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 fibreoptic endoscope, mathematicians write
differential equations (DEs) and study analytical properties of their solutions, which are then interpreted to draw conclusions
about reallife 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 rockforming crystals to the thickness of layers of rock in the ground, or
b) the thicknesstolength ratio of the endoscope.
The project will develop a new approach to the analysis of solutions of parameterdependent 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 twodimensionalplane 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 reallife 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 ``lossfree'' 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.
