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
Professor Marco Marletta
Deputy Head of School
- Email:
- marlettam@cardiff.ac.uk
- Telephone:
- +44 (0)29 2087 5552
Academic staff
Professor Alexander Balinsky
Professor of Mathematical Physics
- Email:
- balinskya@cardiff.ac.uk
- Telephone:
- +44 (0)29 2087 5528
Professor Malcolm Brown
Professor of Computational Mathematics
- Email:
- brownbm@cardiff.ac.uk
- Telephone:
- +44 (0)29 2087 5538
Dr Federica Dragoni
Reader (Associate Professor)
- Email:
- dragonif@cardiff.ac.uk
- Telephone:
- +44 (0)29 2087 5529
Professor Des Evans
Emeritus Professor of Mathematics
- Email:
- evanswd@cardiff.ac.uk
- Telephone:
- +44 (0)29 2087 4206
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.
Speakers | Date | Abstract |
---|---|---|
Jean-Marc Bouclet (Toulouse) | 13 May 2019 | Sharp time decay and resolvent estimates for dispersive equations I will present recent results obtained with Nicolas Burq about local energy decay (and the related resolvent estimates) for the wave, Klein-Gordon and Schrödinger equations on an asymptotically Euclidean background. I'll make most of the talk accessible to a large audience and only touch the technical aspects if time permits. |
Mahir Hadžić (KCL) | 10 May 2019 Room M/0.40 | Stability and instability of self-gravitating relativistic matter distributions We prove that the compactly supported steady states of the Einstein-Vlasov system (representing galaxies) and Einstein-Euler system (representing stars) become linearly unstable when their central redshift is sufficiently large. Our work confirms the scenario of dynamic instability proposed in the 1960s by Zel'dovich and Podurets (for the Einstein-Vlasov system) and by Harrison, Thorne, Wakano, Wheeler (for the Einstein-Euler system). This is a joint work with Gerhard Rein and Zhiwu Lin. |
Mohammed Lemou (Rennes) | 7 May 2019 Room E/2.20 | Nonlinear stability and instability results for gravitational kinetic models I will start by giving a short overview around stability and instability issues in gravitational systems driven by kinetic equations. A well-known conjecture in both astrophysics and mathematics communities was that "all steady states of the gravitational Vlasov-Poisson system which are decreasing functions of the energy, are stable up to space translations". We will explain why the traditional variational approaches are not sufficient to answer this conjecture and briefly present an alternative approach (that we introduced in 2012), which, although nonquantitative, solves this conjecture in the case of the Vlasov-Poisson system. More recently, we proved a quantitative version of this nonlinear stability. This has been achieved by using an extended rearrangement notion of functions and a refined functional inequality on this rearrangement, together with a Poincaré-like inequality. Another advantage of this new approach is its applicability to other systems like the so-called Hamiltonian Mean Field (HMF), where the space domain is bounded and where the decreasing property of the steady states is no more sufficient to guarantee their stability. In fact, an additional explicit criteria is needed for HMF, under which the non-linear stability is proved. This criteria is sharp since a nonlinear instability result can be proved for HMF if it is not satisfied. |
Alberto Debernardi (Bar-Ilan University) | 15 April 2019 | Weighted norm inequalities for integral transforms with splitting kernels Given an integral transform on the positive real line, we say that its kernel is |
Anne Nouri (Marseille) | 8 April 2019 | On some quantum kinetic equations The study of gases at very low temperatures is an important field in contemporary physics. We present different models like a Boltzmann type equation for anyons and the Boltzmann-Nordheim quantum equation for bosons. A study of the well-posedness of their related Cauchy problem is presented. |
Kangwei Li (BCAM) | 1 April 2019 | Extrapolation for multilinear Muckenhoupt class of weights and applications In this talk, I will introduce our recent progress on extrapolation theory. In the linear case, the extrapolation theory is well understood. However, in the multilinear case, the extrapolation was only known for product $A_p$ weights. The multilinear $A_{\vec P}$ weight, which was introduced in 2009, no extrapolation theory was known before. In this talk, I will give a full solution to this problem. As applications, we can improve the weighted estimates for the bilinear Hilbert transform, the multilinear Marcinkiewicz-Zygmund inequality etc. Endpoint cases are also provided. This talk is based on joint work with José María Martell and Sheldy Ombrosi, and joint work with Henri Martikainen, José María Martell, Sheldy Ombrosi and Emil Vuorinen. |
Filip Rindler (Warwick) | 25 March 2019 | Theme & variations on div \mu = \sigma The PDE div \mu = \sigma for (vector) measures \mu and \sigma appears -- sometimes in a slightly hidden way -- in many different problems of geometric measure theory and the calculus of variations, for instance in the structure theory of normal currents, Lipschitz functions, varifolds, and liftings for BV maps. In this talk I will survey a number of recent results about this equation and other related PDEs. As applications, I will discuss the structure of singularities of solutions, dimensional estimates, integral functionals on BV, and several converses of Rademacher's theorem (both in Euclidean and non-Euclidean settings). |
