# 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 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:

# Professor Marco Marletta

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

# Dr Iskander Aliev

## Senior Lecturer

Email:
alievi@cardiff.ac.uk
Telephone:
+44 (0)29 2087 5547

# 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

# Professor Malcolm Brown

## Professor of Computational Mathematics

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

# Dr Mikhail Cherdantsev

## Lecturer

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

# Professor Nicolas Dirr

## Personal Chair

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

# Dr Federica Dragoni

## Senior Lecturer

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

# Dr Suresh Eswarathasan

## Lecturer

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

# Professor Des Evans

## Emeritus Professor of Mathematics

Email:
evanswd@cardiff.ac.uk
Telephone:
+44 (0)29 2087 4206

# Dr Matthew Lettington

## Lecturer

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

# Dr Karl Schmidt

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

## 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 Suresh Eswarathasan.

Speaker

Date

Abstract

Igor Velcic (Univertsity of Zagreb)

3 October 2016

Non-periodic homogenization of von Karman rod equations.

Abstract: In the talk we will start from the equilibrium equations of 3d elasticity and will derive the homogenized equations for von Karman rod. The derivation is done under the non-physical assumption on linear growth of differential and under the assumption that the equilibrium satisfy von Karman scaling of energy. This work is in collaboration with M. Bukal and M. Pawelczyk.

Konrad Swanepoel (London School of Economics)

10 October 2016

Geometric Steiner Trees and Small Subset Sums in Normed Spaces.

Abstract: Shortest networks interconnecting a finite set of points in a normed space can have singularities that are characterized using subdifferentials. This description motivates various questions in the combinatorial geometry of normed spaces, such as the following: How many unit vectors can be found in a d-dimensional normed space such that the sum of any k of them has norm at most 1? I will discuss this problem using convexity, duality, linear algebra and graph theory.

Kevin Hughes (Heilbronn Institute for Mathematical Research)

17 October 2016

Lacunary discrete spherical averages

Abstract: Motivated by work of Calderon and Coifman--Weiss on lacunary (continuous) spherical averages, we investigate lacunary discrete spherical averages of Magyar--Stein--Wainger. This turns out to be surprisingly different from the Euclidean case. We discover a new connection to ergodic theory and discuss a p-adic Seeger--Wainger--Wright theorem. This is joint work with Jim Wright and Jacek Zienkiewicz.

Mathieu Kurzke (University of Nottingham)

24 October 2016

Boundary and interior vortices in thin film micromagnetics.

Ferromagnetic materials are described by a nonlocal and nonconvex variational principle. For certain regimes, it is possible to rigorously derive simplified models using Gamma-convergence. In my talk, I will concentrate on models that contain point defects carrying a topological charge, and will discuss static and dynamical results for these defects.

Euan Spence (University of Bath)

31 October 2016

Is the Helmholtz equation really sign-indefinite?

The usual variational (or weak) formulations of the Helmholtz equation are sign-indefinite (i.e. not coercive). This is often for a good reason, since in bounded domains under certain boundary conditions the solution of the Helmholtz equation is not unique at wave numbers that correspond to eigenvalues of the Laplacian, and thus the variational problem cannot be sign-definite. However, even in cases where the
solution is unique for all wave numbers, the standard variational formulations of the Helmholtz equation are still indefinite when the wave number is large. In this talk I will argue that this indefiniteness is not an inherent feature of the Helmholtz equation, only of its standard formulations.

This talk will give on overview of joint work with Simon
Chandler-Wilde (Reading), Ivan Graham (Bath), Ilia Kamotski (UCL), Andrea Moiola (Reading), and Valery Smyshlyaev (UCL).

Sally Hill (Cardiff University)

7 November 2016

On Sum and Distance Systems, Reversible Squares and Divisor Functions.

Abstract: Sum and distance systems can be thought of as a discrete analogue of the union of a Minkowski sum system with a Minkowski difference system. One of their natural habitats is within a traditional reversible square matrix, where conjugation with a specific orthogonal symmetric involution, always reveals a sum and distance system within the block structure of the conjugated matrix. In fact it can be shown that the block representation is an algebra isomorphism.

Building upon results of Dame Kathleen Ollerenshaw, and David Bree, one finds that for a fixed dimension n, the number of traditional principal reversible square matrices of size n x n can be enumerated using the jth non-trivial divisor function c_j (n). This counts the total number of proper ordered factorisations of the integer n into j factors (i.e. excluding the factor 1, but counting permutations of the factors); if j is greater than the total number of prime factors of n, including repeats, then c_j (n) = 0. The non-trivial divisor function c_j has been far less studied than its multiplicative cousin, the jth divisor function d_j. Further relations concerning these two functions are discussed in the talk.

Alexander Pushnitski (King’s College London)

14 November 2016

Hankel operators and their multiplicative analogues

A Hankel matrix is an infinite matrix of the form {a(n+m)}, where n, m
are non-negative integers. A multiplicative Hankel matrix is an
infinite matrix of the form {a(nm)} (the argument of a is the product
of n and m), where n and m are positive integers.
The theory of Hankel matrices is classical and well established, while
the theory of their multiplicative analogues seems to be in its
infancy. I will attempt to give a survey and comparison of these two
theories. The talk is partly based on my work in progress with
Karl-Mikael Perfekt.

