Seminar 47, Tuesday 22 June 2021, 14:00 (London Time)

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Speaker: Elena Cherkaev (Department of Mathematics, University of Utah, USA)

Title: Quasiperiodic composites: Forward and inverse homogenization

Abstract: Quasiperiodic materials present a novel class of metamaterials that possess extraordinary mechanical, thermal, and electromagnetic properties, such as superconductivity, extremely low thermal conductivity, unusual mechanical properties, and diffraction patterns forbidden by crystallographic symmetries. The properties of such materials critically depend on the microstructure of the media. The talk will discuss the effective properties and homogenized equations governing the effective behavior of quasiperiodic composites. I will also address an inverse homogenization problem of reconstructing information about microstructural parameters from known effective properties. An approach to the inverse homogenization problem is based on the Stieltjes analytic representation which involves a spectral measure of a self-adjoint operator. The spectral measure contains all information about the geometry of the composite and can be uniquely recovered from effective measurements given in an interval of frequency. Stieltjes analytic representation also allows us to determine spectral characteristics and establish an analogy between Anderson transition in quantum transport and transition from ordered to disordered behavior in classical transport in composites without scattering or wave interference effects.

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