Choi-Hong Lai (Greenwich) | 18 March 2019 | A generalised defect correction method and its applications in option pricing The Black-Scholes partial differential equation is well known to be an option pricing tool for an ideal market. This requires the use of constant volatility over the period of time from zero up to the mature time. For realistic markets the volatility may become nonlinear, and the partial differential equation becomes nonlinear. A typical example is Frey and Patie nonlinear volatility model which contains a very small parameter in it. In addition to the nonlinearity a correction term may be added to the Black-Scholes partial differential equation to facilitate discrete hedging. The technique of defect correction method is extended to handle this type of nonlinear partial differential equations. Several examples of the defect correction method are used to motivate the use of defect correction method and the adaptation of it to handle asymptotic expansion and nonlinear correction term is to be discussed. Numerical examples of a European option pricing will be used to demonstrate the algorithm. |
Jessica Guerand (Cambridge) | 11 March 2019 | Quantitative parabolic regularity à la De Giorgi De Giorgi method is a way to prove interior Hölder regularity of solutions of parabolic equations. While in the elliptic case the proof is completely quantitative, in the parabolic case it seems to remain a non-quantitative step: the intermediate value lemma. The purpose of this talk is to present a quantitative version of this step after introducing how it is useful to get Hölder regularity. |
Jan Lang (Ohio State) 3:10 - 4:10 | 4 March 2019 | Approximation of non-compact Sobolev embeddings For limiting noncompact Sobolev embeddings into continuous functions, we study the behavior of approximation, Gelfand, Kolmogorov, Bernstein and isomorphism s-numbers. In the one-dimensional case, the exact values of the above-mentioned strict s-numbers are obtained, and in the higher dimensions sharp estimates for asymptotic behavior of the strict s-numbers are established. As all known results for s-numbers of Sobolev type embeddings are studied mainly under the compactness assumption, our work is an extension of existing results and reveal an interesting behavior of s-numbers in the limiting case when some of them (approximation, Gelfand, and Kolmogorov) have positive lower bound and others (Bernstein and isomorphism) are decreasing to zero. It also follows from our results that such limiting noncompact Sobolev embeddings are finitely strictly singular maps. Based on joint work with Vit Musil. |
Davoud Cheraghi (Imperial) 2:10 - 3:10 | 4 March 2019 | Quasi-periodic dynamics in complex dimension one Quasi-periodic dynamics in one complex variable reveals fascinating interplays between complex analysis and Diophantine approximations. The question of whether a nonlinear perturbation of a linear rotation is conjugate to a linear rotation (linearisation) dates back to more than a century ago, with remarkable contributions by C. Siegel, A. Brjuno, and J.-C. Yoccoz. The behaviour of non-linearisable maps is very complicated. Indeed, there is not a single example of a non-linearisable map whose local behaviour is completely understood. There is major recent advances on this problem which has lead to a complete description of the topological behaviour of typical orbits. This is an introductory talk to demonstrate some of these results. |
Gianne Derks (Surrey) | 25 February 2019 | Existence and stability of fronts in inhomogeneous wave equations Models describing waves in anisotropic media or media with imperfections usually have inhomogeneous terms. Examples of such models can be found in many applications, for example in nonlinear optical waveguides, water waves moving over a bottom with topology, currents in nonuniform Josephson junctions, DNA-RNAP interactions etc. Homogeneous nonlinear wave equations are Hamiltonian partial differential equations with the homogeneity providing an extra symmetry in the form of the spatial translations. Inhomogeneities break the translational symmetry, though the Hamiltonian structure is still present. When the spatial translational symmetry is broken, travelling waves are no longer natural solutions. Instead, the travelling waves tend to interact with the inhomogeneity and get trapped, reflected, or slowed down. |
Stephen Pankavich (Colorado School of Mines) | 18 February 2019 | Classical solutions of the Vlasov-Maxwell-Fokker-Planck system The Vlasov-Maxwell system is a fundamental kinetic model of plasma dynamics. When one includes effects due to collisions with a fixed background of particles, the result is the Vlasov-Maxwell-Fokker-Planck system. The first Lorentz-invariant model, which considers relativistic velocities, was derived by Calogero in 2010. Here, we shall discuss well-posedness results for global-in-time classical solutions of this system and its non-relativistic analogue, each posed in a variety of dimensional settings. Our methods utilize a gain in regularity stemming from the (momentum) diffusion operator to arrive at smooth solutions arising from initial data which lack even weak differentiability properties. |
