1
Introduction
Mathematics and the sciences have developed hand in hand as man has sought to understand the physical universe surrounding him. Over three hundred years ago, Galileo is reported to have said, “The laws of nature are written in the language of mathematics.” Nobel laureate Eugene Wigner entitled his Courant lecture “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” (Wigner, 1960). He emphasized that mathematical concepts turn up in entirely unexpected natural applications and that the enormous usefulness of mathematics borders on the mysterious.
Many mathematical applications are not mysterious. Mathematics has developed out of attempts to explain the world around us: for example, the calculus, differential equations, the Fourier series and the Fourier transform, and statistics all arose out of scientific problems. Not only has the application of these methods advanced engineering and the sciences, but the methods have also become an integral part of mathematics. Sometimes, moreover, mathematics gained insights on its own that greatly influenced other fields: the importance of symmetry groups in coding theory, chemistry, and physics and of nonEuclidean and differential geometry in relativity and string theory.
In this report the Committee on Strengthening the Linkages Between the Sciences and the Mathematical Sciences will highlight a few examples (among many) of the effective interaction between mathematics and other fields: the beginning of modern weather prediction, the development of biostatistics, mathematical economics, and biomedical imaging. Behind these dramatic developments lies a vast array of research activities at the interface of mathematics and other disciplines.
With the advent of the computer, linking mathematics and science is even more imperative. Mathematical modeling has become a third investigative approach in many areas of science, providing an important complement to the classical investigative approaches of theory and experimentation. In a number of sciences, modeling and simulation are the only viable complement to theoretical studies, because many problems cannot be addressed experimentally.
Large volumes of quantitative data with increasingly rich structures necessitate new mathematical approaches for their analysis. The evaluation of astronomical, geophysical, agricultural, climate, weather, economic, and genomic data now involves addressing data sets with one million times more data (i.e., tera and petabyte data sets) than were previously involved in analysis. The daytoday experiments in human gene expression now require techniques utilizing concepts not of 3 dimensions but of 600 dimensions.
Performing largescale computations on complex systems and visualizing and analyzing the results pose new mathematical problems. The solutions to these problems will require new engineering science developments in computer architectures and communications and visualization systems. Future generations of computers will challenge mathematics even more.^{1}
The need for research linkages is well described in Science, Technology, and the Federal Government: National Goals for a New Era (NAS, NAE, and IOM, 1993):
Traditionally, science has been organized into specific disciplines. However, science, by its nature, is in continual flux. New disciplines emerge at the edges or intersections of existing ones. Old disciplines are transformed by new knowledge and new techniques, while new disciplines draw knowledge and techniques from the old.
Furthermore, many of the problems that scientists are now trying to solve require contributions from more than one discipline. For this interdisciplinary research to succeed, scientists must be able to extend their knowledge to new areas and work effectively as members of teams.
The performers and funders of research must allow these dynamics of science to drive its organization. They must remove barriers to emerging areas of research and encourage permeable institutional structures that allow for the flow of interdisciplinary opportunities.
To encourage new linkages between mathematics and other fields and to sustain old ones, the NRC appointed the committee and gave it the following charge:
. . . to examine mechanisms for strengthening interdisciplinary research between the sciences and mathematical sciences, with the principal efforts of the committee being to suggest what are likely to be the most effective mechanisms for collaboration, and to implement them through the Internet, widely circulated reports, and other dissemination activities, such as campus workshops convened by committee members [and to] examine implications for education in the sciences and mathematics and suggest changes in graduate training intended to reinforce efforts to strengthen the dialogue among the sciences.
The committee represents a crosssection of scientists and mathematicians from academia, national laboratories, and industry. The members' backgrounds encompass disciplines from the biomedical, life, physical, engineering, and social sciences, and from different mathematical sciences, including statistics and theoretical computer sciences. The methods used by the committee to arrive at the recommendations in this report include the following:

Examination of case studies documenting the value of crossdisciplinary work involving mathematical sciences;

Personal experience and interviews with peers, administrators, representatives of industry, federal agencies, and private foundations;

Examination of the mechanisms developed by different institutions to overcome some of the barriers to successful crossdisciplinary work;

Consideration of related studies conducted by the NRC, federal agencies, and professional societies since 1980;

Consideration of the status of a variety of efforts under way to enhance the opportunities and climate for crossdisciplinary work; and

Debate among committee members leading to a consensus on the key issues and recommendations of this report.
The committee focused on the first part of its charge. Although it agreed early in the course of its work that it could not credibly propose meaningful reforms of graduate education, it nonetheless felt that such education is an extremely important topic that needs to be addressed. Topdown dictums meant to apply to all of science and/or mathematics and proposed by a group that by its nature had thin representation from some disciplines and none from others would at best be too general to be meaningful. Each community, discipline, or department responds to a specific set of both national and local needs and must determine for itself the role it should play in its community, in other words, its mission. Each organization's mission will largely determine the steps it can take at the graduate level to enhance crossdisciplinary linkages. Accordingly, this report relies instead on an anecdotal approach to provide ideas and inspiration. Those who recognize the value of crossdisciplinary linkages can use the lessons these anecdotes teach in formulating programs that meet their local needs.
Chapter 2 of this report discusses examples of crossdisciplinary research efforts; it distills those features that seem most likely to foster successful interactions and draws conclusions from them. The chapter also lists the major barriers to crossdisciplinary research. Chapter 3 looks at previous studies relevant to crossdisciplinary interactions and compares the findings and recommendations to the committee's own experience. Chapter 4 details the committee's recommendations. The committee believes the benefits to be accrued from activities linking the mathematical sciences with the other sciences could be huge and farreaching. Successful linkages benefit both mathematics and science, and society as a whole will be greatly enriched.
Case studies of mathscience interactions are presented in Appendix A, and a report on the workshop held by the committee to examine factors enabling and inhibiting crossdisciplinary research is presented in Appendix B. Appendix C gives a brief description of earlier studies that considered how to promote mathematics/science linkages. Appendix D is a beginner's guide to federal funding opportunities for projects linking the sciences and mathematical science. Appendix E lists acronyms and abbreviations.
REFERENCES
National Academy of Sciences (NAS), National Academy of Engineering (NAE), and Institute of Medicine (IOM). Committee on Science, Engineering and Public Policy. 1993. Science, Technology, and the Federal Government: National Goals for a New Era. Washington, D.C.: National Academy Press.