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City University, London
UOA 21 - Applied Mathematics
RA5a: Research environment and esteem
This submission covers research in mathematics and theoretical physics conducted within the Centre for Mathematical Science, which forms part of the School of Engineering and Mathematical Sciences. The School was formed in 2002 by the unification of the School of Mathematics, Actuarial Science and Statistics and the School of Engineering, with activity in actuarial science moving to the Business School. As well as the activities described here, the Centre has significant research activity in statistics which is undertaken jointly with engineering and has been submitted to the General Engineering Unit of Assessment. All academic staff in the Centre are research active and have been submitted in the assessment exercise.
Research in mathematics and theoretical physics is conducted in four main groups whose activities during the assessment period have involved the following academic staff of the Centre.
A. Fluid Dynamics and Nonlinear Systems:
Professor P G Daniels
Dr O S Kerr
Dr R A Satnoianu*
B. Quantum Field Theory:
Dr A Fring
Dr M Alvarez
Dr O Castro-Alvaredo
Dr V Caudrelier
Dr C Figueira de Morisson Faria*
Dr C Korff *
C. Statistical Mechanics and Algebraic Representation Theory:
Professor P P Martin*
Dr A G Cox
Dr M De Visscher
Dr R Twarock*
D. Theoretical Spintronics:
Professor J Mathon.
Those marked * have now moved elsewhere. Martin accepted a chair at Leeds shortly before the census date and has been replaced by
Professor J Chuang
whose appointment begins on 1 January 2008. The ongoing academic staffing level at that point will be ten, a net increase of two posts compared with the position at the last assessment exercise. This reflects the University’s strategic investment in the Centre following its improved rating (by both the Applied Mathematics and Physics units of assessment) in the last exercise. Arrangements for a further appointment, to begin in 2009, have already been made. The staff list above includes three EPSRC Advanced Research Fellows, a PPARC Advanced Research Fellow and a Royal Society University Research Fellow. Activities have also been supported by visiting fellows, research assistants and a growing number of research students.
Activities and achievements of the research groups 2001-2007
A. Fluid Dynamics and Nonlinear Systems
This group led by Daniels is concerned with fundamental aspects of transition, structure and pattern formation in fluid flows. Research has involved seven international collaborations and has been supported by a visiting fellow and five research students, including two funded by EPSRC.
Significant progress has been made in identifying the boundary-layer structure of thermally-driven cavity flows, focusing attention, as planned, on high Darcy-Rayleigh number porous media flows generated by horizontal differential heating. Similar structures have now been observed in high Prandtl number furnace melt flows, suggesting that this theory will have wide applicability. Other boundary-layer studies by Daniels include further work with Patterson (James Cook) on the instability of the boundary layer on a suddenly heated vertical plate and with Ratnanather (Johns Hopkins) on the thermal field of a separating wall jet.
Kerr, in collaboration with Zebib (Rutgers), has studied the influence of vibration on double-diffusive horizontal intrusions, including results for the low-gravity environment of crystal-growth experiments on the International Space Station. He has also studied instabilities observed in a sediment-laden salt-stratified fluid under a sloping boundary where the settling particles give rise to a buoyant boundary-layer flow and can lead to destabilization of the flow. This work led to the discovery of a new exact solution to the Navier-Stokes equation. He has also investigated the evolution of intrusions with step-like density profiles in a smoothly stratified environment, an important step towards a full understanding of the nonlinear dynamics of intrusions observed in the oceans.
In pattern formation, studies of large planform phenomenological models of convection by Daniels and two research students have confirmed earlier predictions of the onset and weakly nonlinear development of convection along the diagonals of rectangular planforms, explaining some of the complex patterns observed in large aspect ratio systems. For channels, new results for the wavelength and phase shift have been found for the fully nonlinear regime in collaboration with Skeldon (Surrey). A new class of patterns in reaction-diffusion systems has been identified by Satnoianu in collaboration with Daniels and Kuptsov (Saratov) who held a one-year Royal Society Visiting Fellowship in the Centre.
B. Quantum Field Theory
This group has evolved during the assessment period through new appointments designed to build on and complement the existing activity in statistical mechanics and algebraic representation theory. The group is currently supported by three research students and is led by Fring, who was promoted to a readership in 2005. Research activities centre around the key objective in quantum mechanics and statistical models of computing correlation functions.
A large amount of work by Fring and Castro-Alvaredo has been carried out for models with unstable particles in their spectrum, a feature only recently introduced into integrable models. Various aspects have been investigated, such as their supersymmetric version, form factors, renormalization properties and even applications in condensed matter physics. Results have been extended, partly in collaboration with Korff and Figueira de Morisson Faria, to incorporate boundaries and/or defects in quantum field theory and spin chains.
