London Mathematical Society -- EPSRC Durham Symposium

Mathematical and Computational Aspects of Maxwell's Equations

2016-07-11 to 2016-07-21

Abstracts of Talks

Simon Arridge: *Direct Inversion for Quantitative PhotoAcoustic Tomography*

Quantitative Photoacoustic Tomography (QPAT) is the problem of recovering optical parameters, such as absorption and scattering, from photoacoustic data. It is usually formulated in two steps : firstly the reconstruction of an internal photoacoustic initial pressure image from measured time-series on a surface boundary, and secondly the solution of a non-linear inverse problem from internal data. Whereas the first problem is stable if the measurement surface surrounds the object, in a limited aperture setting this reconstruction can generate artefacts. In this talk we show the advantages of using a direct method to reconstruct internal optical parameters from the observed time-series. Joint work with A. Pulkkinen, B. Cox, T.Tarvainen.

Jonathan Bennett: *The Brascamp-Lieb inequality in harmonic analysis and dispersive PDE*

The Brascamp-Lieb inequality simultaneously generalises important classical inequalities in euclidean analysis such as the multilinear H\"older, Young convolution and Loomis-Whitney inequalities. In the first part of this talk we discuss this well-known inequality from a structural point of view, addressing basic issues such as the finiteness of the constant, and the existence and nature of extremisers. In the second part we describe a natural Fourier-analytic generalisation of the Brascamp-Lieb inequality that is currently seeing far-reaching applications from harmonic analysis to partial differential equations and number theory.

Timo Betcke: *Fast boundary element solution of Maxwell problems with BEM++*

We will present some current ongoing developments in BEM++ for the efficient solution of low to high-frequency Maxwell problems.

Anthony Byrne: *Welcome from the LMS*

Fioralba Cakoni: *A Qualitative Approach to Inverse Electromagnetic Scattering for Inhomogeneous Media*

Since the introduction of the linear sampling method in 1996 followed by the factorization method in 1998 and later the proof of the existence and monotonicity properties of real transmission eigenvalues in 2010, qualitative methods have become a popular approach for solving inverse scattering problems. Interest in this area has exploded and the vast amount of literature currently available is an indication of the myriad directions that this research has taken. In this talk we consider the inverse electromagnetic scattering problem for an inhomogeneous (possibly anisotropic) media and show how to obtain information about the support as well as electric and magnetic properties of the media based on an investigation of the corresponding far field operator. At the foundation of this investigation is the so-called the transmission eigenvalue problem, which is a non-linear and non-selfadjoint eigenvalue problem. In particular, we will discuss the relevance of the transmission eigenvalue problem for Maxwell's equations and present what type of information transmission eigenvalues provide about the inhomogeneity.

Fioralba Cakoni: *Nondestructive testing of the delaminated interface between two materials*

We consider the problem of detecting if two materials that should be in contact have separated or delaminated. The goal is to find an electromagnetic radiation technique to detect the delamination. We model the delamination as a thin opening between two materials of different electromagnetic properties, and using asymptotic techniques we derive a reduced model where the delaminated region is replaced by jump conditions on the electric and magnetic fields. We discuss in details the two-dimensional case of TE polarization. The reduced model has potential singularities due to the edges of the delaminated region, and we show that the forward problem is well posed for a large class of possible delaminations. We then design a special Linear Sampling Method (LSM) for detecting the shape of the delamination assuming that the background, undamaged, state is known. Finally we show, by numerical experiments, that our LSM can indeed determine the shape of delaminated regions. Extensions to Maxwell’s equations will also be discussed. This is a joint work with Irene De Teresa, Houssem Haddar and Peter Monk.

Yves Capdeboscq: *Open Problem Session*

Yves Capdeboscq: *Small inclusions for the Time Harmonic Maxwell Equations.*

I will review recent result on the topic of small perturbative inclusions for the Time Harmonic Maxwell Equation : we show that the general theory developed over a decade ago for conductivity equation can be extended, in similar generality, for the Maxwell Equation, by revisiting elliptic regularity theory in this context.

Maxence Cassier: *Bounds on Stieltjes functions and their applications to fundamental limits of broadband passive cloaking in the quasi-static regime*

In this talk, we derive bounds on Stieltjes functions which generalize those provided by M. Gustafsson and D. D. Söjberg and apply them in the context of broadband cloaking to negatively answer the following question. For the quasi-static approximation of Maxwell’s equation, is it possible to construct a passive cloaking device that will cloak an object over a whole frequency band? Joint work with Graeme W. Milton.

Simon Chandler-Wilde: *Numerical-Asymptotic Approximation at High Frequency*

It is an obvious idea, when approximating time harmonic scattering problems at high frequency, to seek to build into the approximation space the oscillatory phase or phases that are predicted by high frequency asymptotics. This idea is particularly attractive in a boundary integral equation context, where one only needs to approximate (lower dimensional) boundary traces. The hope is that with some clever insertion of asymptotic information into conventional finite element-style spaces, one can construct "numerical-asymptotic" approximation spaces that can approximate accurately with a dimension of approximation space that is independent (or almost independent) of the frequency, across wide frequency ranges of interest. We discuss progress towards this objective at Reading and elsewhere, discussing both rigorous best approximation results and heuristic experiments, for both 2D and 3D problems. Looking forward to the open problems session in the afternoon, we will see that there is much left to do!

