5 FINDINGS AND RECOMMENDATIONS
Findings
The principal finding of this report is as follows:

The mathematical sciences are vital to economic competitiveness. They are a critical, generic, enabling technology.
Six points complement and elaborate on this finding.

Applications of the mathematical sciences arise in all aspects of the product cycle and across the technology base.

Applications also arise from highly diverse areas of the mathematical sciences; they depend on the vitality of research in the mathematical sciences and draw on this research as a technology base.

Computation and modeling recur as central themes. They are a primary route for technology transfer from the mathematical sciences.

Technology transfer, from the research to the industrial sector, is of critical importance for the enhancement of economic competitiveness.

In the mathematical sciences (as elsewhere), technology transfer occurs seriously below its potential. The transfer of technology will be accomplished best if the creators of technology assume the primary responsibility for its transfer. Also helpful is an atmosphere of cooperation among the industrial, governmental, and

academic sectors, an atmosphere in which the central importance of technology transfer is clearly understood by the participants in the process.

Technology transfer, computational and mathematical modeling, and education have an importance to economic competitiveness that is very large relative to the recognition given to these activities by the academic mathematical sciences community.

Manpower and technical training are also crucial for economic competitiveness. The mathematical sciences community has a significant responsibility in this area.
The ability of the mathematical sciences community to deliver shortterm results with any degree of consistency depends crucially on healthy support for its longterm development. Conversely, the more fundamental areas of the mathematical sciences are continually invigorated by interaction with applications. It is an almost universal experience that once an application succeeds, further progress depends on the development of new, and often fundamental, theories. The Board on Mathematical Sciences endorses the principle that shortterm applications and fundamental theories are virtually inseparable. Detailed studies of the health and vitality of the mathematical sciences technology base have been conducted (see, e.g., [19] and [20]). Their primary conclusion was that renewal of the U.S. mathematical sciences was at risk owing to weaknesses in manpower and training, and to a lack of balance in the funding level for the mathematical sciences. Widespread underinvestment in the mathematical sciences was documented in these reports. The recommendations were to increase the priority given to manpower and training and to eliminate the funding deficiencies.

The board endorses the conclusions and recommendations of the reports Renewing U.S. Mathematics: Critical Resource for the Future [19] and Renewing U.S. Mathematics: A Plan for the 1990s [20], to assure future manpower availability, correct funding imbalances, and preserve the vitality of fundamental research in the mathematical sciences.
Engineering and manufacturing research and design depend heavily on computational and mathematical modeling. Both the knowledge

base and the problemsolving approach of the practicing engineer have benefited increasingly from simulation and, absent such simulation, would be totally inadequate for the types of problems that are being addressed today. Increased involvement of the mathematical sciences community in all aspects of production research is a lowcost, highly effective means of improving and accelerating production. Computational and mathematical modeling have been emphasized in many studies, leading to a major policy statement by the U.S. government [21].

The board endorses the conclusions and recommendations of The Federal High Performance Computing Program [21].
The mathematical sciences community has the primary responsibility for collegiate mathematics education of engineers and scientists. They have provided significant portions of the intellectual leadership in efforts to revitalize education at the K12 levels. They have sole responsibility for graduate and professionallevel education in the mathematical sciences. Just at a time when increased use of mathematics across many disciplines has raised the requirements for mathematical reasoning on the part of students, there have been ongoing problems with U.S. students being motivated to learn mathematics. These two events have prompted serious and highlevel examination of the entire educational aspect of the mathematical sciences.

The board endorses the recommendations and conclusions of the MS 2000 project,^{4} in support of education and training in the mathematical sciences. More attention must be given to the challenge of motivating students to study mathematics.
^{4} 
MS 2000 is the shorthand reference to the Committee on the Mathematical Sciences in the Year 2000, a joint project of the Mathematical Sciences Education Board and the Board on Mathematical Sciences, National Research Council. See reports numbered (1), (2), and (12) in Appendix B. 
Recommendations
This report makes two primary recommendations:

The board recommends that the mathematical sciences community significantly increase its role in the transfer of mathematical sciences technology.

The board calls on the mathematical sciences community to put far greater emphasis on and give greater career recognition to activities connected with computational and mathematical modeling, technology transfer, and education.
The mathematical sciences can be viewed as a vital technology base for the economic process. Following from this point of view, the transfer of technology is the delivery of the product. The time for technology transfer is a lengthy process; commonly accepted are estimates that it takes on the order of one or more decades for fundamental discoveries to enter into commercial use or defense applications. In the critically important case of statistical quality control and quality improvement, the time for transfer of methods of statistical experimental design to industrial use has exceeded 70 years and is still far from complete. To ensure that technology transfer occurs, mathematical scientists, engineers, manufacturers, and business leaders must accept the task to be accomplished and plan for the result. Some successful examples of such systematic and planned efforts are documented in this report (see, e.g., §§3.7, 3.9, and 4.3).
Computational and mathematical modeling, technology transfer, and education have a critical and direct connection to economic competitiveness, a connection that should be reflected in the recognition given these activities by the mathematical sciences community. Specific actions, endorsed by this report, for carrying out its two primary recommendations include the following:

Federal and state agencies should ensure that investments in research to improve productivity include the necessary involvement and support of the mathematical sciences.

