These mathematical investigations concern physical behavior of materials and media which is caused by or related to a microscopic structure. The questions to be considered include problems in which the physics is well described by averaging, such as estimating the overall elastic and electrical properties of composite materials or describing the asymptotic behavior of aerosols; and problems in which averaging does not enter, such as calculating the optical properties of diffraction gratings. Interestingly, these problems are not only related by their small-scale structure but also by the fact that similar methods and ideas can be applied to their study. These questions are of theoretical importance in science, and vital in some engineering and industrial applications. The Principal Investigator has recently introduced new methods based on function theory to deal with high-contrast composites and diffraction gratings, and some mathematical constructions related to a new kind of aerosol which arose recently in the photographic industry. Many improvements and further applications of these methods suggest themselves. The present goal is to extend and improve these methods so that they can be used in more complicated design and prediction applications than those permitted by the techniques currently available.
