Asymptotics, Operators, and Functionals

Leading the discussion in 2018:

15 January Freddy Symons (Topic: Uniqueness and cloaking in a boundary-singular two-dimensional inverse​ problem) Abstract: I will present a topic from my recently finished doctoral thesis. The area is uniqueness in inverse recovery for the Schrödinger boundary-value problem in a bounded domain. The crucial new progress is in the situation of a singular Robin boundary condition, of a type first defined by Berry and Dennis in 2008, attached to this set-up. Firstly I will present a rigorous definition of this problem and derive some useful properties, then I will state and explain some new uniqueness theorems. The proof methods combine existing results on the two-dimensional inverse uniqueness problem (with Dirichlet conditions) with some new ideas involving the asymptotics of the negative eigen-values arising from the Berry--Dennis condition.

22 January Aaron Pim (Topic: Asymptotic analysis of minimisation problems from the theory of liquid crystals)

29 January Matthew Lewis (Cardiff) (Topic: On the spectral problem for the p-Laplacian operator with periodic coefficients)

5 February Igor Velčić (Zagreb) (Topic: Regularity of W^{2,2} convex manifolds and homogenization of bending theory of shells) Abstract

12 February Will Graham (Topic: Generalised Leontovich condition and surface waves in lossy dielectrics)

23 February (Friday) Kirill Cherednichenko (Topic: Two-scale series expansions for travelling wave packets in one-dimensional periodic media)

Away: [26 February Conference "Liquid Crystals, Metamaterials, Transformation Optics, Photonic Crystals, and Solar Cells", IMA, Minnesota]

5 March Yulia Ershova (Topic: A new approach to the analysis of quantum graphs and its applications to periodic homogenisation)

12 March (Joint session with BUC-XII) Serena D'Onofrio (Topic: Operator-norm convergence estimates for elliptic homogenisation problems on periodic singular structures)

Away: [22 March University of Helsinki]

Away: [4 April Mathematics and Mathematical Physics Seminar, St.Petersburg]

Away: [23 April Center for Mathematical Modeling, Universidad de Chile, Santiago

            24 April Pontificia Universidad Católica,Santiago de Chile]

30 April Alexander Kiselev (St. Petersburg) (Topic: Asymptotic analysis of Dirichlet-to-Neumann maps for multidimensional media with critical contrast and effective theories for metamaterials I)

Tuesday 8 May Yulia Meshkova (St. Petersburg) (Topic: Operator error estimates for homogenisation of periodic hyperbolic systems) Abstract

                   Andrew Comech (Texas A&M) (Topic: Limiting absorption principle and resonances)

14 May Gennady Mishuris (Aberystwyth) (Topic: Waves and fracture in discrete structures. Slepyan's method: its advantages and limitations.) Abstract

Wednesday 23 May Duvan Henao (Pontificia Universidad Católica de Chile) (Topic: Lower bound for the coalescence load in 2D neoHookean materials) Abstract: This work (joint with V. Cañulef-Aguilar) forms part of the variational analysis of cavitation (the nucleation and sudden burst of voids) in elastomers, made possible by Gent & Lindley '59, Ball '82, Muller & Spector '95, and Sivaloganathan & Spector '00. It is known (Sivaloganathan & Spector '10; H. & Serfaty '13) that the energetically most favourable cavities are spherical; nonetheless, for large external loads it is impossible to create only spherical cavities and at the same time satisfy the constraint of incompressiblity. This raises the question of determining the largest load for which incompressible deformations of an hyperelastic body still exist that open only spherical cavities, itself a lower bound for the load at which cavities start to interact and coalesce. We give a simple geometric answer in a simplified 2D setting, using the flow of Dacorogna & Moser '90.

Tuesday 29 May Kirill Cherednichenko (Topic: New developments in the operator-norm resolvent analysis of thin-plate problems)

4 June Euan Spence (Topic: Frequency‐uniform coercive boundary integral equation for acoustic scattering)

11 June Alexander Kiselev (St. Petersburg) (Topic: Asymptotic analysis of Dirichlet-to-Neumann maps for multidimensional media with critical contrast and effective theories for metamaterials II)

18 June Matthias Täufer (Dortmund) (Topic: Control cost for heat-like equations in the (de-)homogenization limit) Abstract

25 June Kirill Cherednichenko (Topic: Asymptotic analysis of Dirichlet-to-Neumann maps and dispersion relations for composite media I)

2 July Xavier Pellet (Topic: Homogenization of high-contrast Mumford-Shah energies)

9 July Kirill Cherednichenko (Topic: Asymptotic analysis of Dirichlet-to-Neumann maps and dispersion relations for composite media II)

Away: [16 July IMSE2018, University of Brighton]

Away: [3 September Workshop on Calculus of Variations and Applications, University of Zagreb]

10 September David Bird (Topic: Guidance of light in hollow-core optical fibres) Abstract: Conventional optical fibres guide light in a core region that has a higher refractive index than the surrounding cladding. Trapping and guiding light in a low refractive index core (for example, air) is much more challenging, but potential technological applications have driven a sustained research effort in this field. Cladding structures that support a photonic band gap were introduced about twenty years ago, but this has largely been superseded by a surge of recent interest in new designs of hollow-core fibres that have surprisingly simple cladding structures with no band gap, but which nevertheless support core-guided modes with a remarkably low loss. I will discuss the "anti-resonance" guiding mechanism that is believed to operate, and present analytic and numerical results on simple model structures that provide insights into what is currently understood and not understood about these fibres.