Angkana Ruland (Christ Church, Oxford University)

21 November 2016

Backward Uniqueness for the Heat Equation in Conical Domains

Abstract: In this talk I will present backward uniqueness results for the heat equation in conical domains, improving previous results of Sverák and Li. Moreover, under additional assumptions on the initial data the optimal angular dependence is proved. After reviewing the diﬀerent behavior of the heat equation in bounded and un-bounded domains and illustrating the underlying diﬃculties, the central Carleman estimate will be discussed.

Pankaj Vishe (Durham University)

28 November 2016

Quartic forms in 37 variables

Abstract: A projective variety X defined over the rational numbers is said to satisfy the Hasse principle if the presence of an adelic point on X guarantees the presence of a rational point. We prove that a smooth quartic hypersurface X over Q satisfies Hasse Principle as long as the number of variables are greater than or equal to 37. The key ingredient is Kloosterman type extra averaging in conjunction with the van der Corput differencing applied to estimate the ''minor arc contribution'' in the Hardy-Littlewood circle method. This is a joint work with Oscar Marmon.

Jozsef Lörinczi (University of Loughborough)

5 December 2016

Some spectral properties of non-local Schrödinger operators.

Abstract: Non-local Schrödinger operators are of the form H = - L + V, where L is a suitable pseudo-differential operator, and V is a multiplication operator called the potential. In this talk I will discuss some cases when H has discrete spectrum and the eigenvalue problem can be solved explicitly. Then I will review some results on the spatial decay of eigenfunctions for a more general class of non-local Schrödinger operators. The arguments use a combination of analysis and probability.

Arick Shao (Queen Mary University)

23 January 2017

Unique Continuation from Infinity for Waves and Applications.

Abstract: We survey some recent results regarding unique continuation of linear and nonlinear waves from infinity. Time permitting, we discuss some Carleman estimates that were applied in their proofs, and we discuss extensions of these results to geometric settings. Finally, we discuss some applications of these results and estimates to other problems, such as singularity formation.

Leonid Parnovski (University College London)

6 February 2017

Local density of states and the spectral function for almost-periodic operators

I will discuss the asymptotic behaviour (both on and off the diagonal) of the spectral function of a Schrödinger operator with smooth bounded potential when energy becomes large. I formulate the conjecture that the local density of states (i.e. the spectral function on the diagonal) admits the complete asymptotic expansion and discuss the known results, mostly for almost-periodic potentials.

13 February 2017

Delocalising the parabolic Anderson model

The parabolic Anderson problem is the Cauchy problem for the heat equation on the integer lattice with random potential. It is well-known that, unlike the standard heat equation, the solution of the parabolic Anderson model exhibits strong localisation. In particular, for a wide class of i.i.d. potentials, it is localised at just one point. In the talk, we discuss a natural modification of the parabolic Anderson model on Z, where the one-point localisation breaks down for heavy-tailed potentials and remains unchanged for light-tailed potentials, exhibiting a range of phase transitions. This is a joint work with Stephen Muirhead and Richard Pymar.

Amit Einav (TU Vienna)

20 February 2017

Kac’s Model and the (Almost) Cercignani’s Conjecture

Abstract: The validity, and invalidity, of Cercignani’s Conjecture in Kac’s many particle model, is a prominent problem in the field of Kinetic Theory. In its heart, it is an attempt to find a functional inequality, which is independent of the number of particles in the model, that will demonstrate an exponential rate of convergence to equilibrium. Surprisingly enough, this simple conjecture and its underlying functional inequalities contain much of the geometry of the process, and any significant advances in its resolution involves intradisciplinary approach.
In this talk I will present recent work with Eric Carlen and Maria Carvahlo, where we have defined new notions of chaoticity on the sphere and managed to give conditions under which an ‘almost’ conjecture is valid. With that in hand, I will show how Kac’s original hope to conclude a rate of decay for his model's limit equation from the model itself, is achieved.

27 February 2017

Abscissae of functions related to a class of problems in analytic number theory

A number of major problems in analytic number theory, including Dirichlet's Divisor Problem and the Lindelöf Hypothesis, share similar characteristics. In both of these examples, the problem is to find the maximal order of some particular function. In this talk, we will consider this problem in greater generality in terms of abscissae. We will discuss various questions around them and some pertinent results.