Simon Chandler-Wilde: *Open Problem Session*

Kirill Cherednichenko: *Homogenisation of the system of high-contrast Maxwell equations*

I shall discuss the system of Maxwell equations for a periodic composite dielectric medium with components whose dielectric permittivities $\epsilon$ have a high degree of contrast between each other. We assume that the ratio between the permittivities of the components with low and high values of $\epsilon$ are of the order $\eta^2,$ where $\eta>0$ is the period of the medium. We determine the asymptotic behaviour of the electromagnetic response of such a medium in the ``homogenisation limit", as $\eta\to0,$ and derive the limit system of Maxwell equations in ${\mathbb R}^3.$ Our results extend a number of conclusions of the paper [Zhikov ,V. V., 2004. On the band-gap structure of the spectrum of some divergent-form elliptic operators with periodic coefficients, {\it St.\,Petersburg Math.\,J.}] to the case of the full system of Maxwell equations. This is joint work with Shane Cooper (University of Bath).

Andrew Comech: *Point eigenvalues of the Dirac operators*

We consider the point spectrum of non-selfadjoint Dirac operators which arise as linearizations at solitary wave solutions to the nonlinear Dirac equation. It is known (Barashenkov-Pelinovsky-Zemlyanaya, PhysRevLett.80.5117) that point eigenvalues could emerge from the essential spectrum, bifurcating from the embedded thresholds. We prove the following additional results: 1. Eigenvalues can not bifurcate from the region of the essential spectrum beyond the embedded thresholds; 2. Eigenvalues can be born from the essential spectrum before the embedded thresholds, but only from embedded eigenvalues. We give an example of such bifurcations. Results are based on the preprint "On spectral stability of the nonlinear Dirac equation" (with Nabile Boussaid), http://arxiv.org/abs/1211.3336 to appear in Journal of Functional Analysis.

Tomas Dohnal: *Dirac Type Asymptotics for Wavepackets in Periodic Media of Finite Contrast*

Small broad wavepackets in nonlinear equations can be modeled by asymptotic slowly varying envelope approximations. A classical slowly modulated moving frame ansatz for wavepackets centered at a single Bloch wave leads to the constant coefficient nonlinear Schroedinger equation for the envelope and generates wavepackets traveling at a velocity close to the group velocity of the Bloch wave. We study asymptotics of wavepackets centered at two arbitrary counter-propagating Bloch waves. This leads to the first order Dirac equations (coupled mode equations) for the envelopes. We concentrate on the one dimensional periodic cubically nonlinear Schroedinger equation (PNLS) as a working model but explain how the results carry over to the semilinear (cubic) wave equation. We show that under a suitable periodic perturbation of the periodic structure in PNLS the Dirac equations may allow whole families of localized solitary waves parametrized by velocity. We prove a rigorous justification of the asymptotics and provide several numerical tests. We close by discussing the extension to the higher dimensional problem.

Raluca Felea: *The microlocal analysis of some inverse problems in bistatic SAR*

We consider Fourier integral operators (FIOs) which appear in the bistatic Synthetic Aperture Radar (SAR) imaging when the emitter and receiver are moving on different platforms. We use microlocal analysis methods to characterize the scattering operator $F$, which maps the image to the data, and the normal operator $F^*F$ which is used to reconstruct the image. We show that $F$ exhibits certain singularities and that the normal operator can be written as a sum of operators belonging to a class of distributions associated to two cleanly intersecting Lagrangians $I^{p;l}(\Delta; \Lambda)$. We conclude that the image we obtain exhibits artifacts (given by $\Lambda$) which have the same strength as the the bona-fide part of the image. This is joint work with G. Ambartsoumian, V. Krishnan, C. Nolan and T. Quinto.

Alexander Figotin: *Neoclassical Theory of Electromagnetic Interactions, I *

The theory of electromagnetic (EM) phenomena known as electrodynamics is one of the major theories in science. At macroscopic scales the interaction of the EM field with matter is described by the classical electrodynamics based on the Maxwell-Lorentz theory. Many of electromagnetic phenomena at microscopic scales are covered by the so-called semiclassical theory that treats the matter according to the quantum mechanics, whereas the EM field is treated classically. The subject of this presentation is a recently advanced by us neoclassical electromagnetic theory that describes EM phenomena at all spatial scales –microscopic and macroscopic. This theory modifies the classical electrodynamics into a theory that applies to all spatial scales including atomic and nanoscales. The neoclassical theory is conceived as one theory for all spatial scales in which the classical and quantum aspects are naturally unified and emerge as approximations. It is a classical Lagrangian field theory, and consequently it is a local and deterministic theory. Probabilistic aspects of the theory may arise in it effectively through complex nonlinear dynamical evolution. In this, the first part of our presentation, we provide an introduction to our theory including a concise historical review. This is joint work with Anatoli Babin.