Away: [17 September BUC-XV: "Function spaces meet materials science: recent developments in spectral theory and scattering", CIMAT, Guanajuato, México]

24 September Matthias Langer (Strathclyde) (Topic: Quasi boundary triples and extensions of symmetric operators) Abstract: In this talk I will discuss the concept of quasi boundary triples, which can be used to describe extensions of symmetric operators and obtain information about the location of the spectrum of such extensions. In particular, this concept can be applied to elliptic operators with non-local and/or non-self-adjoint boundary conditions and to Schroedinger operators with potentials supported on hypersurfaces.

1 October Aaron Pim (Topic: The analysis of singularities in nematic liquid crystals and some approaches to their regularisation) Abstract: I shall start by introducing the the physical concept of a liquid crystal. I shall then discuss the Oseen-Frank model and solve it asymptotically on a domain with a small circular exclusion. I shall also talk about how the energy of the system changes with respect to the radius and position of the exclusion and what the optimal place for the exclusion is, in terms of energy minimisation. I shall then move onto the Landau-de Gennes model and discuss how it is an advancement of the Oseen-Frank approach. I shall conclude by providing the results of a numerical investigation into the behaviour of the corresponding minimisers (local and global), with respect to the Landau-de Gennes penalisation parameter.

Away: [8 October Workshop "Dynamic Phenomena in Media with Microstructure", Tel Aviv University]

15 October Harry Rainbird (Topic: Conservation laws for systems of differential equations) Abstract: I will start by discussing where conservation laws of differential equations come from and how they are derived. I will then explain how Noether’s theorem is applied to differential equations and show how conservation laws can be constructed without using the theory of Lie groups. I will conclude by deriving conservation laws for some simple examples.

22 October Julius Kaplunov (Keele) (Topic: Floquet-Bloch and Rayleigh-Lamb spectra: comparative asymptotic analysis of thin and periodic structures)

29 October Luis Silva (IIMAS-UNAM, Bath) (Topic: Oversampling, undersampling, and their applications to the approximation of functions in Paley-Wiener and de Branges spaces)

5 November Anton Souslov (Topic: Topological mechanics and acoustics) Abstract: The 2019 Breakthrough Prize in Fundamental Physics was awarded to Charles Kane and Eugene Mele ‘for new ideas about topology and symmetry in physics, leading to the prediction of a new class of materials that conduct electricity only on their surface.’ Over the last few years, the reach of these ideas about topological insulators has been extended to photonics as well as the focus of this talk, mechanics and acoustics. After connecting deformations and waves in materials to the mathematics of topology, I will discuss three different examples of topological mechanics. In the first example, time-reversal symmetry is preserved and the rigidity of an elastic network is characterized by a topological invariant called the polarization. Materials with a uniform polarization display a dramatic range of edge softnesses depending on the orientation of the polarization relative to the terminating surface. I will discuss the design of a 3D material in which topological soft modes are localized in interior regions via defects called dislocation lines. In the last two examples, time-reversal symmetry is broken by including active fluids composed of self-driven components inside the material. In that case, I will present two designs for topological states: one using periodic confinement and another using a bulk fluid without periodic order. In a periodic lattice, the geometry of confinement controls the structure of topological waves. Without periodic order, topological edge waves can arise in a fluid of self-spinning particles undergoing spontaneous active rotation. This can occur because a fluid undergoing rotation experiences a Coriolis force that breaks Galilean invariance and opens a gap at low frequency. I will discuss how all of these examples are connected via the bulk-boundary correspondence principle, which shows how a topological invariant characterizing the bulk of the material has profound consequences on the material’s surface.

12 November David Lafontaine (Topic: Scattering for non-linear waves equations in non-trapping and unstable trapping geometries) Abstract: We are concerned with non-linear wave equations in exterior domains. When solutions exist for all times, it is natural to wonder what they look like in large times. In particular, when they behave linearly in such times, we say that the solutions scatter. The intuition, supported by rigorous criterions, is that such a behavior should occur when all the energy of the wave goes away of the obstacle in infinite time. Thus, we expect scattering in geometries where all the rays of geometrical optics go to infinity: the so called non-trapping geometries, and in weakly trapping geometries as well. This is however a very open problem. We will present some methods and results using arguments inspired by Morawetz in the non-trapping case, and explain how these arguments, combined with more recent ones inspired by Kenig and Merle, should lead to the scattering in some unstable trapping geometries.

19 November Luis Silva (IIMAS-UNAM, Bath) (Topic: Subordinacy theory and the asymptotic analysis of the spectrum of the Schrödinger operator on the half line)

Away: [21 November University of Lyon and University of Saint-Étienne]

Away: [26 November Mathematical Analysis Seminar, Cardiff University]

3 December Kirill Cherednichenko (Topic: Stochastic homogenisation of high-contrast media)

10 December Serena D'Onofrio (Topic: Operator-norm homogenisation estimates for the system of Maxwell equations on periodic singular structures)

17 December Jonathan Bevan (Surrey) (Topic: Estimating critical cavitation loads in nonlinear elasticity)