Castro-Alvaredo has developed further aspects of the boundary form factor programme. She collaborated with Maillet (Lyon) to initiate a study of the computation of spin-chain correlation functions in the presence of impurities and with Cardy (Oxford) and Doyon (Durham) to compute the entropy. Fring has collaborated with Korff on Lie algebraic aspects of the programme, in particular on exactly solvable potentials of Calogero type for q-deformed Coxeter groups and on affine Toda field theories related to Coxeter groups of non-crystallographic type.
Korff has developed Q-operators for spin chains and collaborated with McCoy (Stony Brook) on loop symmetry of integrable vertex models at roots of unity, with Roditi (Rio de Janeiro) on superalgebras at roots of unity and non-Abelian symmetries of integrable models, and with Seaton (Melbourne) on universal amplitude ratios and Coxeter geometry in the dilute A model. Caudrelier has collaborated with the groups in Annecy and York to study impurities in integrable quantum field theory.
Figueira de Morisson Faria has collaborated with Fring in his recent study of the relatively new field of PT-symmetric quantum mechanics and quantum field theory, and also with Fring and Castro-Alvaredo to obtain the first theoretical description of harmonic generation in relativistic quantum field theory with impurities.
Alvarez’s work on magnetic sources in non-orientable spaces was carried out partly in collaboration with Olive (Swansea) and led to a study in collaboration with Martin of Temperley-Lieb algebras in higher dimensions. This development can be seen as a direct extension of work on diagram algebras reported in the last assessment exercise.
C. Statistical Mechanics and Algebraic Representation Theory
A number of important advances were made by this group led by Martin, mainly involving the extension of algebraic Lie theory methods into new areas of mathematics. These extensions have typically been prompted by methods and examples arising in statistical mechanics. The group currently includes two research students.
Cox and Martin collaborated with Graham (Sydney) to determine the structure of standard modules for the blob algebra, providing the first complete sets of decomposition numbers for certain representations of affine Hecke algebras in positive characteristic. As the characteristic zero theory is still a challenging problem, this was a significant advance. Cox, De Visscher and Martin determined the blocks of the Brauer algebra in characteristic zero, thus solving a very long-standing problem arising from the classical works of Weyl and Brauer on invariant theory. The alcove-geometric approach has been extended to various families of Hecke algebra quotients, providing an important combinatorial tool for their study. This work has since been axiomatized in a theory of towers of recollement by Cox, Martin, Parker (Sydney and Leicester) and Xi (Beijing) to develop a framework for treating the representation theory of diagram algebras, a topic largely born out of discoveries reported at the previous assessment.
Cox, partly in collaboration with Parker, has determined a number of other important results in algebraic Lie theory, including the first complete description of Hom-spaces for Weyl modules for an algebraic group outside of the trivial case of SL_2. This and related work sheds new light on the major open problem of determining decomposition numbers for the symmetric groups.
De Visscher has worked with Donkin (York) on projective and injective modules for the Schur algebra, and on construction of quasi-hereditary quotients of finite Chevalley groups and Frobenius kernels (that is, a defining characteristic counterpart to the work of Cline-Parshall-Scott and Takeuchi in non-defining characteristic).
Martin defined generalizations of the blob and partition algebras, and used the latter to compute exact results for statistical mechanical models, giving evidence in support of a rich phase structure for certain such models. In collaboration with Parker, Grimm (OU), Doikou (Annecy), Green (Colorado) and his EPSRC postdoctoral research assistant, he developed a suite of new algebras and representations providing new solutions to the Yang-Baxter and reflection equations, and a general framework within which to construct such solutions.
Planned work on quadratic algebras was not developed due to the dramatic discovery by Twarock of applications of aperiodic tiling techniques in virology. This ground-breaking research, supported by EPSRC, became the major part of her work within the group.
D. Theoretical Spintronics
This group led by Mathon works in the new field of spintronics, an area in which the Nobel Prize for Physics was awarded this year. There are four main effects in spintronics: giant magnetoresistance, oscillatory exchange coupling, spin-dependent tunneling, and current-induced magnetization switching and domain-wall movement. The group, which has included two EPSRC postdoctoral research assistants and a research student, is heavily involved in all these areas. It is currently the only UK group to provide theoretical underpinning of the experimental effort of the EPSRC-funded Spinart Consortium of seven UK universities and the Rutherford Appleton Laboratory.