Alexander Figotin: *Neoclassical Theory of Electromagnetic Interactions, II *

The subject of this presentation is a recently advanced by us neoclassical electromagnetic (EM) theory that describes EM phenomena at all spatial scales –microscopic and macroscopic. In this, the second part of our presentation, we provide: (i) a concise presentation of significant features of our neoclassical theory of EM interactions and (ii) compare it with the well established and experimentally verified classical EM theory and the quantum mechanics. The key element of the neoclassical theory is the concept of balanced charge which is conceived to model an elementary charge such as electron or proton. To keep presentation as simple as possible, we consider a spinless balanced charge. The advanced neoclassical theory implies as a valid approximation the classical Maxwell-Lorentz theory at macroscopic scales. Namely, the dynamics of charges when they are well separated is governed approximately by Newton's equations with the Lorentz forces, and the evolution of the EM fields is governed by the Maxwell equations. The neoclassical theory describes also basic quantum effects at atomic scales, including, in particular, the discreteness of the energy levels in the Hydrogen atom and the main features of the de Broglie wave theory. This is joint work with Anatoli Babin.

Sonia Fliss: *Higher order transmission conditions for the homogenization of interface problems*

This work is a joint work with Xavier Claeys (UPMC, University Paris 6) and Valentin Vinoles (POEMS, University Paris Saclay). The mathematical modelling of electromagnetic metamaterials and the homogenization theory are intimately related because metamaterials are precisely constructed by a periodic assembly of small resonating micro-structures involving dielectric materials presenting a high contrast with respect to a reference medium. We wish to look carefully at the treatment of boundaries and interfaces that are generally poorly taken into account by the first order homogenization. This question is already relevant for standard homogenization (i.e. without high contrast) for which taking into account the presence of a boundary induces a loss of accuracy due to the inadequateness of the standard homogenization approach to take into account the boundary layers induced by the boundary. The objective of this work is to construct approximate effective transmission conditions that would restore the desired accuracy. We have first considered a plane interface between a homogeneous and a periodic media in the standard case without high-contrast. We obtain high order transmission conditions between the homogeneous media and the periodic media. The technique we use involves matched asymptotic expansions combined with standard homogenization ansatz. Those conditions are non standard : they involve Laplace-Beltrami operators at the interface and requires to solve cell problems in infinite periodic waveguides. The analysis is based on a original combination of Floquet-Bloch and a periodic version of Kondratiev technique.

Romina Gaburro: *EIT: anisotropy within reach via curved interfaces*

We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body $\Omega\subset\mathbb{R}^{n}$ when the so-called Neumann-to-Dirichlet map is locally given on a non empty curved portion $\Sigma$ of the boundary $\partial\Omega$. We prove that anisotropic conductivities that are a-priori known to be piecewise constant matrices on a given partition of $\Omega$ with curved interfaces can be uniquely determined in the interior from the knowledge of the local Neumann-to-Dirichlet map. This is joint work with Giovanni Alessandrini and Maarten V. de Hoop.

Allan Greenleaf: *Electrical Impedance Tomography and hybrid imaging*

Electrical Impedance Tomography (EIT) uses surface measurements of electrostatic (or very low frequency) fields to noninvasively image the interior of an object, such as the human body or an industrial part. Mathematically, it is based on Calderon’s inverse conductivity problem. Despite (or perhaps because of) being severely ill-posed, the Calder\’on problem has been a source of many developments over the last 35 years. Due to this ill-posedness, EIT suffers from low spatial resolution, and a number of modalities, taking advantage of coupling between electrostatics and other kinds of waves, have been developed to improve upon the imaging that EIT provides. I will discuss EIT and some of the related hybrid imaging modalities.

Allan Greenleaf: *Virtual Hybrid Edge Detection: recovery of singularities in Calderon’s inverse conductivity problem*

The ill-posedness of the Calderon inverse problem is responsible for the poor spatial resolution of Electrical Impedance Tomography (EIT) and has been an impetus for the development of hybrid imaging techniques (discussed in the first talk), which compensate for this lack of resolution by coupling with a second type of physical wave, typically modeled by a hyperbolic PDE. I will describe how the inverse conductivity problem already contains within itself a mechanism for efficient, high resolution propagation of interior singularities of the conductivity to the boundary, based on propagation of singularities for complex principal type operators. Preliminary numerical simulations indicate that this approach is effective for detecting and resolving complex inclusions in the interior using Dirichlet-to-Neumann data. This is joint work with M. Lassas, M. Santacesaria, S. Siltanen and G. Uhlmann.

