Maxwell Institute

We are part of the Maxwell Institute for Mathematical Sciences partnership with Edinburgh University, and we share our research activities. There are currently 90 academic members of staff and 20 or so post-doctoral research fellows and research assistants in the Maxwell Institute. The departments involved are highly rated: in the 2001 RAE exercise, Heriot-Watt University achieved grade 5s for both Applied Mathematics and for Statistics & OR, whilst the University of Edinburgh achieved 5* in Pure Mathematics, 5 in Applied Mathematics, and 4 in Statistics & OR.

The Maxwell Institute covers a very wide spectrum of research in mathematics and in its applications, and we describe below our main strands of interest in the Heriot-Watt Mathematics department.

Areas of Research - Heriot-Watt Mathematics

Algebra and number theory

The main areas of research of the Algebra and Number Theory group are: representation theory, algebraic combinatorics, noncommutative algebra, quantum algebras, Lie algebras and automorphic forms, commutative algebra, algebraic geometry, algebraic number theory, algorithmic aspects of algebraic curves, combinatorial aspects of network design, geometric group theory, homological algebra, combinatorial and geometric semigroup theory, automata and languages.

The group has close links with the Geometry and Mathematical Physics research groups within the Maxwell Institute. There have also been recent interactions with the Analysis and Statistics groups.

Heriot-Watt researchers: J Howie, ND Gilbert, MV Lawson, AR Prince

Analysis

The Analysis group works on a wide range of topics. Our particular strengths lie in harmonic analysis, spectral theory, linear and nonlinear elliptic, hyperbolic and parabolic PDEs, dynamical systems, stochastic analysis including stochastic nonlinear PDEs, foundations of numerical analysis and applications of the above to problems of physics, continuum mechanics, etc... The group has active links with most of the other groups in the Maxwell Institute including Applied Mathematics, Geometry and Topology, Computational Analysis and Probability. We run a very active joint research seminar and a working group seminar.

Heriot-Watt researchers: L Boulton, ken, J Carr, D Coutand, RJ Knops, kuksin, AA Lacey, SJA Malham, ar134, BP Rynne, W Staubach

Applied mathematics

Research activities in Applied Mathematics cover a wide range of topics, many related to differential equations -- ordinary, partial and stochastic -- and dynamical systems. They include nonlinear waves, such as solitons, with applications to optoelectronics and continuum mechanics, electromagnetic and elastic wave propagation in complex and composite media, stellar dynamics and Hamiltonian mechanics, and fluid, solid and statistical mechanics. There are applications to oil-reservoir modelling, geophysical and astrophysical systems, turbulence, combustion, phase transitions and free-boundary problems, and to more general industrial and social modelling. The mathematical methods employed range from asymptotic techniques, in particular exponential asymptotics and homogenisation, to dynamical-system techniques, analysis and numerical methods.

There are close links between Applied Mathematics and other activities within the Maxwell Institute, such as Computational Mathematics, Mathematical Biology, Mathematical Physics, Pure and Applied Analysis, and Probability and Stochastics. Regular seminars are organised for postgraduate students and staff.

Heriot-Watt researchers: E Buckwar, J Carr, DB Duncan, JC Eilbeck, AA Lacey, GJ Lord, SJA Malham, RJ Knops

Computational mathematics

Research in computational mathematics covers a broad range of different areas and has strong interdisciplinary links. The focus of our work is on integrated modelling, formulation, analysis and numerical algorithms for differential equations, including ODEs, PDEs, integro-differential equations and stochastic DEs. Of particular interest is the development of innovative discretisation methods and on the approximation of spectral properties of differential operators. Research topics are varied; some examples include the use of computer algebra to analyse integrable systems, the development and analysis of geometric integrators, the design of efficient numerical schemes for multiple scale modelling, stochastic PDEs and quantum lattice dynamics.

Applications arise from diverse areas of science and engineering, including biomedical science, finance, fluid dynamics, material science, molecular dynamics, modelling of neurons, oil reservoir simulation, phase transitions and wave propagation.

There are close connections between Computational Mathematics and other activities within the Maxwell Institute, such as Applied Mathematics, Mathematical Biology, Mathematical Physics, Pure and Applied Analysis, and Probability and Stochastics. Collaborations are in place with other major research groups in computational mathematics based throughout the UK and the world.

Heriot-Watt researchers: L Banas, L Boulton, E Buckwar, JC Eilbeck, DB Duncan, GJ Lord, SJA Malham

Geometry and topology

The main areas of research of the geometry and topology group are algebraic geometry, algebraic topology, differential geometry, geometric group theory and surgery theory. More specific research areas include birational geometry (especially of 3-folds), Kaehler geometry, topics in gauge theory, geometry of moduli spaces and high-dimensional manifolds and knot-theory.