The main achievements have been in spin-dependent tunneling and current-induced switching of magnetization. Collaboration with Umerski (OU) and others has resulted in a series of publications on coherent spin-dependent tunneling in junctions with a magnesium oxide barrier and has predicted that coherent spin-dependent tunneling, in contrast to previously-studied incoherent tunneling, results in a twentyfold improvement of the tunneling magnetoresistance. These predictions have subsequently been confirmed experimentally and are of great technological importance, with reading heads based on this system now being fitted routinely to new computers. This new technology has now replaced the older giant magnetoresistance technology for which the Nobel prize was awarded.
An EPSRC-funded collaboration with Edwards (Imperial) has resulted in a new first-principle theory of spin-transfer torque based on the Keldysh nonequilibrium formalism. This has enabled a fully realistic model to be developed for current-induced switching of magnetization in magnetic nanopillars. This theory is now being applied within the Spinart framework to current-induced movement of domain walls. These effects are currently the subject of intense experimental investigation and have great potential for applications.
New work involves an EPSRC-funded collaborative project with Bland (Cambridge) to study the quantum theory of transport of spin and charge across ferromagnet/semiconductor interfaces.
Leadership of research
The School Research Committee formulates the overall research strategy of the School. Following the 2001 exercise the main priority for the Centre has been to focus resources on top quality research by continuing to build coherent highly-qualified research teams on a foundation of individual strengths. Each of the Centre’s four research groups has been led by an international authority in the field and, with the very recent departure of Martin, rapid action has been taken to appoint a professorial replacement (Chuang) of outstanding ability.
Collaborative and interdisciplinary research
Every member of the Centre is involved in collaborative research, both externally and internally, and much of the activity crosses the boundaries of mathematical physics and pure mathematics. The unification with engineering has provided scope for enhancing interaction within the University. An example is a collaboration in laser physics between Figueira de Morisson Faria and Professor K Grattan (Electronic Engineering). A regular series of School research seminars encompasses areas of engineering and mathematics and encourages further interaction.
A high priority over the assessment period has been to increase the number of research students within the Centre. This has been achieved mainly through successful competition for the award of University Research Studentships and by making available fully funded research studentships within the School. The Centre has been awarded six such studentships since 2004, so that presently there are six full-time PhD students in the Centre, together with four more at the writing-up stage. This is a considerable improvement from 2001.
Research students meet with their supervisors on an at least weekly basis. Each student has a primary and a secondary supervisor and progress is monitored by the Senior Postgraduate Tutor. This includes an annual report by the supervisor and an interim report by the student describing the work carried out in their first year of study. All students have their own PC, attend the Centre’s weekly seminar programme and, from 2007/8 onwards, attend the EPSRC-funded London Taught Course Programme in Mathematics, of which the Centre is a founding member. Research students also attend and present work at conferences, such as the BAMC, and at internal research student workshops.
The unification with engineering has had major benefits for the Centre in terms of research facilities. The School operates a rolling IT replacement system so that all staff have state-of-the-art IT facilities. Local support is provided by the School IT Manager and one other full-time member of School staff. Computing facilities are monitored on a regular basis by the School IT Committee. Particular advantages for the Centre’s staff and students include access to a high-power computer cluster for dedicated research use within the School and shared use of School-wide licences for software such as NAG and MATLAB. The Centre has its own library and the University library provides an extensive range of journals in mathematics and physics in hard copy and electronically.
Seminars and visiting researchers
The Centre runs a weekly mathematics seminar that reflects the numerous common interests of staff within the four research groups and provides a regular flow of visitors to the Centre, including speakers such as Cameron (Queen Mary), Cardy (Oxford), Keating (Bristol), Olive (Swansea) and Skylanin (York).
The Centre has enjoyed a continuous stream of visitors engaged in research collaboration, including:
Doikou (Annecy), Doty (Chicago), Graham (Sydney), Green (Colorado), Jimbo (Tokyo), Karowski (Berlin), Kuptsov (Saratov), Li (Beijing), Manojlovic (Algarve), Marsh (Leicester), Mazorchuk (Uppsala), Nagy (Algarve), Parker (Sydney), Ratnanather (Johns Hopkins), Rittenberg (Bonn), Rotter (Dresden), Ryom-Hansen (Talca), Znojil (Prague).
Relationships with research users
There are excellent relations with users of the Centre’s research, who are mostly other academics and scientists. The School’s Industrial Liaison Panel promotes interaction between the electronics industry and the Centre’s research in theoretical spintronics.
Conferences and summer schools
Conferences and workshops organized by members of the Centre include:
City-Leicester-Oxford-Warwick Network Conference in Representation Theory (City 2004),
Annual UK Conference on Conformal Field Theory and Integrable Systems (City 2005),
Workshop on Diagram Algebras and Cellular Algebras (Oxford 2005, with grant funding held at City),
Conference on Themes at the Interface of Physics and Representation Theory (City 2006),
Sixth International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics (City 2007),
Conference on Infinite Dimensional Algebras and their Applications to Quantum Integrable Systems (Faro 2007).