Yuzhao Wang (Birmingham) | 11 February 2019 | Further renormalization and unconditionally global well-posedness of the cubic We first develop a normal form approach to study NLS in Fourier-Lebesgue spaces. By applying an inﬁnite iteration of normal form reductions, we derive a normal form equation which is This is a joint work with Tadahiro Oh at the University of Edinburgh. |
Monica Musso (Bath) | 4 February 2019 | Gluing methods for vortex dynamics in Euler flows A classical problem for the two-dimensional Euler flow for an incompressible fluid confined to a smooth domain is that of finding regular solutions with highly concentrated vorticities around N moving vortices. The formal dynamic law for such objects was first derived in the 19th century by Kirkhoff and Routh. We devise a gluing approach for the construction of smooth N-vortex solutions. We capture in high precision the core of each vortex as a scaled finite mass solution of Liouville's equation plus small, more regular terms. |
Ilya Molchanov (Bern) | 28 January 2019 | The semigroup of metric measure spaces and stable random spaces The family of Polish spaces equipped with probability measures can be turned into a semigroup using the Cartesian product as the semigroup operation. The main result is the Fundamental Theorem of Arithmetics for this semigroup, which establishes an analogue of the prime numbers decompositions for such spaces. Further, random metric measure spaces are considered in view of their infinite divisibility and stability properties with respect to the semigroup operation. A characterisation of stable metric measure spaces is also provided. |
Sabine Bögli (Imperial) | 21 Januaury 2019 | The essential numerical range for unbounded linear operators If a linear operator T is approximated by projection or domain truncation methods, eigenvalues may accumulate at a point that does not belong to the spectrum of T. The occurrence of such a spurious eigenvalue is commonly known as spectral pollution. A useful tool to describe the set of spectral pollution is the notion of essential numerical range W_e(T) which was introduced in the late 1960s for bounded T. We discuss the generalisation of this notion to unbounded operators, including equivalent characterisations and perturbation results. |
Michela Ottobre (Herriot-Watt Uni) | 10 December 2018 | On a class of SDEs with multiple invariant measures In 1968 Hoermander introduced a sufficient condition to ensure hypoellipticity of second order partial differential operators. As is well known, this seminal work of Hormander had deep repercussions both in the analysis of PDEs and in probability theory and a large strand of literature has been devoted to studying ergodic properties of processes which do satisfy the Hoermander condition (HC). While such literature has mostly been concerned with study of convergence to equilibrium for dynamics which admit a unique invariant measure, it is a known fact that Hoermander-type diffusions need not be ergodic, i.e. they need not admit a unique invariant measure. In this talk we will present the UFG condition, which is weaker than the Hormander condition. Such a condition was introduced by Kusuoka and Strook with probabilistic motivations, and, independently, by Sussman, Hermann and Lobry, this time in the field of control theory. We will present new results on the geometry and long time behaviour of diffusion semigroups that do not satisfy the Hoermander condition. We will highlight how, loosely speaking, UFG diffusions constitute a large class of SDEs which exhibit multiple equilibria (invariant measures) and such that it is possible to determine in a systematic way the basin of attraction of each equilibrium state. |
Christian Maes (KU Leuven) 3:10 - 4:10 | 3 December 2018 | Time-symmetric aspects of stationary flow We start by recalling the relation between detailed balance and gradient flow. We generalize it to the case of GENERIC. We discuss how time-symmetric aspects matter crucially when nonequilibrium driving is added. |
David Lafontaine (Bath) 2:10 - 3:10 | 3 December 2018 | About wave and Schrödinger equations in the exterior of many strictly convex obstacles In order to study the non-linear Schrödinger and wave equations, it is crucial to understand the decay of solutions of the associated linear equations. When a trapped trajectory exists, a loss is unavoidable for a first family of a-priori estimates of the linear flow: the so-called smoothing estimates. In contrast, we will show that in the exterior of many strictly convex obstacles, the estimates of space-time norms of solutions, known as Strichartz estimates, hold with no loss with respect to the flat case, as soon as the dynamic of the trapped trajectories is sufficiently unstable. Finally, if time permits, we will say a word about the associated non-linear equations: if the geometry does not induce too much concentration of energy, we expect that the solutions behave linearly in large times. |
