Asymptotics, Operators, and Functionals

Speaking in 2020:

Away: [13 January Operator Theory, Analysis and Mathematical Physics, IIMAS-UNAM, México City]

Away: [20 January BUC-XVI: Recent developments in wave propagation and their application to new materials, CIMAT, Mérida]

Away: [27 January ITT11: Exploring limits in wave transmission, Bath]

3 February Feodor Borodich (Cardiff) (Topic: Extensions of the JKR contact theory to elastic punches of arbitrary shapes, 2D Membranes and thin layers) Abstract

10 February Alexander Kiselev (IIMAS-UNAM, México) (Topic: Non-selfadjoint operators, their spectra, and functional models)

Away: [17 February Quantum Mechanics of Artificial Material Structures, Sochi]

2 March Kirill Cherednichenko (Bath) (Topic: Dispersion relations for stratified media with resonant layers)

9 March Pranav Singh (Bath) (Topic: New ideas in the analysis of convergence in numerical methods based on the Magnus expansion)

16 March Kirill Cherednichenko (Bath) (Topic: Functional model for extensions of symmetric operators and applications to scattering theory) Abstract: I shall discuss the functional model for extensions of symmetric operators and its applications to the theory of wave scattering. In terms of Boris Pavlov's spectral form of this model, we find explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices. On the basis of these formulae, we are able to derive a new representation for the scattering matrix for pairs of such extensions. We use this representation to explicitly recover the coupling constants in the inverse scattering problem for a finite non-compact quantum graph with delta-type vertex conditions. This is joint work with A. V. Kiselev and L. O. Silva.

21 September Karl-Mikael Perfekt (Reading) Title: The plasmonic eigenvalue problem for corners. Abstract: We consider the plasmonic eigenvalue problem for 2D domains having a curvilinear corner, by studying the spectral theory of the Neumann--Poincaré integral operator of the boundary. The goal of the talk will be to see that the corner produces absolutely continuous spectrum of multiplicity 1, possibly with a discrete set of embedded eigenvalues, and no singular continuous spectrum. The method I will describe relies on the existence of a non-self-adjoint version of the problem in which the essential spectrum is "opened up". By applying Fredholm theory to this non-self-adjoint problem we are able to realise the spectral theorem for its natural self-adjoint formulation.

28 September David Bourne (Heriot-Watt) Title: Optimal transport and nonoptimal weather. Abstract: The semi-geostrophic equation is a simplified model of large-scale atmospheric flows. It is used by researchers at the Met Office to check simulations of more complicated weather models, and Alessio Figalli's work on the semi-geostrophic equation is listed in his 2018 Fields Medal citation. In this talk I will discuss the semi-geostrophic equation in geostrophic coordinates (SG). This is a nonlocal transport equation, where the transport velocity is computed by solving an optimal transport problem, or equivalently a Monge-Ampère equation. Using tools from semi-discrete optimal transport theory, we give a new proof of the existence of weak solutions of SG. The proof is constructive and leads to an efficient numerical method. This is joint work with Charlie Egan and Beatrice Pelloni (Heriot-Watt University and the Maxwell Institute for Mathematical Sciences) and Mark Wilkinson (Nottingham Trent University).

5 October Uzy Smilansky (Weizmann) Title: Systematics of spectral shifts in random matrix ensembles. Abstract

12 October Ivan Veselić (TU Dortmund) Title: Quantitative unique continuation estimates and resulting uncertainty relations for Schrödinger and divergence type operators. Abstract: The talk is devoted to quantitative unique continuation estimates and resulting uncertainty relations of solutions of elliptic differential equations and eigenfunctions of associated differential operators, as well as linear combinations thereof. Such results have recently been successfully applied in several fields of mathematical physics and applied analysis: control theory, spectral engineering of eigenvalues in band gaps, and Anderson localisation for random Schrödinger operators. In this talk we will focus on properties of functions in spectral subspaces of Schrödinger operators. At the end we will give some results which apply to more general elliptic second order differential equations.

