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 organize regular workshops and conferences, to the benefit of the
worldwide analysis community. Examples include:

• a workshop on Inverse Problems at the Isaac Newton Institute (2011)
• the EPSRC Matrix and Operator Pencil Network, MOPNET (2009-2012)
• the EPSRC Analysis On Graphs and Applications Network, AGA (2013-2015)
• the Leverhulme-WIMCS thematic year on Dissipative Spectral Theory, including 2 conferences (2012-2013)
• the 2012 South West PDE Day, funded by OxPDE and LMS
• the ICMS Programme on Spectra and Pseudospectra, September 2014

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.
• Viscosity solutions for subelliptic nonlinear PDEs.
• Analysis in Carnot-Caratheodory spaces.
• Asymptotic behaviour of nonlinear PDEs.
• PDEs with random coefficients.