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 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.

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

Prof Marco Marletta

Professor Marco Marletta

Deputy Head of School

Email:
marlettam@cardiff.ac.uk
Telephone:
+44 (0)29 2087 5552

Academic staff

Iskander Aliev

Dr Iskander Aliev

Reader

Email:
alievi@cardiff.ac.uk
Telephone:
+44 (0)29 2087 5547
Photograph of Prof Alex Balinsky

Professor Alexander Balinsky

Professor of Mathematical Physics

Email:
balinskya@cardiff.ac.uk
Telephone:
+44 (0)29 2087 5528

Dr Jonathan Ben-Artzi

Senior Lecturer

Email:
ben-artzij@cardiff.ac.uk
Telephone:
02920 875624
Malcolm Brown

Professor Malcolm Brown

Professor of Computational Mathematics

Email:
brownbm@cardiff.ac.uk
Telephone:
+44 (0)29 2087 5538
Photograph of Mikhail Cherdantsev

Dr Mikhail Cherdantsev

Lecturer

Email:
cherdantsevm@cardiff.ac.uk
Telephone:
+44 (0)29 2087 5549
Dr Nicholas Dirr

Professor Nicolas Dirr

Personal Chair

Email:
dirrnp@cardiff.ac.uk
Telephone:
+44 (0)29 2087 0920
Dr Federica Dragoni

Dr Federica Dragoni

Senior Lecturer

Email:
dragonif@cardiff.ac.uk
Telephone:
+44 (0)29 2087 5529
Dr Suresh Eswarathasan photograph

Dr Suresh Eswarathasan

Lecturer

Email:
eswarathasans@cardiff.ac.uk
Telephone:
+44 (0)29 2087 0935
Photograph of Des Evans

Professor Des Evans

Emeritus Professor of Mathematics

Email:
evanswd@cardiff.ac.uk
Telephone:
+44 (0)29 2087 4206
Photograph of Dr Matthew Lettington

Dr Matthew Lettington

Lecturer

Email:
lettingtonmc@cardiff.ac.uk
Telephone:
+44 (0)29 2087 5670

Dr Baptiste Morisse

Research Associate

Email:
morisseb@cardiff.ac.uk
Karl Schmidt

Professor Karl Schmidt

Professor

Email:
schmidtkm@cardiff.ac.uk
Telephone:
+44 (0)29 2087 6778
Mathematical Analysis

Dr Kirstin Strokorb

Lecturer

Email:
strokorbk@cardiff.ac.uk
Telephone:
+44 (0)29 2068 8833

Seminars

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 Baptiste Morisse.

SpeakersDateAbstract

Choi-Hong Lai (Greenwich)

18 March 2019

To be announced.

Michela Ottobre (Herriot-Watt Uni)

10 December 2018

To be announced.

David Lafontaine (Bath)

3 December 2018

To be announced.

Benjamin Gess (Leipzig)

26 November 2018

To be announced.

Kirill Cherednichenko (Bath)

26 November 2018

To be announced.

Mahir Hadžić (KCL)

19 November 2018

To be announced.

Petr Siegl (Belfast)

12 November 2018

To be announced.

David Beltran (BCAM)

5 November 2018

To be announced.

Maria Carmen Reguerra (Birmingham)

22 October 2018

To be announced.

Sanju Velani (York)

15 October 2018

To be announced.

Frank Rösler (Cardiff)

9 October 2018

To be announced.

Anton Savostianov (Durham)

1 October 2018

Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations

It is well known that long time behaviour of  a dissipative dynamical system generated by an evolutionary PDE can be described in terms of attractor, an attracting set which is essentially thinner than a ball of the corresponding phase space of the system. In this talk we compare long time behaviour of damped anisotropic wave equations with the corresponding homogenised limit in terms of their attractors. First we will formulate order sharp estimates between the trajectories of the corresponding systems and will see that the hyperbolic nature of the problem results in extra correction comparing with parabolic equations. Then, after brief review on previous results on homogenisation of attractors, we will discuss new results. It appears that the Hausdorff distance between the anisotropic attractors and their homogenised counter-parts, in suitable norms, can be estimated via operator norm of the difference of the resolvents of the corresponding elliptic operators. Furthermore, we show that the homogenised attractor admits first-order correction suggested by the natural asymptotic expansion. The corrected homogenised attractors, as expected, are close to the anisotropic attractors already in the strong energy norm. The corresponding quantitative estimates on the Hausdorff distance between the corrected homogenised attractors and anisotropic ones, with respect to the strong energy norm, are also obtained. Our results are applied to Dirchlet, Neumann and periodic boundary conditions. This is joint work with Shane Cooper.

Marco Marletta (Cardiff University)

Jiang-Lun Wu (Swansea)

Federica Dragoni (Cardiff University)

Peter Hintz (Berkeley)

Dmitri Finkelshtein (Swansea)

27 September 2018

South Wales Analysis and Probability Seminar

9:30-10:00 Coffee and registration
10:00-10:45 Marco Marletta (Cardiff)
10:45-11:30 Jiang-Lun Wu (Swansea)
11:30-11:50 Coffee
11:50-12:35 Federica Dragoni (Cardiff)
12:35-14:05 Lunch
14:05-15:05 Colloquium: Peter Hintz (Berkeley)
15:05-15:30 Coffee
15:30-16:15 Dmitri Finkelshtein (Swansea)

Past events

Mathematical Analysis Seminars 2017-18

Mathematical Analysis Seminars 2015-16

Bath - WIMCS analysis meetings

25/09/2015

These meetings are sponsored by an LMS Scheme Three Grant.

Bath - WIMCS Cardiff meeting

South-West Network in Generalised Solutions for Nonlinear PDEs meetings

12/02/2016

These meetings are organised by Cardiff University, University of Reading and the University of Bath.

University of Bath Meeting.

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.

New trends in non linear PDEs