Dirk Hundertmark: *The solution of the Gevrey smoothing conjecture for the non-cutoff homogenous Boltzmann equation for Maxwellian molecules*

It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties as the heat equation with a fractional Laplacian. In particular, the weak solution of the non-cutoff homogenous Boltzmann equation with initial datum in $L^1_2(\mathbb{R}^3)\cap L\log L(\mathbb{R}^3)$, i.e., finite mass, energy and entropy, should immediately become Gevrey regular. So far, the best available results show that the solution becomes $H^{\infty}$ regular for positive times. Gevrey regularity is also known for weak solutions of the linearised Boltzmann equation, where one studies solutions close to a Maxwellian distribution, or under additional decay assumptions on the solutions. The main problem for establishing Gevrey regularity is that, in order to use the coercivity results on the non-cutoff Boltzmann collision kernel, one has to bound a non-linear and non-local commutator of the Boltzmann kernel with certain sub-exponential weights. We prove, under the sole assumption that the initial datum is in $L^1_2(\mathbb{R}^3)\cap L\log L(\mathbb{R}^3)$, that the weak solution of the homogenous Boltzmann becomes Gevrey regular for strictly positive times. The main ingredient in the proof is a new way of estimating the non-local and non-linear commutator. Joint work with Jean-Marie Barbaroux, Tobias Ried, and Semjon Wugalter.

Steven Johnson: *Nonlinear resonances and the SALT model of lasing*

Although the theoretical description of lasing has been studied for many decades, only recently has it become practical to accurately model the lasing process in complex microstructured lasers, such as random lasers or photonic-crystal laser cavities. A key enabling factor is the SALT (steady state ab-initio laser theory) description pioneered by Tureci, Stone, and others starting in 2006, which reduces the complex time-dependent Maxwell-Bloch equations to a much simpler frequency-domain nonlinear eigenproblem for steady-state lasing modes. The SALT equations themselves resisted general solution for several years, but recently we have developed efficient numerical approaches to solving SALT for complex 3d structures. Moreover, given this numerical foundation, a whole host of new analytical and semi-analytical results become possible, via perturbation theory around the SALT modes. This includes a new understanding of degenerate lasing modes, and a new generalized theory of the laser linewidth arising from quantum fluctuations in the laser. In this talk, we review the mathematical description of lasing and explain why a nonlinear eigenproblem results above threshold, and outline several new developments in laser theory that have been enabled by SALT.

Patrick Joly: *Introduction to Perfectly Matched Layers and wave propagation electromagnetic dispersive media *

This talk will be devoted to an elementary introduction to the two main topics that will be addressed in the second conference talk. First, I will give a general presentation of linear mathematical models for electromagnetic dispersive media together with a hopefully complete and comprehensive description of their analysis. I will give in particular a definition of negative index materials and illustrate some of their properties through various numerical simulations. Second, I will recall the basic foundations of the technique of perfectly matched layers for time domain wave propagation problems in unbounded media. I will give some elements about their stability analysis, a major issue, and show some illustrative numerical simulations.

Patrick Joly: *Time domain Perfectly Matched Layers for wave propagation in dispersive media *

This work analyzes the Perfectly Matched Layers(PMLs) for a large class of dispersive media, particularly for Negative Index Metamaterials (NIMs). Unfortunately, classical PMLs lead to instabilities due to the presence of backward waves, whose phase and group velocities have opposite signs. This is a well-known result for non-dispersive media. In the more general case of dispersive media, which include NIMs, we perform an analysis of a large class of PMLs (which incudes classical PMLs) and deduce a necessary and sufficient stability criterion. We use this criterion to show the inherent instability of classical PMLs in NIMs and to propoose new stable PMLs in this case. We illustrate numerically for the Drude model the instability of the classical PMLs and the stability of the new ones.

Ilia Kamotski: *On the gaps in the spectrum of the Maxwell operator*

We demonstrate the existence of the gaps in the spectrum of the periodic Maxwell operator with medium contrast coefficients. We provide the estimates on the width and location of the gaps and their dependence on the geometry of the media. Co-authors: S. Cooper (University of Bath), V. Smyshlyaev (UCL)

Yaroslav Kurylev: *Geometric Whitney problem and interior inverse spectral problem*

This is a joint work with Ch. Fefferman, S. Ivanov, M. Lassas and H. Narayanan. We start with a geometric Whitney problem of an approximation of a (probably discrete) metric space by a Riemannian manifold of a bounded sectional curvature. We later find the connection between this problem and an interior Gel'fand inverse spectral problem with incomplete data. In the later problem we are given a subdomain $\Omega$ of an unknown Riemannian manifold $(M, g)$ and a finite number of eigenvalues, $\lambda_k, \, k=0, \dots, K,$ and the restrictions to $\Omega$ of the eigenfunctions of the Laplace operator on $(M, g)$, i.e. $\phi_k|_\Omega,\, k=0, \dots, K$. Moreover, $\lambda_k, \, \phi_k|_\Omega$ are known with some error. The Gel'fand interior inverse spectral problems is to recover a Riemannian manifold $(M', g')$ which is Gromov-Hausdorff close to $(M,g)$ from $\lambda_k, \phi_k|_\Omega.$

