Research

Our main directions of research 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 focus

Spectral theory

In 1966 Mark Kac asked the programmatic question “Can one hear the shape of a drum?” - ie 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.