Actuarial and financial mathematics

Research in actuarial mathematics includes stochastic process modelling of diseases and disability, risk theory, insurance solvency, pricing and hedging of guarantees, and mortality analysis including the modelling of future longevity.

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.

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.

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.

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.

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.

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.

Mathematical physics

The Edinburgh Mathematical Physics Group consists of 12 permanent academic staff, 3 postdoctoral research fellows and 8 postgraduate students, as of January 2010. 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. We have our own preprint series which contains most of our research output.

Optimization

The The Edinburgh Research Group in Optimization (ERGO) is a loose association of researchers in the University's School of Mathematics, Business Studies, Chemical Engineering and Agriculture/IERM departments, together with the Edinburgh Parallel Computing Centre (EPCC) and the Optimization group in the Department of Mathematics and Computer Science at the University of Dundee. ERGO also has links with Quadstone and Edinburgh Petroleum Services (EPS). The main research activities of the ERGO Group are, in no particular order, large-scale optimization, global optimization, parallel computing and sparse matrix techniques applicable in optimization with a focus on the solution of real-world applications. Our regular research activities include ERGO seminars. Our recent publications and preprints contain most of our research output. We run the MSc in Operational Research with a focus on computational techniques of optimization.

Probability and stochastics

Research in Probability and Stochastics is focussed mainly on Stochastic Differential Equations and Stochastic Partial Differential Equations, including theoretical results, methods for numerical solution, and applications. There is also work on problems in analysis related to probability theory.

There are links with the following research areas within the Maxwell Institute: Analysis, Applied Mathematics, Computational Mathematics, Financial and Actuarial Mathematics, Mathematical Biology, Optimisation and Statistics.

Statistics

Bayesian inference, computer-intensive methods, automated pattern recognition, sampling and resampling and multivariate analysis, with applications in forensic science, the mathematical and statistical evaluation of evidence, the earth sciences, agriculture, biology, ecology and applications of stochastic analysis to economics and finance. There is a regular seminar. There has been a long and fruitful collaboration with Biomathematics and Statistics Scotland(BioSS). Colin Aitken is one of the key members of the Joseph Bell Centre for Forensic Statistics and Legal Reasoning which conducts research at the interface of statistics, law, forensic science and artificial intelligence and involves the Centre for Law and Society and the Artificial Intelligence Applications Institute of the University of Edinburgh, the Division of Law of Glasgow Caledonian University and the Lothian and Borders Forensic Science Laboratory.