Paul Ledger: *Characterising the shape of and material properties of hidden conducting targets in metal detection*

Recent work has demonstrated that the response of a conductive object to a low frequency alternating magnetic field, in which the eddy current approximation is valid, can be approximated using a rank-2 tensor dependent only on the object's shape, size, conductivity and permeability. We will present the connection between an asymptotic expansion of the perturbed magnetic field, as the object's size tends to zero, and the response from metal detection signals that are predicted by engineers. Furthermore, we will discuss the properties of these tensors and their connection with the Polya--Szego tensor, which is associated with the description of small inclusions in electrical impedance tomography and small objects in electromagnetic scattering. The talk will comprise of new analysis, as well as numerical and experimental examples of these polarizability tensors, and the relevance of these findings to the location of anti-personnel land mines, and distinguishing them from other buried objects.

Ulf Leonhardt: *Transformation optics*

Transformation optics has been inspired by concepts taken from general relativity, showing that ideas from from general relativity can be put to practical use for engineering problems. For example, invisibility cloaking relies on optical transformations of space that are related to coordinate transformations. Such transformations are implemented with suitable optical materials, using the idea that optical materials behave like space-time geometries. The coordinate invariance of general relativity guarantees that the transformation - the optical illusion - remains undetectable, at least in principle, making a perfect cloaking device. The lecture gives an introduction to transformation optics with cloaking and perfect imaging as examples.

Ulf Leonhardt: *Cosmology in the laboratory*

In transformation optics ideas from general relativity have been put to practical use for engineering problems. This lecture asks the question how this debt can be repaid. It shows how insights from Maxwell's electromagnetism in moving media shed light on the quantum physics of the event horizon. It also discusses recent progress in the understanding of the forces of the quantum vacuum that perhaps may help in solving the enigma of dark energy.

Yangjie Liu: *Controlling electromagnetic waves in a class of invisible materials*

In this contribution talk, we propose a general methodology to manipulate the amplitude of an electromagnetic wave in a pre-defined way, without introducing any scattering. This leads to a whole class of isotropic spatially varying permittivity and permeability profiles that are invisible to incident waves. The theory is illustrated through various numerical examples, including the non-magnetic case. The implementation of the required material properties using metamaterials is discussed, as well as extensions of the method for controlling the phase of electromagnetic fields.

Graeme Milton: *Analytic Properties of the Dirichlet to Neumann Map for Electromagnetism*

The electrodynamic response of any inhomogeneous body is governed by the frequency dependent Dirichlet to Neumann map. If instead to looking at the map which takes the tangential electrical field to the tangential magnetic field, one looks at the map which takes the tangential electrical field to the cross product of the normal to the body and the magnetic field, then we prove this is a operator valued Herglotz function of the component moduli (permittivities and permeabilities) multiplied by the frequency that is analytic when these products all lie in the upper half of the complex plane. As a result the Dirichlet to Neumann map has an integral representation and one can derive universal bounds limiting the transient or time-harmonic electrodynamic response of bodies. We anticipate that using these bounds in an inverse way will lead to entirely new methods for imaging what is inside a body. This joint work with Maxence Cassier and Aaron Welters is Chapter 4 of the new book "Extending the Theory of Composites to Other Areas of Science" edited by Graeme W. Milton, and published by Milton-Patton Publishing.

Graeme Milton: *Cloaking: where Science Fiction meets Science*

Cloaking involves making an object partly or completely invisible to incoming waves such as sound waves, sea waves or seismic waves, but usually electromagnetic waves such as visible light, microwaves, infrared light, or radio waves. Camouflage and stealth technology achieve partial invisibility, but can one achieve true invisibility from such waves? This lecture will survey some of the wide variety of ideas on cloaking: these include transformation based cloaking, non Euclidean cloaking, cloaking due to anomalous resonance, cloaking by complementary media, active interior cloaking and active exterior cloaking. Beautiful mathematics is involved.

Andrea Moiola: *Space-time Trefftz discontinuous Galerkin methods for wave problems*

We present a discontinuous Galerkin (DG) method for linear wave propagation problems in time-domain. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space-time) mesh. The DG scheme is defined for unstructured meshes whose internal faces need not be aligned to the space-time axes. The Trefftz approach can be used to improve and ease the implementation of explicit schemes based on ``tent-pitched'' meshes. We show that the scheme is well-posed, quasi-optimal and dissipative, and prove a priori error bounds for general Trefftz discrete spaces. A concrete discretisation can be obtained using piecewise polynomials that satisfy the wave equation elementwise, for which we show high orders of convergence. The acoustic wave equation, Maxwell equations and more general hyperbolic systems are considered.

