the mathematical sciences have a vested interest in the maintance of a strong mathematical sciences enterprise for our nation. And because that enterprise must be healthy in order to contribute to the supply of well-trained individuals in science, technology, engineering, and mathematical (STEM) fields, it is clear that everyone should care about the vitality of the mathematical sciences.
This chapter discusses how increasing interaction with other fields has broadened the definition of the mathematical sciences. It then documents the importance of the mathematical sciences to a multiplicity of fields. In many cases, it is possible to illustrate this importance by looking at major studies by the disciplines themselves, which often list problems with a large mathematical sciences component as being among their highest priorities. Extensive examples of this are given in Appendix D.
Over the past decade or more, there has been a rapid increase in the number of ways the mathematical sciences are used and the types of mathematical ideas being applied. Because many of these growth areas are fostered by the explosion in capabilities for simulation, computation, and data analysis (itself driven by orders-of-magnitude increases in data collection), the related research and its practitioners are often assumed to fall within the umbrella of computer science. But in fact people with varied backgrounds contribute to this work. The process of simulation-based science and engineering is inherently very mathematical, demanding advances in mathematical structures that enable modeling; in algorithm development; in fundamental questions of computing; and in model validation, uncertainty quantification, analysis, and optimization. Advances in these areas are essential as computational scientists and engineers tackle greater complexity and exploit advanced computing. These mathematical science aspects demand considerable intellectual depth and are inherently interesting for the mathematical sciences.
At present, much of the work in these growth areas—for example, bioinformatics, Web-based companies, financial engineering, data analytics, computational science, and engineering—is handled primarily by people who would not necessarily be labeled “mathematical scientists.” But the mathematical science content of such work, even if it is not research, is considerable, and therefore it is critical for the mathematical sciences community to play a role, through education, research, and collaboration. People with mathematical science backgrounds per se can bring different perspectives that complement those of computer scientists and others, and the combination of talents can be very powerful.