Analysis and Differential Equations

Cardiff has one of the largest research groups in spectral theory of differential operators in the world. Our work on spectral theory informs and uses the results of vigorous activity in closely related areas such as function spaces, integral operators and geometry. Our overall strength across analysis creates a vibrant environment, attractive to visitors and to top-calibre new appointees alike, in which collaborative research, both inside and outside Cardiff, thrives. We work closely with the research groups at, among others, Birmingham (Alabama), Karlsruhe, Moscow State, Weizmann Institute, St. Petersburg, Bern, Santiago, Pisa, Montréal, McGill, Toulon and Marseilles.

We have been exceptionally active in organising workshops and conferences, to the benefit of the worldwide spectral theory community. Examples are:

• a forthcoming workshop on Inverse Problems at the Isaac Newton Institute (2011);
• EPSRC funded UK Matrix and Operator Pencil Network, MOPNET
• a recent six-month programme on Analysis on Graphs at the Isaac Newton Institute;
• EPSRC funded UK Spectral Theory Network, 2001-04;
• triennial international Gregynog Workshops on Differential Equations since 1983, supported by EPSRC;
• LMS supported workshop on “Analytical and numerical aspects of periodic PDE”, Cardiff, 2007.

Key Research Areas

The main directions of research in the Analysis and Differential Equations Group include:

• Analytic and Computational Spectral Theory
• Spectral theory of differential operators on domains, manifolds and graphs, including operators arising in mathematical physics and non-self-adjoint problems;
• Homogenization for (mainly elliptic) PDEs;
• Function spaces & integral operators,
• Computational spectral theory
• Inverse problems
• Applications in mathematical bioinformatics; internet security and risk management; archaeology; image recognition; oceanology.

Applications

We work on a wide range of applications of differential equations and spectral theory, including problems arising in  elasticity, hydrodynamics, electromagnetism, mathematical bioinformatics, internet security.