Gibbs measures of Kleinian group actions, which would yield to new essential spectral gap estimates for Laplacian on hyperbolic manifolds.

Title

Open Maps

Group Analysis and Dynamical Systems
Supervisor
Description

The study of open maps is an exciting relatively new area of the theory of dynamical systems. The characterisation of the holes involves geometry (‘shape’) and measure theory (‘size’), so their study involves a complex interplay between dynamics, geometry and analysis.

The standard dynamical approach is to assume that the survivor set (= the points whose orbits do not fall into a hole) is “sufficiently large” and investigate ergodic and geometric properties of the induced map. The novelty of the proposed project is to take a step back and give sufficient conditions that the survivor set is indeed "sufficiently large" (uncountable, say) for the induced map to be meaningful. This involves a detailed analysis of the class of holes under investigation.

The classes of maps under investigation include - but are not limited to - expanding maps of the interval, algebraic toral automorphisms and subshifts.

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.

Title

Thermodynamic Quantum Chaos and large networks

Group Analysis and Dynamical Systems
Supervisor
Description

Quantum chaos is a field that is aimed to study the properties of eigenfunctions of the Laplacian (stationary quantum states) on a Riemannian manifold using the chaotic properties of the underlying geodesic flow in high energy, that is, in the large eigenvalue limit. In this sense the field is connecting quantum mechanics to classical mechanics. A key result in the field is the Quantum Ergodicity Theorem of Shnirelman, Zelditch and Colin de Verdière, which is an equidistribution result of the eigenfunctions for large eigenvalues when the geodesic flow is ergodic. In our recent work we have been attempting to study the theory for a problem of Thermodynamic Quantum Ergodicity (TQE), where instead of large energy, we fix an energy window and vary the geometric properties of the manifold such as volume or genus. The project would aim to develop TQE, in particular for the context of Lie groups and variable curvature manifolds. Moreover, we would attempt to find connect the ideas from TQE to discrete analogues such as spectral theory of large networks, which are well-developed by the recent works of Anantharaman, Brooks, Le Masson, Lindenstrauss, Sabri, and others.

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