The group has links with the following research groups within the Maxwell Institute: Algebra, Analysis, and Mathematical Physics.

Heriot-Watt researchers: J Howie, paul, richardw

Mathematical biology

Research in mathematical biology concerns the application of mathematics to cell biology, medicine, ecology and evolution. Some of our work is focussed on specific applications and is done in collaboration with experimental biologists or field ecologists. Other work is more theoretical in nature, developing fundamental modelling techniques with potential applications to a wide range of biological problems. Our models include ordinary and partial differential equations, and spatially discrete models such as cellular automata.

Heriot-Watt researchers: KJ Painter, JA Sherratt, AR White

Mathematical physics

The Edinburgh Mathematical Physics Group consists of 15 permanent academic staff, 6 postdoctoral research fellows and 13 postgraduate students. Our main areas of research are string and M-theories, classical and quantum integrable systems, topological quantum field theories, low-dimensional quantum gravity, statistical mechanics of random surfaces; although our interests are varied and span a wide range of topics in modern mathematics and physics. We have close ties with the Algebra and Topology/Geometry groups. Our regular research activities include a weekly seminar series, as well as student seminars and journal clubs on a variety of topics, most recently generalised geometry and D-branes. We have our own preprint series which contains most of our research output.

Heriot-Watt researchers: JC Eilbeck, DA Johnston, A Konechny, O Penrose, BJ Schroers, RJ Szabo, RA Weston

 


STAFF RESEARCH INTERESTS

Dr L Banas: Modelling and numerical analysis of nonlinear time-dependent PDEs

Dr L Boulton: Spectral theory and mathematical physics.

Dr E Buckwar: Numerical analysis of stochastic differential equations

Prof J Carr, FRSE: Differential equations, dynamical systems

Dr D Coutand: Analysis of free boundary and interface problems in continuum theories

Prof DB Duncan: Numerical analysis of evolutionary partial differential equations, particularly in wave propagation problems

Prof JC Eilbeck, FRSE: Numerical analysis (partial differential equations), mathematical biology (reaction-diffusion equations, neurophysics), nonlinear waves and solitons (integrable and nonintegrable classical and quantum systems), computer algebra.

Dr F Genoud: Nonlinear analysis and differential equations

Dr ND Gilbert: Group theory, topology

A Gripton: Bayesian inference in complex systems, fluid mechanics and hydrodynamics, Kernel-based classification

Prof J Howie, FRSE: Combinatorial group theory, low-dimensional topology

Prof DA Johnston: Statistical mechanics, random surfaces, conformal field theory, string theory

Dr A Konechny: String theory, conformal field theory, renormalization group flows, noncommutative geometry

Prof RJ Knops, FRSE: Qualitative behaviour (in particular dynamic and spatial stability) of solutions to equations in (nonlinear) continuum theories; related techniques for ill-posed problems

Prof AA Lacey, FRSE: Differential equations, asymptotic analysis, mathematical modelling, industrial mathematics

Dr MV Lawson: Semigroup theory, particularly inverse semigroups and their applications; category theory; automata theory

Dr S Loisel: Domain decomposition methods, …

Prof GJ Lord: Numerical analysis, dynamical systems, applied computational mathematics

Dr SJA Malham: Navier-Stokes analysis and reaction diffusion

Dr GR McGuire: Mathematical education

Dr KJ Painter: Mathematical modelling in biological and medical systems; pattern formation

Prof O Penrose, FRS, FRSE: Statistical mechanics, kinetics of phase transitions

Dr AR Prince: Group theory, combinatorics

Dr S Rodriguez-Lopez: Analysis

Prof BP Rynne: Differential equations, nonlinear analysis

Dr D Rule: Analysis and PDEs

Dr C Saemann: Mathematical Physics: noncommutative geometry, twistor geometry and integrability, string and M-theory

Dr BJ Schroers: Quantum gravity, gauge theory, topological solitons, quantum groups

Prof JA Sherratt, FRSE: Mathematical modelling of biology and medicine; waves and patterns in reaction diffusion equations

Dr W Staubach: Real and complex analysis

Dr M Strauss: Analysis

Prof RJ Szabo: Mathematical Physics: string theory, noncommutative geometry, K-theory, quantum field theory

Dr RA Weston: Solvable lattice models of statistical mechanics, integrable quantum field theories

Dr AR White: Population dynamics, ecosystem modelling

Dr MA Youngson: Functional analysis, operator algebras