Mathematical Sciences: Microscopic Structure and Physical Behavior of Materials and Media

这些数学研究涉及由微观结构引起的或与微观结构相关的材料和介质的物理行为。要考虑的问题包括用平均法很好地描述物理的问题，例如估计复合材料的总体弹性和电学性质或描述气溶胶的渐近行为；以及不考虑平均法的问题，如计算衍射光栅的光学性质。有趣的是，这些问题不仅与它们的小规模结构有关，而且还与类似的方法和想法可以应用于它们的研究这一事实有关。这些问题在科学上具有重要的理论意义，在一些工程和工业应用中也是至关重要的。首席研究员最近介绍了基于函数理论的新方法来处理高对比度复合材料和衍射光栅，以及与最近在照相业中兴起的一种新型气溶胶有关的一些数学结构。这些方法的许多改进和进一步的应用不言而喻。目前的目标是扩展和改进这些方法，以便它们可以用于比现有技术所允许的更复杂的设计和预测应用中。

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.