Programs in industrial mathematics, jointly supported by industry and government, should be established in our colleges and universities and should include grants for small science and for individual investigators.
The economic competitiveness of the United States can be substantially improved by increased involvement of the mathematical sciences community. Numerous agencies currently sponsor programs that have successfully encouraged interaction between academic and government laboratory researchers. This report proposes a similar effort to stimulate increased interaction between universities and industry.
The large federally funded science and technology centers already have a mandate for technology transfer. However, many industrial problems in mathematics are of moderate size, appropriate for small science and individual investigators. Thus the board recommends a program in industrial mathematics that includes small science and individual investigators, with the goal of furthering economic competitiveness through increased contributions by mathematical scientists. By uniting existing fragmented efforts, such a program would establish industrial mathematics as a subfield of the mathematical sciences. By expanding these efforts, this program would focus the creative energies of the mathematical sciences on the vital national goal of economic competitiveness.
The mathematical sciences should also take full advantage of existing programs. Many states have industrial development programs that have paid favorable attention to wellconceived initiatives for technology transfer from the mathematical sciences. Similarly, the federal government, in addition to sponsoring programs that support a base technology, sponsors research programs oriented to the solution of problems. The mathematical sciences community should compete aggressively in such programs.

The board urges that strong and meaningful consideration for hiring, retention, promotion, and tenure be given for achievements in research and education supporting industrial mathematics. The emphasis given to computational mathematics, modeling, and applications should be in balance with that accorded to other areas of mathematics.

The board calls on university administrators to encourage adoption by their mathematical sciences units of criteria and procedures that promote strengthening of the ties between universities and industry.
Our national requirements for the creation of new mathematical sciences technology, as well as for effective access to this technology, call for a balance between theory and applications. Applications should be judged not only on the basis of their relation to theory, but also for their success in solving problems important to modern industrial practice, among other societal goals. Important new mathematics and entire new areas of mathematics (mathematical programming, combinatorial optimization, queuing theory, mixed elliptichyperbolic partial differential equations) have been developed to solve practical problems and, in the process, to understand the mathematical structure that underlies them.

The board recommends that industry, government, and university cooperative research and education programs be encouraged and funded.
Cooperation among industry, government, and universities benefits all three. Such cooperation can take many forms, several of which exist to some extent at the present time. A simple and widespread form of interaction is a consulting relationship between a faculty member and an industrial firm. For example, a problem that must be solved may require specialized expertise that a firm does not have and probably cannot afford to maintain inhouse. Using an external consultant is very costeffective, and he or she usually gains intellectually by being exposed to interesting problems that are then often addressed in a more general form in the individual's subsequent research. Higher levels of interaction involve support for graduate students and the development of ideas into computer algorithms and computer codes. A common outgrowth of such a relationship, when successful, is the employment of the student, after his or her graduation, by the industrial firm. At this point, an important step in completing technology transfer to the industrial sector has taken place. A wellfocused program can attract the support of an industrywide consortium. Work that appears to be applied mathematics from an academic perspective may still be

viewed as rather basic by industrial managers. Industrial firms often have difficulty organizing cooperative research focused on the basic problems of their industry and may find it advantageous to participate in a consortium to accomplish this goal.

The board recommends development of course materials to support the teaching of modeling and of industrial applications of mathematics.
These course materials could be designed as short segments for insertion into existing courses, where they would add to the range of practical illustrations available for use in the present curricula. Such material exists today and has been developed systematically by the UMAP^{5} and CoMAP^{6} projects. The thrust of this recommendation is that curricular reform should include the expansion, modernization, and revitalization of this material, as well as its integration into the mainstream of undergraduate education. A followup goal would be to develop material to support full elective courses in industrial mathematics at the advanced undergraduate and graduate levels.
The purpose of this course material would be to narrow the gap between academic mathematics and the industrial uses of mathematics. The course material should broaden students' intellectual horizons as well as their technical training. It should increase their potential usefulness in an industrial organization and should increase their ability, as future teachers of mathematics to engineers, to motivate the use of mathematics in an industrial context.
Innovations that would distinguish such course materials from the usual offerings would be modeling, case studies, and industrial examples. Modeling is the process of converting a realworld problem into a mathematical problem. The end result is a mathematical formulation of the problem, for example, an equation to be solved. In problem solving, modeling is often the most important step. Case studies drawn from industrial practice would show how theory is used in practice and

would emphasize a problemsolving rather than a deductive approach. Each case study would be relatively short and selfcontained. In this way the breadth of applicability of mathematics and the value of the other material in students' courses could be illustrated.
An excellent way to start the teaching of modeling and industrial mathematics would be to hold summer schools for advanced graduate students and junior faculty, similar to the summer schools available in theoretical physics.

The board urges the mathematical sciences professional societies to promote intellectual activity in problem solving and modeling to strengthen the industrial use of mathematics.
Technology transfer can occur only with the full support of the people who are actively engaged in the process. The board proposes that the mathematical professional societies include technology transfer within their mission and encourage, through workshops, minisymposia, and plenary lectures, more interaction among mathematicians in industry, universities, and government laboratories. Special conferences on theoretical areas of mathematics could include the industrial perspective as well. Mathematicians in industry should be encouraged to serve in larger numbers on the editorial boards of professional journals. There is evidence that these activities are actually occurring, but the mathematical sciences community needs to pay far more attention to these issues.