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Postgraduate research opportunities

Find out more about our postgraduate research opportunities below.

Wave propagation on graphs

Supervisor: Dr Jens Bolte  (Jens.Bolte@rhul.ac.uk)

Quantum graphs are models for the quantum mechanical motion of a particle along the edges of a graph. They involve a Schroedinger equation with a Laplacian acting on functions on the edges of a graph. Early versions of quantum graphs were used by Pauling as simple models for organic molecules; other variants have since been established in a variety of network models. More recently, quantum graphs were mainly studied in the framework of quantum chaos where one is interested, among other questions, in correlations of quantum energy eigenvalues and in the propagation of wave packets.

The project involves studies of the wave equation as an alternative to the Schroedinger equation and the propagation of quantum wave packets. In contrast to the Schroedinger equation, the wave equation is hyperbolic and should therefore produce a dispersionless wave propagation with constant velocity.

There are many unresolved problems that can be considered in this context such as proving causality on a general graph and proving a suitable Egoroy theorem. Eventually one might be interested in proving quantum ergodicity on a graph. 

Candidates interested in such a project should have a good knowledge of quantum mechanics and/or PDEs. To get some impression of the field one can look at the following references:

  • S Gnutzmann and U Smilansky ‘Quantum graphs: applications to quantum chaos and universal spectral statistics.’ Advances in Physics 55 (2006). P 527-625.
  • J Bolte and S Endres ‘The trace formula for quantum graphs with general self-adjoint boundary conditions’ Ann.H Poincare 10 (2009) p189-223.
  • R.Schrader ‘Finite propagation speed and causal free quantum fields on networks’ J. Phys. A:Math, Theor. 42 (2009) 495401.

Representations of symmetric groups

Supervisor:  Dr Mark Wildon  ( mark.wildon@rhul.ac.uk )

The representation theory of the symmetric group is an active area of research that sees an attractive interplay between algebra and combinatorics. There are important open questions about representations both in characteristic zero and in prime characteristic that could be the subject of a PhD thesis.

In characteristic zero, an important open problem is Foulkes' Conjecture. This concerns the permutation representations of the symmetric group of degree mn acting on set partitions of a set of size mn into either m sets each of size n, or n sets each of size m. If m < n then Foulkes' Conjecture asserts that the former representation is smaller than the latter, in a sense made precise by looking at character multiplicities. Despite considerable attention, the conjecture has only been proved when m ≤ 4, so any partial results would be extremely welcome!

One possible project  would build on [1] and [3] to get a better idea of the 'small' and 'large' irreducible characters contained in Foulkes characters. Another possibility is to consider some natural generalizations of Foulkes characters defined using wreath products of symmetric groups. These generalized Foulkes characters have not been much studied, and there are many open questions.

In prime characteristic, the main open problem is to determine the decomposition matrices of symmetric groups. Roughly put, these matrices record how representations in characteristic zero decompose when they are defined over fields of prime characteristic. In [5] I showed that in odd characteristic, the rows of these matrices are distinct from one another. A good starting project in this area would be to extend this result to alternating groups.

For further details, or to discuss funding opportunities, please email me.

References and further reading:

  1. E. Giannelli, On the decomposition of the Foulkes Module, Archiv der Mathematik, 100 (2013) 201—214.

  2. G. D. James, The representation theory of the symmetric groups, volume 682 of Lecture Notes in Mathematics. Springer-Verlag, 1978.

  3. R. Paget and M. Wildon, Set families and Foulkes modules, J. Algebraic Combin, 34 (2011) 525–544.

  4. B. E. Sagan, The symmetric group: representations, combinatorial algorithms, and symmetric functions, volume 203 of Graduate Texts in Mathematics. Springer, 2nd edition, 2001.

  5. M. Wildon, Character values and decomposition matrices of symmetric groups. J. Algebra, 319 (2008) 3382–3397. 

Group-based cryptography

Supervisor: Simon Blackburn (s.blackburn@rhul.ac.uk)

Over the past 10 years, various proposals have attempted to use group theory to construct secure cryptosystems. Most of these proposals have either been broken or are currently impractical, but there are many interesting combinatorial and computational problems that remain.

The project would aim to understand and then break one or more group-based cryptosystems, or to explore some of the pure combinatorial and computational problems motivated by the cryptographic applications. 

To be successful on this project, a good undergraduate mathematics degree is essential, including a sound knowledge of undergraduate group theory and general algebra. Some knowledge of Mathematica, Maple, Sage or a similar system, and knowledge of combinatorics, is desirable but not essential.

References: S.R. Blackburn, C. Cid and C. Mullan, 'Group theory in cryptography', Groups St Andrews 2009 in Bath, Volume 1, C.M. Campbell, M.R. Quick, E.F. Robertson, C.M. Roney-Dougal, G.C. Smith and G. Traustason (Eds) (Cambridge University Press, Cambridge, 2011) 133-149. See http://arxiv.org/abs/0906.5545


Statistics, Probability Theory and their Applications

Supervisors:  Dr Alexey Koloydenko (Alexey.Koloydenko@rhul.ac.uk

                         Dr Teo Sharia (t.sharia@rhul.ac.uk)

                         Dr Vadim Shcherbakov  (Vadim.Shcherbakov@rhul.ac.uk)

New approaches to inference in hidden Markov models. Hidden Markov models (HMMs) have become indispensable in signal processing and communications, speech recognition, natural language modelling, and computational biology and bioinformatics, and continue to find new applications, for example, in information security.  Recent developments in HMM theory and practice include discovery of limiting Viterbi processes and various hybrid methods of inference about hidden path and model parameters. These have opened up opportunities for further theoretical and applied investigation. Depending on the candidate's background and aspirations, this project can focus more on the theoretical or applied component. A possible application is computational biology benefiting from a collaboration with the Computer Science Department of Royal Holloway. The project will also be benefiting from our continuing collaboration with the Institute of Mathematical Statistics of Tartu University (Estonia).     