Benjamin Gess (Leipzig) 3:10 - 4:10 | 26 November 2018 | Random dynamical systems for stochastic PDE with nonlinear noise. In this talk we will revisit the problem of generation of random dynamical systems by solutions to stochastic PDE. Despite being at the heart of a dynamical system approach to stochastic dynamics in infinite dimensions, most known results are restricted to stochastic PDE driven by affine linear noise, which can be treated via transformation arguments. In contrast, in this talk we will address instances of stochastic PDE with nonlinear noise, with particular emphasis on porous media equations driven by conservative noise. This class of stochastic PDE arises in particular in the analysis of stochastic mean curvature motion, mean field games with common noise and is linked to fluctuations in non-equilibrium statistical mechanics. |
Kirill Cherednichenko (Bath) 2:10 - 3:10 | 26 November 2018 | Periodic PDEs with critical contrast: unified approach to homogenisation and links to time-dispersive media. I shall discuss a novel approach to the homogenisation of high-contrast periodic PDEs, which yields an explicitly construction of their norm-resolvent asymptotics. A practically relevant outcome of this result is that it interprets composite media with micro-resonators as a class of time-dispersive media. This is joint work with Yulia Ershova and Alexander Kiselev. Random dynamical systems for stochastic PDE with nonlinear noise. In this talk we will revisit the problem of generation of random dynamical systems by solutions to stochastic PDE. Despite being at the heart of a dynamical system approach to stochastic dynamics in infinite dimensions, most known results are restricted to stochastic PDE driven by affine linear noise, which can be treated via transformation arguments. In contrast, in this talk we will address instances of stochastic PDE with nonlinear noise, with particular emphasis on porous media equations driven by conservative noise. This class of stochastic PDE arises in particular in the analysis of stochastic mean curvature motion, mean field games with common noise and is linked to fluctuations in non-equilibrium statistical mechanics. |
Mahir Hadžić (KCL) | 19 November 2018 | Cancelled |
Petr Siegl (Belfast) | 12 November 2018 | Spectral instabilities of Schrödinger operators with complex potentials. We present an overview of recent results on pseudospectra and basis proper- |
Nicolas Dirr (Cardiff) Elaine Crooks (Swansea) Tristan Pryer (Reading) Carlo Mercuri (Swansea) Gui-Qiang Chen (Oxford) 11:00 - 5:30 | 9 November 2018 | LMS South-West Network on Generalised Solutions for Nonlinear PDEs 11.30-12.00 Welcome coffee. |
David Beltran (BCAM) | 5 November 2018 | Local smoothing estimates for Fourier Integral Operators and wave equations The sharp fixed-time Sobolev estimates for Fourier Integral Operators (and therefore solutions to wave equations in Euclidean space or compact manifolds) were established by Seeger, Sogge and Stein in the early 90s. Shortly after, Sogge observed that a local average in time leads to a regularity improvement with respect to the sharp fixed-time estimates. Establishing variable-coefficient counterparts of the Bourgain—Demeter decoupling inequalities, we improved the previous known local smoothing estimates for FIOs, and we show, in particular, that our results are sharp in both the Lebesgue and regularity exponent (up to the endpoint) in odd dimensions. This is joint work with Jonathan Hickman and Christopher D. Sogge. |
Maria Carmen Reguerra (Birmingham) | 22 October 2018 | Sparse bounds for Bochner-Riesz operators Sparse operators are positive dyadic operators that have very nice boundedness properties. The L^p bounds and weighted L^p bounds with sharp constant are easy to obtain for these operators. In the recent years, it has been proven that singular integrals (cancellative operators) can be pointwise controlled by sparse operators. This has made the sharp weighted theory of singular integrals quite straightforward. The current efforts focus in understanding the use of sparse operators to bound rougher operators, such a oscillatory integrals. Following this direction, our goal in this talk is to describe the control of Bochner-Riesz operators by sparse operators. |
Sanju Velani (York) | 15 October 2018 | Inhomogeneous Diophantine Approximation on M_0 -sets with restricted denominators Let F ⊆ [0,1] be a set that supports a probability measure µ with the property that | b µ(t)| ≪ (log|t|) −A for some constant A > 0. Let A = (q n ) n∈ N be a sequence of natural numbers. If A is lacunary and A > 2, we establish a quantitative inhomogeneous Khintchine-type theorem in which (i) the points of interest are restricted to F and (ii) the denominators of the ‘shifted’ rationals are restricted to A. The theorem can be viewed as a natural strengthening of the fact that sequence (q n x mod1) n∈ N is uniformly distributed for µ almost all x ∈ F. Beyond lacunary, our main theorem implies the analogous quantitative result for sequences A for which the prime divisors are restricted to a finite set of k primes and A > 2k. |
Frank Rösler (Cardiff) | 8 October 2018 | Norm-resolvent convergence in perforated domains. For several different types of boundary conditions (Dirichlet, Neumann and Robin), we prove norm-resolvent convergence for the operator −∆ in the perforated domain Ω without balls of small radius, to the limit operator −∆+µ on L 2 (Ω), where µ ∈ C is a constant depending on the choice of boundary conditions. |
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 |
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.
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.