Research in the group includes work into: the theory and application of nonlinear ordinary and partial differential equations (with emphasis on rigorous results, applications and computations), mathematical control theory, nonlinear elasticity, numerical analysis, mathematical biology, industrial mathematics, differential and algebraic geometry and group theory. There are increasing links with the probability group, including a weekly workshop on reaction diffusion equations and with the schools of chemical and mechanical engineering (see the STOP PRESS announcement of a new PDRA position.) The group has also been active in looking at nonlinear problems arising in industry, maintains close collaborations with several companies and will be hosting the European Study Group with Industry in 1997.
Here are some examples of what is going on
Active research into studies of the existence, uniqueness, regularity and qualitative properties of the solutions of nonlinear partial differential equations governed by variational principles, especially problems which arise in the studies of nonlinear elasticity and of water wave and vortex phenomena. Analysis of fourth order problems.
Investigations of nonlinear parabolic problems with singularities and interfaces, especially solutions which blow-up in a finite time. Applications of bifurcation theory. Construction of exact solutions using techniques from group theory and the analysis of the evolution of parabolic problems on low dimensional manifolds. Studies of the existence and stability and the numerical approximation of self-similar solutions. Applications in biology. Investigations of the existence, uniqueness and symmetry of systems governed by nonlinear elliptic equations, especially equations involving critical Sobolev exponents. Close interaction is maintained with the probability group headed by Prof. Williams exploring deep links between the application of martingale methods for studying branching processes and analytic methods for such reaction diffusion equations as the Fisher equation.
Here studies centre on methods for infinite dimensional control, system uncertainty, non-smooth systems and the analysis of differential inclusions (nonlinear differential equations with discontinuities) arising in models of control systems. The group is part of an active EU Network in control theory.
There has recently been an increase in both the perceived and the actual potential of the mathematical approach to problems in biology, both because of the increased awareness of the power of mathematical modelling and analysis within the biological community, and the increased power of modern mathematics, in particular in dealing with the strongly nonlinear and extremely complex systems that abound in biology. Mathematics can aid biology in two particular ways.
The first is in the study of order in its most general sense. Here we include self-organisation or the emergence of order from complex disordered systems through very simple laws, and morphogenesis or the formation of structures in initially very simple systems. The archetypal features of biological organisation from biochemistry through to community ecology are complexity and stability. It is also essential that much of this organisation is controlled intrinsically in a way that is robust and capable of self-repair. Mathematics is useful and essential here precisely because of the simplicity of the rules and the complexity of the system or the pattern, which otherwise makes progress extremely difficult. Two aspects are morphogenesis, represented at Bath for example by tumour formation, and self-organisation and emergent properties, represented for example by studies of ant colonies.
The second is in areas where classical controlled experimentation is difficult or impossible. Unlike most of physics and almost all of chemistry, for example, there are many areas in medicine and in biology, particularly in behaviour, evolution, population dynamics and conservation, where problems in experimentation arise because of the absence of controls, unacceptable time scales, or ethical difficulties. Then the most rigorous form of investigation is analytical modelling leading to the testing of the model's predictions through perturbation experiments and new observational techniques. This involves a continuous cycle of theorising, modelling and field or laboratory analysis in which theories and models are refined or refuted. The strength of the procedure relies to a great extent on the statement of the model in rigorous mathematical terms because this promotes rigorous a-priori prediction that can be tested and falsified. In short, mathematics is the necessary language of much of biological investigation, just as it is, for fundamentally similar reasons, in astrophysics. Bath has interests here in conservation biology, evolution of sex, medical statistics, neuroscience and pain.
The group is part of the Centre for Mathematical Biology which forms an interdisciplinary link between the School of Mathematics and the School of Biology and Biochemistry. The Centre organises joint workshops and seminars and actively encourages joint research work.
The study of cracks forming in elastic materials leads to very interesting nonlinear problems involving singularities. The group also studies composite materials, nonlinear waves and diffraction asymptotics. These problems have many important applications - including recent work (with Prof. Spence) on catalytic convertors.
The numerical analysis group at Bath has been very active in both the application of numerical methods to study nonlinear systems and in using dynamical systems theory to analyse the behaviour of numerical methods for both ordianry and partial differential equations. Areas of research include: methods for bifurcation problems, domain decomposition, adaptive methods, application of dynamical systems, inertial manifolds.
For more information, see the article in this issue.
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Last Updated: 1st May 1996.