On-line estimation in times series models. The aim of the project is to develop new parameter estimation methods for some important classes of statistical models using ideas of stochastic approximation theory. Stochastic approximation is a method to detect a  root of an unknown function when the latter can only be observed with random errors. The project aims to develop procedures  that are recursive and, unlike some other methods, do not require storing all the data. These procedures would naturally allow for on-line implementation, which is particularly convenient for sequential data processing.  In particular, this project would   focus on development of  new recursive procedures  for parameter estimation in autoregressive time series models, and explore possibilities  of extending these ideas to derive new estimation procedures in ARMA models.

Statistical classification of human skin tissue using innovative imaging modalities. Raman spectroscopy is on its way to becoming the most effective imaging modality for computer aided diagnosis and surgery of skin cancer.  Operating with small amounts of data, pilot studies focused on reliable detection of tumour, and had to employ simple classification approaches. As more data are now being acquired and annotated, more specialized classification of skin tissue becomes possible without jeopardizing the overall reliability.  In this project, statistical and machine learning approaches would need to be applied in order to realize the anticipated gains in classification accuracy and reliability. The project is to be closely coordinated with a multidisciplinary team of researchers and medical practitioners from Nottingham University (Nottingham, UK). A successful candidate would need to contribute an increasingly strong expertise in statistical and machine learning, including hands-on data analysis skills. Familiarity with Matlab and confidence in programming in Matlab or a similar environment are highly desirable. The project will also benefit from the links with the Computer Science Department of Royal Holloway, which has an internationally recognized track record in machine learning.

Statistical analysis and modeling of complex environments with application to Diffusion Weighted Magnetic Resonance Imaging (DWMRI).  Diffusion Weighted Magnetic Resonance Imaging (DWMRI) extends the conventional MRI by measuring diffusion. Resulting diffusion profiles in turn allow us to estimate local structure of analysed matter, such as the human brain. Advanced statistical methods, such as inference on non-Euclidean manifolds, have been applied to make this estimation possible. Depending on the candidate's background and interests, this project can focus more on development and investigation of new statistical models and approaches inspired by DWMRI or on extensions of the existing models and methods and their novel applications to particular brain studies. In the latter case, links with the Brain and Behaviour Group of the Psychology Department of Royal Holloway and with St George's Medical School, University of London, should be particularly beneficial. 

Invariance and Symmetry in Statistical Modeling.  Algebraic statistics emerged fairly recently and now covers a broad range of problems where methods of abstract algebra and computational algebraic geometry offer unique insights into probabilistic and statistical models and methods. One particular development takes advantage of the theory of polynomial invariants of finite groups in order to incorporate respective modes of invariance into probabilistic models and related statistical inference procedures. This project would investigate relevance and suitability of these ideas for various classes of statistical models and approaches, as well as particular applications, such as in statistical image analysis and computer vision.    


For details of possible PhD research opportunities in Statistics please click here.

Time Dependent Quantum Systems

 Supervisors: Prof. Pat O’Mahony (p.omahony@rhul.ac.uk) and Dr. Francisca Mota-Furtado (f.motafurtado@rhul.ac.uk)

To describe the evolution of a quantum system subject to time dependent forces requires the solution of the time dependent Schroedinger equation subject to some specified initial conditions. While this is possible analytically in a few cases, such as for a harmonic oscillator with a time dependent frequency, one generally has to resort to numerical methods.  We are interested in understanding the effect of strong time dependent laser fields on atoms and molecules. 

Super strong laser fields of the order of the atomic unit of field strength combined with pulses as short as atto-seconds (10-18 seconds) lead to the creation of novel states of matter allowing one to study the states of atoms, molecules and solids on many different time scales. Attoseconds are on the same time scale as it takes an electron in the ground state of hydrogen to orbit the proton. So the recent experimental realisation of such short pulses makes it possible to study electron motion in real time. 

The project involves solving the Schroedinger equation for atoms such as hydrogen when they are subject to a short low frequency (mid-infrared) intense femtosecond laser pulse. This is an area of current interest where recent experiments have found new and unexpected effects. It is very challenging theoretically as the wave packet created by the low frequency pulse spreads rapidly over space giving rise to multiple physical effects such as harmonic generation and very high energy electrons.

There is a strong overlap with classical theory also and semi-classical theory is used to understand the underlying physical processes.   Understanding how very short intense pulses interact with multi-electron systems in atoms, molecules and in solid state systems is a major goal of our future research.

We have international collaborators in this area and we are part of an EU COST programme running from 2013 to 2017 which allows for exchange visits between many EU research centres and us.

The project would suit a student with an interest in Applied Mathematics/Theoretical Physics and in numerical computation.

 References

  • A. Hamido, J. Eiglsperger, J. Madroñero, F. Mota-Furtado, P. O’Mahony, A. Frapiccini and B. Piraux, Phys. Rev. A 84, 013422 (2011). 'Time scaling with efficient time-propagation techniques for atoms and molecules inpulsed radiation fields'
  •  C.I. Blaga et al, Nature Physics, 5, 335 (2009). 'Strong field photoionization revisited'.
 
 
 
 

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