If we insert into one material grains of another material, we get a composite material which may exhibit properties that are radically different from those of its constituents. Fluid with magnetic particles or electrically charged particles will enhance the effects of brake fluid and shock absorbers in the car. Over the last decade, mathematicians have made important discoveries in the study of composites. They have developed new tools in functional analysis, PDE, and numerical analysis, by which they have been able to estimate or compute the effective properties of composites. Considering the vast demand for understanding and analysis of these new materials, the mathematical achievements up to now may be viewed only as a very modest beginning.
Mathematical challenges abound even for standard materials which have been studied for some years.
For example, when a uniform elastic body is subjected to high pressure, cracks will form. Where and how the cracks initiate, how they evolve, and when they branch out into several cracks are questions that are still being researched. Mathematics in Biology Mathematical models are also emerging in the biological and medical sciences. The highly publicized genome project has important aspects which require statistics, pattern recognition, and large-scale optimization methods. Less publicized but longer-term challenges are developing in other areas of biology such as physiology.