Mathematicians in the group have been undertaking high quality research in mathematics since the 1980s. The present members of this group have published a number of research papers in prestigious international mathematical journals in the last four years. As well as pursuing individual projects, they have enjoyed and benefited from collaboration with colleagues and users in the UK and abroad.
Mathematics projects involve:
- Lie and Banach algebras, the theory of operator Lipschitz functions and the theory of group representations
- mathematical biology and dynamical systems
- operadic and simplicial methods, with applications in K-theory, cohomology of categories and mathematical physics
The members of this mathematics group have been peer review assessors of articles submitted to leading mathematical journals such as:
- the Journal and Proceedings of the London Mathematical Society
- the Journal of Functional Analysis
- the Transactions of the American Mathematical Society
- the Journal of Nonlinear Analysis
- Mathematical Reviews
They are also members of Editorial Boards of the Eurasian Mathematical Journal and the Journal of Mathematical Research and Applications. They were invited to give lectures at various mathematical conferences and have also given talks at many British universities and abroad.
Operator, Banach and C*-algebras
The work in this area is concentrated on the development of the general approach, allowing you to apply the tools and the results of the lattice theory to various aspects of operator theory, the theory of Banach algebras and to the structural analysis of C*-algebras. The methods developed by the members of the team shed new light on the structure of Banach and C*-algebras and use a number of important results obtained by other mathematicians who work in this area of research. In particular, the team is trying to characterise the structure of:
- GCR type C*-algebras
- CCR type and dual C*-algebra
- The structure of the set of projections in operator W*-algebras
Group representations and the theory of operator Lipschitz functions
The team is actively working on the investigation and understanding of non-unitary representations of nilpotent groups. The first step in this direction is a description of the cohomology group of nilpotent groups and neutral cohomology cocycles. This research is intrinsically linked with the theory of extension of representations of groups and of their double extensions. This naturally leads to the development of the theory of J-unitary representations of nilpotent groups on the spaces with indefinite metric which is quite widely used in Quantum mechanics.
The team is also involved in intensive investigation of the structure of Lp-spaces of operators from Schatten ideals and obtain analogues of Clarkson-McCarthy inequalities for these spaces. This work requires development of various important analytical tools which allow to make sharp estimations for a number of inequalities for Schatten class operators.
Qualitative and asymptotic behaviour of solutions of some classes of ordinary and functional differential equations
Our work in this area involves theoretical analysis on various behaviour of solutions of some class of differential equations, with or without delays. This includes existence of a fixed point, local and global stability of a fixed point, a fixed point as a global attractor or a global repellor restricted to a certain manifold, vanishing components, periodic solutions, bifurcation, oscillation, limit cycles, and behaviour near limit cycles.
Mathematical biology and dynamical systems
In theoretic biology, many population dynamics are modelled mathematically by dynamical systems (discrete or continuous). Thus, effective investigation of species evolution relies on our knowledge about the solution behaviour of the model. Our research in this area is mainly on qualitative and asymptotic behaviour of some type of biological models, eg Lotka-Volterra and Kolmogorov systems, in response to prediction for long-term future of the species coexistence or extinction.