Mathematical foundations of metamaterials: homogenisation, dissipation and operator theory

In order to understand how various physical media behave under specific conditions, for example: a) how the earth surface
is deformed during an earthquake, or b) what image resolution one can achieve with a fibre-optic endoscope, mathematicians write
differential equations (DEs) and study analytical properties of their solutions, which are then interpreted to draw conclusions
about real-life objects. Part of this activity involves analysis of DEs that additionally depend on a parameter. In the above examples
this parameter could be: a) the ratio of the size of the rock-forming crystals to the thickness of layers of rock in the ground, or
b) the thickness-to-length ratio of the endoscope.
The project will develop a new approach to the analysis of solutions of parameter-dependent DEs, based on recent achievements
in the mathematical theory of ``operators'' and their spectra, which could roughly be thought of as the sets of the operator ``values''.
In general, the spectrum of an operator is found in the two-dimensional-plane of ``complex'' numbers. However, for many operators
representing DEs, the spectrum is a subset of a straight line in this plane. It turns out that considering such an operator as a member of a
wider operator family, whose spectra are not necessarily situated on the same line, brings about a lot of technical benefits, in a similar way
analysis in the complex plane helps understanding real numbers. In the last 50 years or so, many elegant mathematical results about
operators (and about DEs as their particular case) have been obtained by using this analogy. We will exploit these results in order to
improve our understanding of the behaviour of families of DEs. As a particular source of such families we will study equations representing
composites, i.e. media that have several simpler constituent parts. Many objects around us are composites, for example, wood, porous
rocks, foams, bubbly liquids, reinforced resins, polycrystal metals. Mathematical statements that we aim at will provide new information
about such real-life objects concerning, for example, their acoustic properties, or the way in which they interact with an electromagnetic
field. From the physics point of view, members of this wider operator family admit some dissipation (i.e. loss of energy) in comparison to
the original ``loss-free'' setup. The project will provide a general mathematical framework for such dissipative extensions in the case of DEs
describing composites, yielding a new analytic approach to the study of their effective properties.

The proposed research has two aspects in terms of evaluating its future impact. On the one hand, it has a strong
fundamental component: it is situated on the interface between areas of analysis that have become classical in the
course of development of mathematics in the 20th century. The related concepts and questions (the relationship between
miscroscopic and macroscopic properties of physical media around us, their scattering properties, the structure of the
mathematical objects that describe them) emerge at different levels of dissemination of science knowledge.
They have found their place in basic undergraduate and postgraduate courses and also offer many exciting problems at the
cutting edge of basic research. From this perspective our results will have impact on a fairly large community right from the
start of our work. This community will include those who pursue a professional path in areas related to engineering,
materials science, communication. For example, this could be someone involved in manufacturing non-destructive
testing devices, or someone working on internet technologies. Progress in these areas relies on good understanding
of the basics that underpin them, and this project will increase such understanding.
On the other hand, our mathematical results will have impact on the research and development of new real-world
applications in the above areas. This aspect is likely to materialise in the longer run, 7-10 years from the start of the project
onwards. However, the public will be informed about our progress towards these technological advances even before they
happen. We believe it will be of great benefit to all to be able to learn about science ``in the making'', and it will help
us adjusting our research in order to maximise impact. The project will also make an excellent case for the role of fundamental
mathematics research in shaping our life, which is of benefit to those involved in the education process.