# Mathematical Analysis Research Group

Cardiff Analysis spans an exciting range of topics including spectral theory and spectral geometry, related areas such as analytic number theory and microlocal analysis, through to inverse problems, imaging, nonlinear PDEs, deterministic and stochastic homogenisation.

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

## Staff academaidd

## Yr Athro Alexander Balinsky

Professor of Mathematical Physics

- balinskya@caerdydd.ac.uk
- +44 (0)29 2087 5528

## Yr Athro Malcolm Brown

Professor of Computational Mathematics

- brownbm@caerdydd.ac.uk
- +44 (0)29 2087 5538

## Myfyrwyr Ôl-raddedig

## Seminars

The Analysis and PDE Seminars currently run online on Mondays from 14:10 to 15:00

The Analysis and PDE Seminars schedule is updated regularly. The programme organiser and contact is Dr Matteo Capoferri Please get in touch if you have any questions or would like to receive the weekly announcements.

Join the next seminar via Zoom

Date | Speaker | Abstract |
---|---|---|

Monday 5 October 2020 14.10-15.00 | Frank Rosler (Cardiff) |
The question of whether there exists an approximation procedure to compute the resonances of any Helmholtz resonator, regardless of its particular shape, is addressed. A positive answer is given, and it is shown that all that one has to assume is that the resonator chamber is bounded and that its boundary is C^2. The proof is constructive, providing a universal algorithm which only needs to access the values of the characteristic function of the chamber at any requested point. This is joint work with Jonathan Ben-Artzi and Marco Marletta. |

Monday 12 October 14.10-15.00 | Carla Cederbaum (Tubingen) |
A Riemannian manifold is said to be “static (vacuum)” if it is scalar flat and carries a positive, harmonic function related to the Ricci curvature of the manifold via a degenerate elliptic system (1) of PDEs. It is well-known since the 1960’s that static Riemannian manifolds must be isometric to a so-called Schwarzschild manifold if they satisfy suitable mixed Dirichlet/Neumann boundary conditions at an “inner” boundary and at “infinity". This is known under the name of “static black hole uniqueness” in General Relativity. We will show that this result persists even upon dropping the system of PDEs (1) and replaces vanishing by non-negative scalar curvature, provided one assumes suitable additional boundary conditions at the inner boundary and at infinity. The new boundary conditions can be viewed as combining Dirichlet and Neumann conditions. I will first introduce the relevant geometric concepts and PDEs and give some facts and intuitions about the Schwarzschild manifold. Then I will sketch the proof which applies and extends ideas developed by Bunting and Masood-ul-Alam in the 1980’s for three-dimensional static Riemannian manifolds. The proof also relies on Schoen and Yau’s positive mass theorem as well as on a Ricci flow result by McFerron and Szekelyhidi. Finally, I will discuss several applications and generalizations which are joint work with Gregory J. Galloway and with Sophia Jahns and Olivia Vivcanek Martinez, respectively. |

Monday 19 October 14.10-15.00 | Analysis: Tuomas Sahlsten (Manchester) |
There have been a number of recent advances in the statistics of the eigenvalues and eigenfunctions of the Laplacian on large (random) graphs. We are exploring how well ideas from graphs help in the context of (random) surfaces and manifolds. Several models exist for random surfaces where spectral theory of the Laplacian has been studied, such as Brooks-Makover’s model, Weil-Petersson/Mirzakhani’s model and more recently a random covering model used by Magee-Naud-Puder in the study of spectral gaps. We will give an overview of the current progress and some of our recent and ongoing works on quantum chaos on random surfaces. In part joint work with C. Gilmore, E. Le Masson and J. Thomas. |

Monday 26 October 14.10-15.00 | Analysis: Alexei Stephanenko (Cardiff University) |
Dissipative barriers are a class of non-self-adjoint perturbations of linear operators that arise in numerical analysis. The aim of this talk is to provide a description of eigenvalues under such perturbations, with an emphasis on Schrodinger operators on the half-line. I will start with an overview of the dissipative barrier method for computing eigenvalues in spectral gaps. Abstract spectral inclusion and pollution results, based on enclosures for the limiting essential spectrum, will then be presented. This is followed by more precise results for the case of Schrodinger operators. In the final part of the talk, I will present bounds for the maximum magnitude and the number of eigenvalues of perturbed Schrodinger operators. These will be compared to existing bounds in the literature. |

Monday 2 November 14.10 | Analysis: Euan Spence (Bath University) |
It is well known that when the geometry and/or coefficients allow stable trapped rays, the solution operator of the Helmholtz equation grows exponentially through a sequence of real frequencies tending to infinity. This talk (based on the paper with my co-authors David Lafontaine and Jared Wunsch) shows that, even in the presence of the strongest-possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. |

Monday 9 November 14.00 -15.00 | Analysis: Raffaele Grand (Cardiff University) | To be announced |

Monday 16 November 14.10 -15.00 | Analysis: Marcus Waurick (Hamburg/Graz) | To be announced |

Monday 23 November 14.10 -15.00 | Analysis: Stephen Shipmanc (LSU) | To be announced |

Monday 30 November 14.10-15.00 | Analysis: Matteo Novaga (Pisa) | To be announced |

Monday 7 December 14.10-15.00 | Analysis: Jacob Stordal Christiansen (Lund) | To be announced |

Monday 14 December 14.10-15.00 | Analysis: Monique Dauge (Rennes) | To be announced |

Monday 14 December 14.10-15.00 | Analysis: Walter Strauss (Brown) | To be announced |

## Past events

- Mathematical Analysis Seminars 2019-20
- 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.

### 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.

### 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.