Peter Monk: *Stekloff Eigenvalues in Inverse Scattering*

We consider a proposed method for non-destructive testing in which small changes in the (possibly complex valued) refractive index of an inhomogeneous medium of compact support are to be determined from changes in measured far field data due to incident plane waves. Discussing first the Helmholtz equation, the problem is studied by considering a modified far field operator whose kernel is the difference of the measured far field pattern due to the scattering object and the far field pattern of an auxiliary scattering problem with the Stekloff boundary condition imposed on the boundary of a domain containing the scattering object. It is shown that scattering data can be used to determine the Stekloff eigenvalues corresponding to this domain. Extensions to Maxwell’s equations will also be presented.

Hoai-Minh Nguyen: *Cloaking and superlensing using negative index materials*

Negative index materials are artificial structures whose refractive index are negative over some frequency range. These materials were first investigated theoretically by Veselago in 1964 and their existence was confirmed experimentally by Shelby, Smith, and Schultz in 2001. There are two main difficulties in the study of negative index materials: 1) the ellipticity and the compactness are lost in general due to sign changing coefficients of modelling equations; 2)the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others as the loss (the viscosity) goes to 0, might occur. In this talk, I will present schemes for cloaking and superlensing using negative index materials and highlight the analysis.

Lauri Oksanen: *Open Problem Session*

Dmitry Pelinovsky: *Open Problem Session*

Dmitry Pelinovsky: *Introduction to nonlinear PDEs on graphs*

We discuss the nonlinear Schrodinger (NLS) equation on metric graphs subject to the Kirchhoff boundary conditions. Bifurcations of standing localized waves are considered by modern methods of the dynamical system theory. We prove existence of distinct families of small-amplitude standing localized waves on such metric graphs as tadpole, dumbbell and periodic graphs. For the periodic metric graph, homogenization of the NLS equation is discussed by using reductions of PDEs to the discrete nonlinear maps.

Dmitry Pelinovsky: *Validity of the NLS approximation on periodic graphs*

We consider a nonlinear Schrodinger (NLS) equation on a spatially extended periodic graph. With a multiple scaling expansion, an effective amplitude equation can be derived in order to describe slow modulations in time and space of an oscillating wave packet. Using Bloch wave analysis and Gronwall's inequality, we estimate the distance between the macroscopic approximation which is obtained via the amplitude equation and true solutions of the NLS equation on the periodic quantum graph. Moreover, we prove an approximation result for the amplitude equations which occur at the Dirac points of the system.

Michael Plum: *Spectrum generated by waveguides in photonic crystals*

We consider the propagation of polarized electromagnetic waves in 2D photonic crystals, the periodic structure of which is perturbed by a linear waveguide aligned with one of the coordinate axes. We show that the waveguide generates additional spectrum inside the band gaps of the unperturbed, fully periodic problem, and that this happens for arbitrarily small perturbations. This is joint work with B.M. Brown, V. Hoang, and I. Wood.

Mikko Salo: *Recent progress in the Calderon problem*

The inverse conductivity problem, posed by A.P. Calderon in 1980, consists in determining the electrical conductivity of a medium from voltage and current measurements on its boundary. This problem is the mathematical model for Electrical Impedance Tomography, and it has been the basis for the study of related inverse problems for time-harmonic Maxwell equations. We will survey known results and discuss recent harmonic analysis, PDE and geometric techniques that have been applied in this problem.

Mikko Salo: *Inverse problems for time-harmonic Maxwell equations*

This talk will consider the inverse problem of determining the permittivity and permeability of a medium from boundary measurements of fixed frequency electromagnetic fields. In the zero frequency limit, this problem reduces to the inverse conductivity problem posed by A.P. Calderon in 1980. We will discuss recent results and open questions, in particular related to matrix coefficients and partial data.

Gideon Simpson: *Coherent Structures and Shocks in a Periodic Nonlinear Maxwell System*

The primitive equations governing wave propagation in spatially varying optical fibers are the nonlinear Maxwell equations, though these are often reduced to the nonlinear coupled mode equations (NLCME). NLCME describes the evolution of the slowly varying envelope of an appropriate carrier wave. They are known to possess solitons, which may be of use in optical transmission. In this talk, we numerically demonstrate the mathematical inconsistency between NLCME and Maxwell, while still finding localized solutions for prepared data. The inconsistency can be corrected through the inclusion of infinitely many harmonics leading us to consider the extended nonlinear coupled mode equations (xNLCME). This system introduces new questions on the existence of localized states. Lastly, we consider when a spatially varying index of refraction will be sufficient to inhibit shock formation. This is joint work with D.E. Pelinovsky and M.I. Weinstein.

Valery Smyshlyaev: *Coercivity of high-frequency scattering problems *

Scattering problems are problems on the spectrum, and their solutions are understood as limited absorption limits. This implies a great difficulty in controlling inverses of the relevant operators, especially at high frequencies as required e.g. for justification of high frequency asymptotics. The latter was intensively advanced starting from 1960s using integral equations and non-standard “multiplies” techniques, and later using microlocal analysis. Recent advances in asymptotic-numerical hybrids for high-frequency scattering have posed new challenges for analysis. In particular, for Galerkin methods for boundary integral equations proving error bounds requires stronger results than bounding the inverses, namely bounding the operators’ coercivity constants. It appears that a non-trivial modification of the multipliers’ techniques of Morawetz & Co allows achieving this for certain scatterers. Joint work with Euan Spence (Bath) and Ilia Kamotski (UCL).