General research support funds
An annual budget is allocated to the Centre to support its seminar programme and attendance by staff and research students at conferences. Other funds are made available for specific research purposes via the School Research Committee, for example to support the Centre’s membership of the Lighthill Institute. An annual library budget is also allocated to the Centre. Funding for IT hardware and software, including the provision of PCs for staff and incoming research students, is administered by the School IT Committee.
All new members of staff are assigned an academic mentor (a member of senior staff not already directly associated to their research activity) to provide ongoing independent support. All staff take part in an annual appraisal process in which they record their achievements, discuss progress with their appraiser and present targets for the short, medium and long term. The University provides an extensive range of training courses that promote good practice in areas ranging from grant applications to research management and supervision.
The University’s sabbatical system provides opportunities for research leave, and during the assessment period has supported both Daniels and Kerr.
Early career researchers
Despite the youth of some of the recent appointees to the Centre, no appointments have been made of academic staff without strong research experience. The strategy is to implement a light teaching load in the first year or two in lectureship, and to ensure an effective logistical infrastructure, mentoring support, and scope for internal research collaborations.
Staff changes and the effect on research culture
Six of the ten academic staff going forward (Alvarez, Castro-Alvaredo, Caudrelier, Chuang, De Visscher and Fring) are appointments made since the beginning of the assessment period. All of these exciting researchers are young in proportion to their achievements and can be expected to provide sustainability of research output and leadership over the long term. The recruitment of talented young research-active staff has had a profound influence on the research culture of the Centre and the age profile has changed significantly since the last exercise. Half of the Centre’s academic staff are now below 40 years of age and 80 per cent are below 50. When considering this profile against only one member of staff retiring by 2010, the present age distribution guarantees not only sustainability of the Centre’s research leadership and activities but justifiable expectation of growth in both research output and quality at international level. The Centre has been able to retain its core strengths in fluid dynamics, theoretical spintronics and algebraic representation theory, and at the same time to create new activity in quantum field theory, and perhaps most notably, at the interface between representation theory, statistical mechanics and quantum field theory. Martin will continue collaborative research with members of the Centre and the appointment of Chuang to the new Chair in Mathematics is a significant development for the future of the Centre.
Main objectives for the next five years are as follows.
A. Fluid Dynamics and Nonlinear Systems
Asymptotic solutions for plume flows in porous media and high Prandtl number fluids will be extended to problems with net vertical heat transfer, where the possibility of reverse flow within the plume, and its complete collapse for certain thermal conditions, has been identified. Research on double-diffusive intrusions will be extended to incorporate the evolution of the background state. For large intrusions, of interest to oceanographers, the main objective will be to incorporate the double-diffusive dynamics that drive the intrusions and a realistic model of the interaction with the ambient fluid. In pattern formation, results for phenomenological models will be extended to real physical systems, including the Rayleigh-Benard system and the Lapwood system in three dimensions.
B. Quantum Field Theory
Activities will remain focused on the central question of computing correlation functions in quantum field theory and spin chains. Particular emphasis will be given to theories involving impurities, and will be greatly supported by the recent appointment of Caudrelier. Besides the challenge of overcoming persistent difficult technical problems in this area, the group will address the computation of important physical quantities such as the entropy emerging in the context of information theory and quantum computing. Some aspects of this research will be extended beyond the limits of integrability and to higher dimensions. The research in PT-symmetry will be developed to encompass more aspects of integrable quantum field theory.
C. Statistical Mechanics and Algebraic Representation Theory
The expansion of Lie methodologies to more general contexts is a major programme which has only just begun. However, even now it is evident that it will have significant impact on the study of Hecke and affine Hecke algebras, on diagram algebra theory and in algebraic statistical mechanics. It is planned to continue work on the most important examples in these areas, and to develop further a general framework for rationalizing this study. The arrival of Chuang will substantially enhance these activities, and also extend them to include categorification and its role in the representation theory of finite groups, and also to the study of operads and homotopy algebras.
D. Theoretical Spintronics
The work on spin-dependent tunneling and current-induced switching and domain wall movement will continue within the framework of the Spinart Consortium, with EPSRC funding in place to support a postdoctoral research assistant for a further two years. A new Spinart project is currently being planned and the work in metal-based spintronics will also continue. A major new EPSRC grant has just been awarded for a study of the quantum transport of spin and charge across ferromagnet/semiconductor interfaces and will fund another postdoctoral research assistant within the group for the next three years.
