Asymptotics, Operators, and Functionals

Speaking in 2022:

24 January Sven Gnutzmann (Nottingham) Title: Trace formula for metric graphs with piecewise constant potentials and multi-mode graphs. Abstract: This is joint work with Uzy Smilansky. In a recent preprint (arxiv:2201.06963[quant-ph]) we generalize the scattering approach to quantum graphs to quantum graphs with with piecewise constant potentials and multiple excitation modes. The new and challenging aspect is the exact inclusion of some evanescent modes which results in non-unitary scattering. By introducing an effective reduced scattering picture which is based on unitary matrices we are then able to introduce new exact trace formulas in the more general setting. In this seminar talk I will present the main aspects of the derivation based on a very simple example: an interval with a potential step. I will then sketch the main ideas of the general approach and explain the resulting trace formula in some detail.

7 February Euan Spence (Bath) Title: Old and new bounds on solutions of the Helmholtz equation proved by integrating by parts. Abstract: A classic technique in PDE theory is that of multiplying by a carefully-chosen test function and integrating by parts. This method was famously used to prove bounds on the Helmholtz equation in the 1960s and 1970s by Cathleen Morawetz. Much-more sophisticated methods now exist for proving bounds on the Helmholtz equation, but (perhaps surprisingly) the multiplier method can still be used to prove new results. In this talk I will review the classic multiplier method, and then discuss a recent application of it to Helmholtz problems in the paper "Scattering by finely-layered obstacles: frequency-explicit bounds and homogenization", co-authored with Théeophile Chaumont-Frelet (INRIA, Nice).

14 February Miles Wheeler (Bath) Title: Some linear and non-linear problems in steady water waves. Abstract: We will discuss two recent results on steady water waves. The first, joint with Vladimir Kozlov and Evgeniy Lokharu, concerns the non-existence of so-called embedded solitary waves. The proof combines exponential asymptotics at infinity with a maximum principle argument for a novel auxiliary function. The second result, joint with Vera Hur, constructs waves with non-graphical surfaces by perturbing exact solutions with zero gravity. Here the main difficulties are linear, and our approach involves reducing a non-local Riemann–Hilbert-type problem to a complex ODE.

7 March Elisa Davoli (TU Wien) Title: Homogenization of high-contrast composites under differential constraints. Abstract: In this talk, we will derive, by means of variational techniques, a limiting description for a class of integral functionals under linear differential constraints. The functionals are designed to encode the energy of a high-contrast composite, that is, a heterogeneous material which, at a microscopic level, consists of a periodically perforated matrix whose cavities are occupied by a filling with different physical properties. Our main result provides a Γ-convergence analysis as the periodicity tends to zero, and shows that the variational limit of the functionals at stake is the sum of two contributions, one resulting from the energy stored in the matrix and the other from the energy stored in the inclusions. As a consequence of the underlying high-contrast structure, the study is faced with a lack of coercivity with respect to standard topologies which we tackle by means of two-scale convergence techniques. In order to handle the differential constraints, instead, we establish new results about the existence of potentials and of constraint-preserving extension operators for linear, k-th order, homogeneous differential operators with constant coefficients and constant rank. This is joint work with Martin Kruzik and Valerio Pagliari.

4 April Massimo Lanza de Cristoforis (Padova) Title: A singularly perturbed transmission problem for the Helmholtz equation. Abstract

9 May Petr Siegl (TU Graz) Title: Diverging eigenvalues in domain truncations of Schrödinger operators with complex potentials. Abstract

23 May Daniel Peterseim (Augsburg) Title: On the localization problem in numerical homogenization. Abstract

13 June Andrey Piatnitski (UiT Narvik and IITP Moscow) Title: Homogenization of nonlocal convolution type operators in periodic and random media. Abstract: The talk will focus on homogenization problems for nonlocal convolution type operators with integrable kernels.Assuming that the coefficients of these operators have a periodic or random statistically homogeneous microstructure we will show that, under natural coerciveness and moment conditions, the homogenization result holds, and the effective operator is differential and elliptic.

28-30 June (Mini-courses)

Luis Silva (IIMAS-UNAM) Title: A modern approach to the classical moment problem. Abstract: This course deals with the classical moment problem from the viewpoint of the theory of functional models for symmetric operators and de Branges spaces. The starting point is a closed, regular, symmetric operator, from which a de Branges space is constructed. The generalised moment problem, as well as the classical moment problem as its particular case, will be introduced. We discuss the determinate-indeterminate dichotomy and its implications for the general case. In colclusion, we will touch upon some aplications in spectral theory and the analytical sampling theory.

Miguel Ballesteros (IIMAS-UNAM) Title: Spectral and scattering theories for discrete Schroedinger operators and applications in biology. Abstract: We present the basic concepts of scattering and spectral theories for Schroedinger operators on the discrete real line. The Schroedinger operator ("Hamiltonian") is the sum of the discrete Laplacian ("free Hamiltonian") and a multiplication operator ("potential"). We study generalized eigenvectors of the free Hamiltonian, which are referred to as Free Jost Solutions (FJS). Scattering theory is presented in terms of the Jost solutions of the Hamiltonian, which are the generalized eigenvectors behaving asymptotically like FJS. Time permitting, in the last part of the course we will address a project on mathematical modelling in ecology inspired by discrete Schroedinger operators on two-dimensional finite lattices. This project involves fieldwork in the rainforest, which I will also describe together with additional mathematical tools required to analyse the model.

7 September Lyonell Boulton (Heriot-Watt) Title: Second order Sobolev embeddings and non-linear bi-Laplacians. Abstract

Away: [17 October Arbeitsgemeinschaft: Quantitative Stochastic Homogenization, Mathematisches Forschungsinstitut Oberwolfach, Germany]

Away: [24 October Basque Center for Applied Mathematics, Bilbao, Spain]

31 October Aaron Pim (Oxford) Title: A strange term coming from the boundary data. Abstract: Historically there has been much focus on limiting behaviour of the Laplace operator in a perforated domain; Murat and Cioranescu proved that, for certain scaling regimes, the sequence of solutions to the homogeneous Dirichlet problem converges to the solution to the Helmholtz equation, with Helmholtz coefficient dependent on the geometry of the perforations. In this talk, I shall derive the limiting behaviour of the solutions to Poisson's equation, in a perforated domain, subject to inhomogeneous Robin boundary conditions. The goal is to understand how the inhomogeneity of the boundary data affects the limiting problem, a question that had yet to be answered. This research has great application in the modelling of metamaterials, with the inhomogeneous boundary conditions representing intermolecular forces.

7 November Alexander Kiselev (Bath) Title: Derivation of instantaneous frequencies of tectonic plates from gravimetric data. Abstract: We present a mathematically rigorous procedure to obtain spectrograms of tectonic plate eigenmodes in an extremely low-frequency band, corresponding to oscillation periods of 15 minutes to 8 hours. The data is sourced from superconducting gravimeters (IGETS network). The motivation for this research stems from the proposal by Pavlov et al to use this spectral data in monitoring the state of active zones located along plate boundaries.

14 November Alexander Kiselev (Bath) Title: An example of a phase transition in a periodic tubular structure. Abstract