Mathematical Analysis Research Group
We are one of the leading analysis groups in the UK. From the 1960s through to the 1990s our main areas of expertise were spectral theory, operator theory, function spaces and linear partial differential equations, together with the applications of all these ideas in mathematical physics.
From 2007 onwards we have expanded our interests to encompass new areas including homogenisation, inverse problems and imaging, nonlinear partial differential equations, deterministic and stochastic homogenisation. The group has been restructured to encompass the traditional Cardiff expertise in analytic number theory and to include topics at the interface between analysis and number theory, such as spectral geometry.
We are a very international group, encompassing researchers and academics from Germany, Israel, Italy, Russia and the USA, as well as the United Kingdom. Our international collaborations reflect this: we have ongoing projects with, among others, Bern, Birmingham (Alabama), Colorado School of Mines, Florence, Karlsruhe, Montréal, McGill, Padova, Pisa, Santiago, St Petersburg and the Weizmann Institute, as well as universities in the UK, in particular our colleagues in WIMCS and the GW4 group.
Our main directions of research in the Analysis and Differential Equations Group include:
- spectral theory, applications and numerical methods
- quantum mechanics, inverse problems
- asymptotic and variational methods for nonlinear partial differential equations, in particular (stochastic) homogenisation
- geometric and stochastic partial differential equations
- combinatorial and analytic number theory, special functions
- applications of analytical methods (e.g. image processing, medical genetics, scaling limits for interacting particle systems)
In 1966 Mark Kac asked the programmatic question “Can one hear the shape of a drum?” - i.e. is a planar domain determined uniquely (up to congruence) by the spectrum of its Dirichlet Laplacian?
This question has since been answered in the negative; but spectral geometry - the study of how geometric and topological properties of domains and manifolds are reflected in the spectra of associated differential operators - is a flourishing mathematical area with ramifications to number theory and physics.
Our other interests in spectral theory include:
- operators of mathematical physics, for example stability questions involving qualitative and quantitative estimates and semiclassical asymptotics for operators derived from the Dirac operator of relativistic quantum mechanics, such as the Brown-Ravenhall operator, and extensions to a quantum field theoretic setting;
- non-self-adjoint problems, including spectral approximation, spectral pollution and operator with special block structure;
- inverse spectral problems, including imaging problems and Maxwell systems;
- boundary triples and their applications to spectral theory of (systems of) PDEs.
Geometric and Stochastic PDEs
Many physical problems, in particular those involving phase transition, nucleation and evolution equations for free boundaries and interfaces, involve mathematical models in which there is sufficient disorder on a sufficiently small length scale that the most effective analysis of their macroscopic solutions is through the analysis of partial differential equations with stochastic coefficients and scaling limits.
We work on several topics in these areas, including
- interfaces in heterogeneous and random media and associated nonlinear PDEs
- interacting Stochastic Processes and their scaling limits; stochastic nonlinear PDEs; nonlinear PDEs and Stochastic Processes
- homogenization and Gamma-convergence;
- scaling limits of singularly perturbed differential equations.
Another very modern approach to certain classes of nonlinear partial differential equations is through ideas from geometry. We have particular interests in problems involving sub-Riemannian manifolds, which are not isomorphic to Euclidean space at any length scale. In these contexts we have been able to adapt techniques from calculus of variations, and generalise notions of convexity, allowing us to treat subelliptic and ultraparabolic PDEs. These equations occur in many unexpected and new applications, including modelling the first layer of the visual cortex and problems in finance related to pricing Asian options.
Analytic Number Theory
Cardiff Number Theory was founded by Professor Christopher Hooley FRS and is still one of our most popular areas for doctoral study.
We are active in research on many classical topics: prime numbers, the Riemann zeta function, Dirichlet polynomials, exponential sums, Dedekind sums, Kloosterman sums, the modular group, Maass wave forms, the Selberg and Kuznetsov trace formulae, lattice points in the plane and the Gauss circle problem, different configurations of lattice points inside a moving shape, as well as integer points close to hypersurfaces and polytopes.
We are also interested in problems at the interface between number theory and spectral theory, usually arising from spectral geometry,
Head of group
Professor of Mathematical Physics
- +44 (0)29 2087 5528
Professor of Computational Mathematics
- +44 (0)29 2087 5538
Dr Kirill Cherednichenko|
(University of Bath)
(ENS - Paris)
Dr Yury Korolev|
(Queen Mary London)
Professor Geoffrey Burton|
(University of Bath)
|Variational problems involving rearrangements of functions|
Dr Alessio Martini|
(University of Birmingham)
All seminars are held at 3:10pm in Room M/2.06, Senghennydd Road, Cardiff unless stated otherwise.
Programme organiser and contact: Dr Mikhail Cherdantsev