Analysis is one of the core mathematical disciplines, offering a wide range of research topics ranging from investigations in complex analysis, geometric analysis, functional analysis, measure theory, and numerical analysis to applications in mathematical physics, PDE, and other areas of mathematics.
Our research group is one of the largest in the department, and covers many of these different research strands.
Complex Analysis, Operator Theory and Applications (Prof. Hedenmalm, Prof. Chandler-Wilde, Prof. Virtanen, Dr. Arroussi, Dr. Hagger, Dr. Miihkinen)
Our work in this area encompasses the study of bounded linear operators on (analytic) function spaces, asymptotic analysis of matrices and their determinants, (quasi)conformal mappings, random matrix theory (RMT) and applications to mathematical physics.
Of particular interest are Toeplitz, Hankel, singular integral, and layer potential operators on various (analytic) function spaces such as Hardy, Bergman, Fock, energy and Lebesgue spaces, and their most important properties, such as boundedness, compactness, Schatten class membership, and other spectral properties. Our work related to RMT and mathematical physics involves (soft) Riemann-Hilbert problems, (planar) orthogonal polynomials, asymptotic study of Toeplitz determinants, random normal matrices, quantum spin chain models and entanglement. Layer potentials are also studied in the context of linear elliptic PDE.
Nonlinear Partial Differential Equations (Dr Nikos Katzourakis)
At the centre of our research in this area lies the rigorous mathematical analysis of nonlinear partial differential equations, including fully nonlinear systems, differential inclusions and the Calculus of Variations. We focus particularly on the vectorial case and on supremal functionals, as well as PDE-constrained optimisation and its automotive engineering and climate applications.
Spectral Theory (Professor Michael Levitin, Professor Simon Chandler-Wilde, Dr. Sugata Mondal)
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, spectra and pseudospectra of random matrices and operators, and spectral geometry of hyperbolic manifolds, which includes analysis of PDE questions around qualitative questions on low lying eigenfunctions on domains on constant curvature space forms.
Linear Wave Equations (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.