### Nonlinear Partial Differential Equations (Dr Nikos Katzourakis)

At the centre of our research in this area lies the rigorous mathematical analysis of non-linear partial differential equations, including free-boundary problems for non-linear partial differential equations, analysis of non-linear PDE models of large scale atmospheric flow, fully non-linear elliptic PDE's and connections to variational problems, boundary value problems for integrable nonlinear equations.

### Applied and Computational Complex Analysis (Dr Jani Virtanen)

The focus of the research is this area is the application of complex analytical techniques to the study of integrable systems, boundary value problems, water waves, orthogonal polynomials and random matrix theory.

### Spectral Theory (Professor Michael Levitin, Professor Simon Chandler-Wilde)

Our research interests are mainly in foundations, applications and numerical methods of spectral theory. Most of our research deals with the spectral analysis of differential operators, including spectral problems in waveguides, spectral geometry, spectral properties of non-selfadjoint operators and operator pencils. We also work on spectral properties of boundary integral operators, and on spectra and pseudospectra of random matrices and operators.

### Operator Theory (Professor Simon Chandler-Wilde, Dr Jani Virtanen)

The central theme of operator theory is the investigation of linear mappings between (infinite dimensional) Banach spaces. The linear mappings under consideration can be very concrete operators, such as Toeplitz operators, or can be studied in an abstract framework.

Of particular interest are Toeplitz, Hankel and singular integral operators on various (analytic) function spaces such as Hardy, Bergman and Fock spaces, and layer potential operators in the theory of linear elliptic PDEs operators on energy and L^p spaces, and their most important properties such as boundedness, compactness, and other spectral properties.

### Linear Wave Equations (Dr Nick Biggs, Dr Peter Chamberlain, Professor Simon Chandler-Wilde)

Of interest are all aspects of propagation and scattering of linear waves, often investigated now with the aid of perturbation methods, in particular the propagation and trapping of waves in waveguides which are slowly-varying in some sense (e.g. containing a bulge whose width varies slowly along the waveguide, or a bend whose curvature varies slowly along the waveguide).

A more recent topic of interest is that of edge waves trapped in the vicinity of a periodic but slowly-varying boundary.

A third area of interest is the derivation of so-called embedding formulae, which allow solutions of certain scattering problems to be expressed in terms of solutions to other related scattering problems. These formulae are interesting in their own right, but also allow the computation of a wide range of solutions to be carried out extremely efficiently.

A final area of interest is work on analysis of high frequency problems, understanding the transition from wave-based to particle-based models of scattering, and proving rigorous results about solution behaviour, norms of resolvent operators, etc, that tease out the complicated interplay between the frequency parameter and the geometry.