ESTEEM INDICATORS DURING THE ASSESSMENT PERIOD
- Alvarez spent the majority of the assessment period as holder of a PPARC Advanced Research Fellowship.
- Castro-Alvaredo gave invited seminars in Wuppertal, York and Bristol and talks at international conferences and workshops in Sao Paulo, Florence, Sozopol, Santiago de Compostela, Annecy, Lyon, Durham and Faro.
- Castro-Alvaredo was co-organizer (with Fring) of the Annual UK Conference on Conformal Field Theory and Integrable Systems (2005) and of the Sixth International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics (2007).
- Caudrelier was awarded a three-year EPSRC Postdoctoral Research Fellowship for a proposal judged outstanding by the referees.
- Caudrelier gave an invited talk at the Meeting on Integrable and Conformal Field Theory in Durham and invited seminars in Jussieu, Montpellier, Orsay, Pisa, Durham, Heriot-Watt, Kent and York.
- Cox gave invited talks at conferences and workshops in Oberwolfach, Gregynog, Antwerp, Warwick, Oxford and Banff, and invited seminars in Aarhus, Cambridge, Kent, Leeds, Manchester, UMIST, Oxford, Sydney and Uppsala, the last two as part of extended visits funded by Sydney University and the Swedish Research Council.
- Cox gave a series of six lectures as one of four main invited speakers at the 2005 International Summer School on Representation Theory in Differential Geometry and Physics in Porto Novo.
- Daniels became an invited member of the Editorial Board of the Quarterly Journal of Mechanics and Applied Mathematics in 2004 following completion of a ten-year period as an Executive Editor of the journal.
- Daniels was one of two invited review speakers at the 2003 Meeting on Pattern Formation and Nonlinear Dynamics in Leeds and one of six invited speakers at the Meeting in Honour of Professors S N Brown and M E O’Neill at University College London.
- De Visscher gave invited talks in Oxford, Queen Mary, Manchester, Leicester, East Anglia and Kent.
- Figueira de Morisson Faria was appointed to a University Research Fellowship and during her period with the Centre won an EPSRC Advanced Research Fellowship to carry out research in theoretical laser physics.
- Fring is an invited member of the Advisory Board of the Journal of Physics A and a Guest Editor for the special issues on Pseudo-Hermitian Hamiltonians in Quantum Physics and Infinite Dimensional Algebras and their Applications to Quantum Integrable Systems.
- Fring presented invited talks at conferences in Nor Amberd, Tianjin, Sao Paulo, Moscow, St Marienthal, Faro, Florence, Swansea, Edinburgh, Bologna, Prague and Stellenbosch, and invited seminars in Trieste, Bayreuth, Hamburg, Beijing, Xian, Santiago de Compostela, Faro, Berlin, St Petersburg, Oxford, Durham, Heriot-Watt, Lisbon and York.
- Fring directed a subnode of the EC-funded network EUCLID: Integrable Models and Applications: from Strings to Condensed Matter and was part of an INTAS collaboration involving Yerevan, St Petersburg and Annecy.
- Kerr was an invited contributor to the final reports of the International Council for Science Scientific Committee on Oceanic Research Working Group on Double-Diffusion in Oceanography, published in Progress in Oceanography.
- Kerr was an invited Visiting Scientist at the Department of Oceanography, Dalhousie University and the Department of Mechanical and Aerospace Engineering, Rutgers University.
- Korff held a Royal Society University Research Fellowship and was awarded an EPSRC grant to fund a visit by Jimbo (Tokyo).
- Martin was a member of the 2005 and 2006 EPSRC Advanced and Senior Fellowship Filter Committees.
- Martin was an invited speaker at the 2001 British Mathematical Colloquium.
- Mathon’s collaboration with Edwards (Imperial) and Umerski (OU) resulted in a string of grants, including four from EPSRC, one from Argone National Laboratory, one as a Visiting Fellow of the JSPS (Japan) and one as a leading member of the Spinart Consortium awarded an EPSRC grant of approximately two million pounds to carry out research in Spintronics at Finite Temperature.
- Mathon’s publication with Itoh, Inoue and Umerski ‘Theoretical Study of Quantum Oscillations of TMR in Junctions with a Nonmagnetic Layer’ was awarded the Distinguished Publication Award of the Magnetic Society of Japan.
- Mathon is an elected member of the Advisory Committee of the Tri-annual International Conference on Magnetism of Thin Films and Surfaces.
- Twarock won an EPSRC Advanced Research Fellowship during her period with the Centre to carry out research in mathematical virology.