Competitive pressures throughout business and government, coupled with the broad expansion of computer analysis and data management, have extended the applications of mathematics to every domain of human activity. These deep challenges, which are vitally important to the nation, offer novel opportunities for research in mathematics. Without increased openness by mathematicians to problems of other disciplines, mathematics may miss opportunities to contribute to and gain from these developments. The missed opportunities would extend beyond mathematics. Excellent mathematical ideas developed in biology would probably not be helpful to financial modeling, say, because the jargon used in the two fields is very different.
Mathematics can help standardize developments in one field for use by others.
Two examples include finite element methods, developed by structural engineers, and sparse matrix methods, developed by power systems engineers and economists. In both cases, standardization and generalization of the results by mathematicians have led to applications in many other fields. The following graphics further illustrate this point. In an ideal world, mathematics has a clean flow from its core out to applications and from applications back to the core. This flow facilitates the adaptation of mathematical concepts from one field, such as physics, to economics, and vice versa.