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:
- convex analysis
- analysis in sub-Riemannian manifolds
- inverse problems and imaging
- nonlinear partial differential equations
- deterministic and stochastic homogenisation.
Our work includes the traditional Cardiff University expertise in analytic number theory and topics at the interface between analysis and number theory, such as spectral geometry.
We are an international group with researchers and academics from Germany, Israel, Italy, Russia and the USA, as well as the United Kingdom. Our international collaborations reflect this with ongoing projects with 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.
Recent past members of the group include Michael Levitin, Igor Wigman and Kirill Cherednichenko.
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 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.
We are grateful to our research sponsors for the steady flow of funding which they have continued to provide over many years. Our past sponsors include the Leverhulme Trust, the European Union Marie Curie program and the Royal Society, as well as the EPSRC. Our current sponsors include the London Mathematical Society and the EPSRC who finance the following projects:
Head of group
Professor of Mathematical Physics
- +44 (0)29 2087 5528
Professor of Computational Mathematics
- +44 (0)29 2087 5538
Emeritus Professor of Mathematics
- +44 (0)29 2087 4206
All seminars are held at 14:10 in Room M/2.06, Senghennydd Road, Cardiff unless stated otherwise.
The programme organiser and contact is Dr Suresh Eswarathasan.
Baptiste Morisse (Cardiff University)
2 October 2017
Well-posedness for systems of PDEs in Gevrey regularity Abstract
Systems of first-order, quasilinear PDEs form a wide and interesting topic. Many physical equations which arise in hydrodynamic are of this form (Euler equation, KdV, Burgers...). I will first introduce the basic notions for the study of such systems, give examples and then point out what are the borderline cases - which we call weakly hyperbolic or weakly elliptic systems. In order to deal with such cases, I will explain the importance of working in spaces of highly regular functions: the Gevrey spaces.
Erez Nesharim (University of York)
9 October 2017
The t-adic Littlewood conjecture is false
Littlewood conjecture and the p-adic Littlewood conjecture are famous open problems in number theoy. Their positive characteristic analogues are well known. In a joint work with Faustin Adiceam and Fred Lunnon we show that the so called "t-adic" Littlewood is false over $F_3((1/t))$. Our counter example is concrete, but the proof uses computers. It is based on a generalisation of Dodson's condensation algorithm for computing the determinant of a matrix.
Michael Magee (Durham University)
23 October 2017
Uniform spectral gap in number theory
I'll begin with Selberg's eigenvalue conjecture. This is an analog of the Riemann hypothesis for a special family of Riemann surfaces that feature heavily in number theory, for example in Wiles' proof of the Taniyama-Shimura conjecture. I'll explain how in the last 10-15 years, number theorists have had to turn to Anosov dynamics to obtain the approximations to Selberg's conjecture that became relevant to emerging 'thin groups' questions about Apollonian circle packings and continued fractions. I will explain what my contributions to this area were. Finally if I have time, I'll explain how I am pushing these techniques into the setting of Teichmuller dynamics in the pursuit of yet more interesting number theory questions.
Asma Hassannezad (University of Bristol)
30 October 2017
Higher order Cheeger type inequalities for the Steklov eigenvalues
In 1970 Cheeger obtained a beautiful geometric lower bound for the first nonzero eigenvalue of the Laplacian in term of an isoperimetric constant. Inspired by the Cheeger inequality, Cheeger type inequalities for the first nonzero Steklov eigenvalue have been studied by Escobar, and recently by Jammes. The generalization of the Cheeger inequality to higher order eigenvalues of the Laplacian in discrete and manifold settings has been studied in recent years. In this talk, we study the higher-order Cheeger type inequalities for the Steklov eigenvalues. It gives an interesting geometric lower bound for the k-th Steklov eigenvalue. It can be viewed as a counterpart of the higher order Cheeger inequality for the Laplace eigenvalues, and also as an extension of Escobar’s and Jammes’ results to the higher order Steklov eigenvalues. This is joint work with Laurent Miclo.
Yulia Ershova (U. Bath)
6 November 2017
A unified operator-theoretical approach to high-contrast homogenisation in models of chain-type graphs
I am going to demonstrate how the new abstract unified approach to the problems of high-contrast homogenisation developed in cooperation with Kiselev, Cherednichenko, and Naboko works in rather simple models of graphs periodic along one axis (chain-type graphs). I will consider three examples of such graphs which show that the above-mentioned new approach is indeed free from the crucial additional assumption that the stiff component of the media is to be connected. The result which our approach yields comprises not only the operator asymptotics in norm-resolvent topology but also sheds new light on the intricate relationship between the norm-resolvent limits of high-contrast media and the corresponding models of time-dispersive media.
Shane Cooper (Durham University)
13 November 2017
A general framework for the homogenisation of high-contrast problems
In homogenisation theory, the important questions of error estimates and (the very much related) spectral convergence are fraught with difficulties: one challenge is to establish the correct limit with respect to some topology that ensures spectral convergence, then one must handle the fact that the limits are in general defined in different function spaces to the prelimit.
These issues appear because conventionally one first establishes a convergence result then asks questions of error estimates and/or spectral convergence. The problem is we first passed to the limit!
In this talk we present a novel framework to study the asymptotic behaviour of (a large class of) second-order linear elliptic PDE systems with periodic coefficients whose ellipticity constant degenerates in the limit of small period. We determine, under one-or-two readily verifiable assumptions, the leading-order behaviour of the (variational) solution with respect to the (small period) parameter. Error estimates, uniform in right-hand-side, are readily deduced in this process of determination.
Amongst other things, this work explains the differences between the convergence results/spectral asymptotics of classical and high-contrast homogenisation problems documented in literature. Additionally, we are compelled to revisit the central concept of averaging over the periodic reference cell in homogenisation theory. I shall present an example in the context of Magnetic-Schrodinger equations where this notion is not applicable.
This is joint work with Ilia Kamotski (UCL) and Valery Smyshlyaev (UCL).
Yves Capdebosq (Oxford University)
27 November 2017
Michiel Van Den Berg (University of Bristol)
4 December 2017
Peter Varju (University of Cambridge)
11 December 2017
Trevor Wooley (Bristol University)
5 February 2018
Ahmed Jama (Cardiff University)
5 March 2018
Isabelle Gallagher (U. Paris Diderot)
12 March 2018
Yaroslav Kurylev (UCL)
30 April 2018
Bath - WIMCS analysis meetings
These meetings are sponsored by an LMS Scheme Three Grant.
South-West Network in Generalised Solutions for Nonlinear PDEs meetings
These meetings are organised by Cardiff University, University of Reading and the University of Bath.
New trends in non linear PDEs
20/06/2016 - 25/06/2016
The aim of this workshop is to get together researchers within recently very active research areas connected to nonlinear partial differential equations (PDEs), in particular where these cross boundaries of mathematical disciplines.