19 October Christophe Prange (CNRS) Topic: About some recent developments in the regularity theory of the Navier-Stokes equations. Abstract

26 October Martin Kružík (Prague) Title: Derivation of von Kármán plate theory in the framework of three-dimensional viscoelasticity. Abstract

2 November Marta Lewicka (Pittsburgh) Title: Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models. Abstract

9 November Valeriy Slastikov (Bristol) Title: Transverse domain walls in thin ferromagnetic strips. Abstract

16 November Patrick Joly (INRIA) Title: Wave propagation in fractal trees: Theoretical and Numerical aspects. Abstract: This work is dedicated to an efficient resolution of the wave equation in fractal trees (with application to wave propagation in a human lung). Thanks to self-similarity, it is possible to avoid computing the solution at deeper levels of the tree by using transparent boundary conditions. The corresponding Dirichlet-to-Neumann operator is defined by a functional equation for its symbol in the frequency domain. In this work, we analyse an approximate transparent condition based on rational approximation of the symbol. The error and complexity analysis relies on Weyl-like estimates of eigenvalues of the weighted Laplacian and related eigenfunctions .

23 November Gerd Grubb (Copenhagen) Topic: Recent results on fractional-order operators. Abstract

30 November Christophe Prange (CNRS) Topic: Boundary layers in homogenization: quantitative asymptotic analysis and large-scale regularity.Abstract: We will focus on two related problems: (i) quantitative analysis of boundary layers in periodic homogenization and (ii) asymptotic analysis and regularity in rough bumpy domains. The talk is based on joint works with S. Armstrong (NYU), M. Higaki (Kobe University), C. Kenig (The University of Chicago), T. Kuusi (University of Helsinki) and J.C. Mourrat (NYU).

7 December Pierluigi Cesana (Kyushu) Title: Mathematical models and ideas for disclinations. Abstract: In this talk, we describe some recent results on the modelling of rotational mismatches at the level of a crystal lattice and on the modelling of self-similar-type martensitic microstructure, two phenomena which appear to be strongly interconnected. First, we introduce an energy functional defined over a triangular lattice accounting for nearest-neighbour interactions. We design special rotational-type boundary value problems on the lattice so that the minimisers necessarily exhibit non-homogeneous rotations. We are interested in the asymptotics of the energy minima and minimisers as the lattice spacing vanishes which we compute exactly with Gamma-convergence. We perform some numerical calculations for the discrete model and show that both the shape of the solutions as well as the values of the energies are in agreement with classical results for positive and negative disclinations. This is a collaboration with P. van Meurs (Kanazawa). Then, depending on time we present a probabilistic model for the description of martensitic microstructure as an avalanche process. A martensitic phase-transformation is a first-order diffusionless transition occurring in elastic crystals and characterised by an abrupt change of shape of the underlying crystal lattice. It is the basic activation mechanism for the Shape-Memory effect. Our approach to the analysis of the model is based on an associated general branching random walk process. Comparisons are reported for numerical and analytical solutions and experimental observations. This is a joint project with John M. Ball and Ben Hambly (Oxford).

14 December Maria Esteban (Ceremade) Title: Best constants in functional inequalities involving Aharonov-Bohm magnetic potentials. Abstract: In this talk I will present various results concerning functional inequalities of the Gagliardo-Niremberg and Sobolev type but with the Schrödinger operator perturbed by an Aharonov-Bohm magnetic potential. The determination of the best constants in the inequalities is very important for their application in mathematical physics. In various works, done in collaboration with J. Dolbeault, A. Laptev and M. Loss, we have studied the case of the magnetic rings in full detail. This is an 1d problem for which we obtained optimal results. In the 2d case, with the above authors plus D. Bonheure, we have proved optimal results in some intervals for the considered parameters but not for all, and some interesting questions remain open. The 3d case is mainly open. If there is time, I will also present some results concerning the Dirac operator perturbed by an Aharonov-Bohm magnetic potential and a Coulomb electrostatic one.