Asymptotics, Operators, and Functionals

Speaking in 2024:

26 February Yi Sheng Lim (IIMAS-UNAM) Title: An operator-asymptotic approach to periodic homogenization for equations of linearized elasticity. Abstract

4 March Alexander Kiselev (Bath) Title: Feasible mathematical mechanisms of metamaterials. Abstract: Using two explicitly solvable models, both based on quantum graphs, I will try to reveal and discuss two rather different, yet related, mathematical mechanisms leading to the phenomenon of negative group velocity, or, in other words, to metamaterials. In the former, the system is being controlled by an external magnetic field, whereas in the latter the contrast in the material properties of two components of a composite is crucial. In both cases, the result is quantitative, with the error controlled via an order-sharp norm-resolvent estimate. I will argue that, under the hood, “negative” effects in both cases are caused by the fact that the effective description of the medium becomes non-scalar, and will demonstrate how this is related to some well-known concepts of complex analysis, and in particular, to the theory of Herglotz matrix- and operator-valued functions and the closely related area of generalized resolvents. A part of the talk will be based on joint research with K. Cherednichenko (Bath) and Yu. Ershova (Texas A&M); yet another part is joint work with K. Ryadovkin (St.Petersburg).

18 March Kirill Cherednichenko (Bath) Title: Three problems of boundary homogenisation: from asymptotic expansions to effective boundary conditions. Abstract: The question of the effect of the boundary or an interface on the behaviour of solutions to PDEs with rapidly oscillating data can take different guises: apart from the classical problem of homogenisation in a bounded spatial region, the shape of the boundary can be oscillatory or a "defect manifold" can be embedded into the microstructure. I shall give a compare-and-contrast overview of this subject area, with the aims of upgrading the existing results to operator-norm convergence and deriving higher-order boundary conditions.

25 March Phil Trinh (Bath) Title: An introduction to exponential asymptotics and its challenges. Abstract: I provide a very informal introduction to some techniques in exponential asymptotics, as applied to both old and new problems in differential equations. This is an opportunity to also share some of the challenges ahead at generalising methods to more advanced ODEs and PDEs that arise in problems in hydrodynamics and fluid mechanics.

24 June Elena Cherkaev (Utah) Title: Stieltjes representation for quasiperiodic operators Abstract: Quasiperiodic geometries are characterized by a long-range order in the absence of periodicity; they can be constructed using a cut-and-projection method that restricts or projects a periodic function in a higher dimensional space to a lower dimensional subspace cut at an irrational projection angle. The talk deals with Herglotz (Stieltjes) function representations and quasiperiodic partial differential operators constructed by the cut-and-projection method. The partial differential operators with quasiperiodic coefficients govern the behavior of the fields in composite materials with quasicrystalline structures. Such quasiperiodic composite materials possess self-similarity on large scales and lack translational symmetry. Using equations for the periodic problem in the higher dimensional space established in the homogenization process and cut and projected partial differential operators, we develop the Stieltjes/ Herglotz analytic representation of the homogenized anisotropic transport coefficients. This representation links the microgeometry of the composite material to the spectral measure of a related self-adjoint operator and determines the spectral characteristics of fields in quasicrystalline composites. Pade approximations of the Herglotz function provide bounds for the effective properties of materials with quasiperiodic structures.

23 September Stephen Shipman (Louisiana State)