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) Title: Algebraic aspects of periodic tight-binding models. Abstract: Certain aspects of the spectrum of multi-layer electronic media are informed by the algebraic structure of the Fermi surface. In Bernal-stacked bi-layer graphene, these include creation of non-symmetry-induced bound defect states, localization of defects, and effects of magnetic potentials. I will discuss progress and interesting questions related to the role of commutative algebra in electronic media.

30 September Eric Hester (Bath) Title: Modelling multiphase matter: from microparticles to mega-icebergs Abstract: How can we model multiphase systems, and simulate them efficiently? I'll start with three examples from my research, boat dynamics in dead water, melting icebergs in salty oceans, and phase-separating polymers in microparticle experiments. The same patterns recur. A seemingly simple partition into PDEs and boundary conditions belies the murky interface between them. This diffuse interface in turn motivates a host of new numerical schemes. The immersed-boundary method, volume-penalty techniques, and phase-field models are a handful of examples. The bulk of my talk will discuss how I analyse these methods. Signed-distance coordinates give a straightforward vector calculus around arbitrary submanifolds, and multiple scales matched asymptotics describes the resulting solutions to arbitrary order. I'll discuss straightforward computational tools to automate the asymptotic analysis, as well as efficient spectral schemes to directly discretise the corresponding PDEs, emphasising how our mathematical tools can improve accuracy and alleviate stiffness. I'll conclude with some bigger questions for multiphase methods.