Euan Spence: *Wavenumber-explicit numerical analysis of high-frequency scattering problems*

In this talk I will discuss recent work on numerical analysis of high-frequency scattering problems, the novelty being to make the estimates explicit in the wavenumber. I will emphasise how this analysis both uses and extends the (non-numerical) analysis of these scattering problems. This talk will complement the talks by Simon Chandler-Wilde and Valery Smyshlyaev.

Tatiana Suslina: *Spectral approach to homogenization of periodic differential operators*

The talk is devoted to the operator-theoretic (spectral) approach to homogenization suggested by M.~Birman and T.~Suslina. In $L_2(\R^d;\C^n)$, we consider matrix elliptic second order differential operators $A=A(\x,\D)$ admitting a factorization of the form $A= b(\D)^*g(\x)b(\D)$. Here an $(m\times m)$-matrix-valued function $g(\x)$ is bounded, positive definite, and periodic with respect to some lattice $\Gamma \subset \R^d$. Next, $b(\D)=\sum_{l=1}^d b_l D_l$, where $b_l$ are constant $(m \times n)$-matrices. It is assumed that $m \ge n$ and that the symbol $b(\bxi)=\sum_{l=1}^d b_l \xi_l$ has rank $n$ for any $0 \ne \bxi \in \R^d$. We study the operator $A_\eps := A(\x/\eps,\D)$ for small $\eps>0$. It turns out that the resolvent $(A_\eps +I)^{-1}$ converges in the $L_2$-operator norm to the resolvent of the \textit{effective operator} $A^0=b(\D)^*g^0 b(\D)$. Here $g^0$ is a constant positive \textit{effective matrix}. We prove that $$ \| (A_\eps +I)^{-1} - (A^0 +I)^{-1}\|_{L_2(\R^d) \to L_2(\R^d)} \le C \eps. \eqno(1) $$ Also, we obtain more accurate approximation of the resolvent: $$ \| (A_\eps +I)^{-1} - (A^0 +I)^{-1} - \eps K(\eps)\|_{L_2(\R^d) \to L_2(\R^d)} \le C \eps^2, \eqno(2) $$ and approximation of the resolvent in the norm of operators acting from $L_2(\R^d;\C^n)$ to the Sobolev space $H^1(\R^d;\C^n)$: $$ \| (A_\eps +I)^{-1} - (A^0 +I)^{-1} - \eps K_1(\eps)\|_{L_2(\R^d) \to H^1(\R^d)} \le C \eps. \eqno(3) $$ Here $K(\eps)$ and $K_1(\eps)$ are the so called \textit{correctors}; they contain rapidly oscillating factors and so depend on $\eps$. Estimates (1)--(3) are order-sharp. The method is based on the scaling transformation, the Floquet-Bloch theory, and the analytic perturbation theory. By the scaling transformation, the problem is reduced to approximation of the operator $\eps^2 (A+\eps^2 I)^{-1}$. The operator $A$ admits expansion in the direct integral of the operators $A(\k)$ acting in $L_2(\Omega;\C^n)$ (where $\Omega$ is the cell of the lattice $\Gamma$). The operator $A(\k)$ is given by the expression $b(\D+\k)^* g(\x) b(\D+\k)$ with periodic boundary conditions. This operator family is studied by means of the analytic perturbation theory. It is possible to approximate the resolvent $(A(\k) + \eps^2 I)^{-1}$ in terms of the spectral characteristics near the bottom of the spectrum. This shows that homogenization can be treated as a \textit{spectral threshold effect}. General results are applied to specific operators of mathematical physics.

Tatiana Suslina: *Homogenization of a stationary Maxwell system with periodic coefficients*

