PhD Research

The Department of Mathematics runs an active PhD research programme on a wide variety of topics. Currently, PhD researchers are working on topics such as discrete mathematics, algebra, cryptography, number theory, quantum chaos, and quantum information.  An overview of the research interests of the department can be found in the Mathematics Research Pages. In normal circumstances all academic staff are happy to take on new PhD students, although from time to time a member of staff may temporarily stop accepting new students if they become overloaded.

We welcome applications from prospective PhD students wishing to propose their own research projects, or who wish to develop a detailed project in conjunction with their supervisor.  However, we also welcome applications from prospective students wishing to work on specific projects of particular interest to members of academic staff, some of which have funding. A short list of such projects is given below.  Prospective applicants are recommended to contact the listed supervisor directly, either before or in parallel with making a formal application to the college admissions office.

1.  Quantum information and foundations of quantum theory

Supervisor:  Dr Jonathan Barrett (Jon.Barrett@rhul.ac.uk)

Prospective PhD students with a strong background in mathematics or theoretical physics, and a deep interest in quantum information theory, the foundations of quantum theory, or both, are encouraged to email Jonathan Barrett (Jon.Barrett@rhul.ac.uk) to discuss ideas for a PhD thesis.

Possible projects are as follows.

• Cryptography and Relativity. Standard quantum cryptography schemes guarantee security against eavesdroppers who are constrained by the laws of quantum physics. Recent work shows that a different physical principle, the impossibility of signalling faster than light in special relativity, can also guarantee security. A further advantage is that users do not need to trust the equipment they are using, hence this is sometimes called "device-independent cryptography". The project will develop this idea, to see how well it works in practice. It might also explore whether other principles of fundamental physics are relevant to cryptography, and other kinds of information processing. For a non-technical account, see:
  • [1] A. Ekert, "Less Reality, More Security", Physics World, September 2009.

  For a technical article, see:

  • [2] J. Barrett, L. Hardy and A. Kent, Phys. Rev. Lett. 95, 010503 (2005).
• Information Processing in General Probabilistic Models. Mathematically, one can write down probabilistic models different from quantum theory, which do not describe Nature yet which are perfectly consistent. The project will investigate the connections between physical principles, which these models may or may not satisfy, and the types of information processing that different models allow. The aim is to discover which features of quantum theory are shared by most models, and which, on the other hand, are unique to quantum theory. This will help us to understand why quantum theory has the structure it does. See:
  • [3] J. Barrett, "Information processing in generalized probabilistic theories", Phys. Rev. A, 032304 (2007).

2.  Zeta functions of groups and rings

Supervisor:  Dr Benjamin Klopsch (Benjamin.Klopsch@rhul.ac.uk)

Zeta functions are analytic functions with remarkable properties.  They play a crucial role in the proofs of many significant theorems in mathematics, such as Dirichlet's theorem on primes in arithmetic progressions, the Prime Number Theorem and the proofs of the Weil conjectures and the Taniyama-Shimura conjectures.  More recently algebraists have defined zeta functions to study infinite groups in terms of their distributions of subgroups, conjugacy classes or representations.  Over the last twenty years, a substantial theory has emerged with many interesting theorems and a growing supply of open problems and conjectures.  Of particular interest are zeta functions of p-adic Lie groups and of arithmetic groups.  The study of their zeta functions has been supported by methods from Lie theory, algebraic geometry and the theory of profinite groups.

To start a PhD project in this area one should have a good background in group theory and an open mind toward algebraic number theory (or possibly the other way around).  Further expertise regarding the subject would be gained in the first year of the PhD, for instance, by computing explicit zeta functions of interesting groups.  Depending on the personal interest and background of the PhD student, one can then focus on particular aspects of zeta functions of a particular family of groups, e.g. the abscissae of convergence of representation zeta functions associated to crystallographic groups. A first impression of the subject area can be gained from the following sources and the references given therein.

3. 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 as, e.g., proving causality on a general graph and proving a suitable Egorov 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 consult the following references.

4. 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 be develop efficient methods for computing the characters of the Foulkes representations. This should involve a nice mixture of algebra, combinatorics and computing, with potentially a big pay-off if a counterexample is discovered. Another possibility, which could be pursued in parallel, would be to build on [2] to get a better idea of the 'small' irreducible characters contained in the Foulkes characters.

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 [4] 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 try to extend this result to alternating groups.

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

References and further reading:

1. [1] G. James, The representation theory of the symmetric groups, volume 682 of Lecture Notes in Mathematics. Springer-Verlag, 1978.
2. [2] R. Paget and M. Wildon, Set families and Foulkes modules. J. Alg. Combinat., 34 (2011) 525–544.
3. [3] B. E. Sagan, The symmetric group: representations, combinatorial algorithms, and symmetric functions, volume 203 of Graduate Texts in Mathematics. Springer, 2nd edition, 2001.
4. [4] M. Wildon, Character values and decomposition matrices of symmetric groups. J. Algebra, 319 (2008) 3382–3397.

5.  Forms in many variables

Supervisor:  Dr Rainer Dietmann (Rainer.Dietmann@rhul.ac.uk)

Recent years have seen a resurgence of activity in Analytic Number Theory, in particular in the Analytic treatment of Diophantine equations. The revised edition of Davenport's book [4] gives a good and accessible introduction to that area, and Browning's monograph [1] is a good survey to many of the more recent developments. One area of particular interest is 'forms in many variables'. The starting point is Minkowski's classical result that any rational indefinite quadratic form in at least five variables has a non-trivial rational zero. Finding analogous results for higher degree forms or for systems of forms is an area of ongoing research and offers many interesting potential PhD projects. Possible projects could be to establish bounds on the smallest non-trivial zero of forms (see [2] for the case of cubic forms), or to discuss interesting new examples of systems of forms, for example one quadratic and one cubic form, or to extend recently established results on 'random'
Diophantine equations (see [3]) to new classes of equations, or to systems of equations.

Students wishing to do a Ph.D. in this area should have a good background in Real Analysis and Elementary Number Theory; previous exposure to Analytic Number Theory or the Hardy-Littlewood circle method is useful, but not absolutely necessary.

References:

[1] Browning, 'Quantitative Arithmetic of Projective Varieties', Birkhäuser, 2009

[2] Browning, Dietmann, Elliott, 'Least zero of a cubic form', arXiv:0907.5261, to appear in Math. Ann.

[3] Brüdern, Dietmann, 'Random diophantine equations of additive type', arXiV:1004.5527

[4] Davenport, 'Analytic Methods for Diophantine equations and Diophantine inequalities', 2nd edition edited by Browning, Cambridge University Press, 2005.

6.  Statistics projects

Supervisors:  Dr Alexey Koloydenko (Alexey.Koloydenko@rhul.ac.uk) and Dr Teo Sharia (t.sharia@rhul.ac.uk)

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

7. Information security and cryptography projects

Supervisors: see list.

For details of possible PhD research opportunities in Information security and cryptography please look here.