Mariya Ptashnyk (University of Dundee)

13 March 2017

Multiscale modelling and analysis of plant tissue biomechanics

In this talk we will consider a microscopic modelling and multiscale analysis of plant tissue biomechanics. Plant tissues are composed of cells surrounded by cell walls and connected by a cross-linked pectin network of middle lamella. Plant cell walls must be very strong to resist high internal hydrostatic pressure and at the same time flexible to permit growth. It is supposed that calcium-pectin cross-linking chemistry is one of the main regulators of plant cell wall elasticity and extension. Hence in the microscopic model for plant cell wall and tissue biomechanics we will consider the influence of the microscopic structure and chemical processes on the mechanical properties of plant tissues. The interplay between the mechanics and the chemistry will be defined by assuming that the elastic properties of cell walls depend on the chemical processes (i.e. on the density of calcium-pectin cross-links) and chemical reactions depend on mechanical stresses within cell walls (i.e. the stress within cell walls can break the cross-links). The microscopic model will constitute a strongly coupled system of reaction-diffusion-convection equations for chemical processes in plant cells, the equations of poroelasticity for elastic deformations of plant cell walls and middle lamella, and Stokes equations for fluid flow inside the cells. To analyse the macroscopic behaviour of plant cell walls, as well as for effective numerical simulations, the macroscopic models for plant cell wall and tissue biomechanics will be derived using homogenization techniques. In the multiscale analysis we will distinguish between periodic and random distribution of cells in a plant tissue. The numerical solutions of the macroscopic model will demonstrate the patterns in the interactions between mechanical stresses and chemical processes.

Igor Wigman (King's College London)

20 March 2017

Nodal intersections of random toral eigenfunctions against a test curve

This talk is based on joint works with Zeev Rudnick, and Maurizia Rossi.

We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard 2-dimensional flat torus (“arithmetic random waves”) with a fixed reference curve. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry.

Our first result prescribes the asymptotic behaviour of the nodal intersections variance for generic smooth curves in the high energy limit; remarkably, it is dependent on both the angular distribution of lattice points lying on the circle with radius corresponding to the given wavenumber, and the geometry of the given curve. For these curves we can prove the Central Limit Theorem. We then construct some examples of exceptional "static" curves where the variance is of smaller order of magnitude, and the limit distribution is non-Gaussian.

Harsha Hutridurga (Imperial College)

27 March 2017

A new approach to studying strong advection problems.

In this talk, I will be attempting to give an overview of a new weak convergence type tool developed by myself, Thomas Holding (Warwick) and Jeffrey Rauch (Michigan) to handle multiple scales in advection-diffusion type models used in the turbulent diffusion theories. Loosely speaking, our strategy is to recast the advection-diffusion equation in moving coordinates dictated by the flow associated with a mean advective field. Crucial to our analysis is the introduction of a fast time variable. We introduce a notion of "convergence along mean flows" which is a weak multiple scales type convergence -- in the spirit of two-scale convergence theory. We have used ideas from the theory of "homogenization structures" developed by G. Nguetseng. We give a sufficient structural condition on the "Jacobain matrix" associated with the flow of the mean advective field which guarantees the homogenization of the original advection-diffusion problem as the microscopic lengthscale vanishes. We also show the robustness of this structural condition by giving an example where the failure of such a structural assumption leads to a degenerate limit behaviour. More details on this new tool in homogenzation theory can be found in the following paper:

T. Holding, H. Hutridurga, J. Rauch. Convergence along mean flows, SIAM J Math Anal., Volume 49, Issue 1, pp.222--271 (2017).

In a sequel to the above mentioned work, we are preparing a work where we address the growth in the Jacobain matrix -- termed as Lagrangian stretching in Fluid dynamics literature -- and its consequences on the vanishing microscopic lengthscale limit. To this effect, we introduce a new kind of multiple scales convergence in weighted Lebesgue spaces. This helps us recover some results in Freidlin-Wentzell theory. We address some well-known advective fields such as the cellular flows, the cat's eye flows and some special class of the Arnold-Beltrami-Childress flows. This talk aims to present both these aspects of our work in an unified manner.

Hendrik Weber (University of Warwick)

3 April 2017

Recent progress in singular stochastic PDE

This talk is concerned with stochastic partial differential equations (SPDE) driven by a singular noise term, such as space-time white noise. Following Hairer’s groundbreaking work on regularity structures,
research on these equations has witnessed an enormous activity over the last years and the aim of the talk is to give a survey of some of the recent results.

First I will explain how non-linear SPDE arise as scaling limits of discrete models from statistical mechanics. Then I will outline how the phenomenon of metastability can be observed in such an equation. Finally, I will try to convey an idea of the type of argument which allows to analyse these SPDE.

Taryn Flock (University of Birmingham)

8 May 2017

A sharp X-ray Strichartz inequality for the Schrodinger equation

We explore a natural interplay between the solution to the time-dependent free Schrodinger equation and the (spatial) X-ray transform-- proving a variant on a Strichartz estimate. Our estimates are sharp in the sense that we identify the best constant C and show that a datum f achieves equality in the estimate if and only if it is an isotropic centered gaussian. In higher dimensions, we prove similar results where the X-ray transform is replaced by the more general k-plane transform. In the process, we obtain sharp L^2(\mu) bounds on Fourier extension operators associated with certain high-dimensional spheres, involving measures supported on natural co-k -planarity" sets. This is joint work with Jonathan Bennett, Neal Bez, Susana Gutierrez, and Marina Iliopoulou.

## Past events

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