We study homogenization of a stationary Maxwell system with periodic coefficients in $\R^3$. Assume that the dielectric permittivity $\eta(\x)$ and the magnetic permeability $\mu(\x)$ are positive definite and bounded $(3\times 3)$-matrix-valued functions, periodic with respect to some lattice. Then $\eta^\eps(\x):=\eta(\x/\eps)$ and $\mu^\eps(\x):=\mu(\x/\eps)$ oscillate rapidly for small $\eps >0$. Consider the system $$ \left\{ \begin{matrix} {\rm curl}\, (\mu^\eps)^{-1} \z_\eps - \w_\eps = -i \q, \cr {\rm curl}\, (\eta^\eps)^{-1} \w_\eps + \z_\eps = i \r, \cr {\rm div}\, \w_\eps =0, \quad {\rm div}\, \z_\eps =0. \end{matrix} \right. \eqno(1) $$ Here $\w_\eps$ and $\z_\eps$ are the electric and magnetic displacement vectors; $\q$ and $\r$ are given divergence-free vector-valued functions in $L_2(\R^3;\C^3)$. We consider also the electric and magnetic fields $\u_\eps = (\eta^\eps)^{-1} \w_\eps$, $\v_\eps = (\mu^\eps)^{-1} \z_\eps$. It is known that, as $\eps \to 0$, the solutions $\w_\eps$ and $\z_\eps$ of system (1) \textit{weakly converge} in $L_2(\R^3;\C^3)$ to the solutions $\w_0$ and $\z_0$ of the \textit{effective system} $$ \left\{ \begin{matrix} {\rm curl}\, (\mu^0)^{-1} \z_0 - \w_0 = -i \q, \cr {\rm curl}\, (\eta^0)^{-1} \w_0 + \z_0 = i \r, \cr {\rm div}\, \w_0 =0, \quad {\rm div}\, \z_0 =0. \end{matrix} \right. \eqno(2) $$ Here $\eta^0$ and $\mu^0$ are the constant \textit{effective matrices}. Also, $\u_\eps$ and $\v_\eps$ weakly converge to $\u_0=(\eta^0)^{-1} \w_0$ and $\v_0=(\mu^0)^{-1} \z_0$, respectively. We find \textit{uniform approximations} in the $L_2(\R^3)$-norm for all physical fields $\w_\eps$, $\z_\eps$, $\u_\eps$, $\v_\eps$ with sharp-order error estimates. Approximations are given in terms of the \textit{effective fields} (the solutions of system (2)) and some correction terms containing rapidly oscillating factors and weakly converging to zero. These terms can be enterpreted as the \textit{zero order correctors}. For instance, for $\w_\eps$ we have $ \| \w_\eps - \w_0 - \widetilde{\w}_\eps\|_{L_2(\R^3)} \le C \eps (\|\q\|_{L_2(\R^3)} + \|\r\|_{L_2(\R^3)}), $ where $\widetilde{\w}_\eps$ is the corresponding corrector. The method is based on the extension of the Maxwell system and on the study of the second order elliptic operator $$ L(\eta^\eps,\mu^\eps) = (\mu^\eps)^{-1/2} {\rm curl} (\eta^\eps)^{-1}{\rm curl} (\mu^\eps)^{-1/2} - (\mu^\eps)^{1/2} \nabla {\rm div} (\mu^\eps)^{1/2} $$ acting in $L_2(\R^3;\C^3)$ (and also the similar operator $L(\mu^\eps,\eta^\eps)$). Homogenization problem for $L(\eta^\eps,\mu^\eps)$ is investigated by using the operator-theoretic approach.

Dmitri Vassiliev: *First order systems of PDEs on manifolds without boundary: understanding neutrinos and photons*

In layman's terms a typical problem in this subject area is formulated as follows. Suppose that our universe has finite size but does not have a boundary. An example of such a situation would be a universe in the shape of a 3-dimensional sphere embedded in 4-dimensional Euclidean space. And imagine now that there is only one particle living in this universe, say, a massless neutrino or a photon. Then one can address a number of mathematical questions. How does the neutrino field (solution of the massless Dirac equation) or the electromagnetic field (solution of the Maxwell system) propagate as a function of time? What are the eigenvalues (energy levels) of the particle? Are there nontrivial (i.e. without obvious symmetries) special cases when the eigenvalues can be evaluated explicitly? What is the difference between the neutrino (positive eigenvalues) and the antineutrino (negative eigenvalues)? Why is the photon its own antiparticle? What is the nature of spin? Why do neutrinos propagate with the speed of light? Why are neutrinos and photons so different and, yet, so similar? The speaker will approach the study of first order systems of partial differential equations from the perspective of a spectral theorist using techniques of microlocal analysis and without involving geometry or physics. However, a fascinating feature of the subject is that this purely analytic approach inevitably leads to differential geometric constructions with a strong theoretical physics flavour.

Alexander Watson: *Dynamics of wavepackets in spatially inhomogeneous crystals by multi-scale analysis*

We study the dynamics of wavepackets in crystals whose structure is spatially inhomogeneous. We make the assumption that inhomogeneities occur over a length scale which is long compared to the lattice period so that we may treat the two scales as approximately independent. We work mainly in the setting of Schrodinger's equation, where the crystal structure is modeled by a `two-scale' potential which varies periodically on the `fast' scale and smoothly on the `slow' scale, but our methods can be applied also to Maxwell's equations where the crystal structure is modeled by a `two-scale' matrix of constitutive relations. Phenomena which result from spatial variation of the crystal structure are: the anomalous velocity of wavepackets due to Berry curvature of the Bloch spectral band (responsible for the spin Hall effect of light), and Landau-Zener-type inter-band transitions, in the presence of spectral band crossings. This is joint work with Michael Weinstein and Jianfeng Lu.

Ian Wood: *Welcome*

Ian Wood: *Wrap-up session*

Workshop: *Inverse Problems*