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Appendix A Executive Summary of the 1984 Reports I. BACKGROUND The Ad Hoc Committee on Resources for the Mathematical Sciences was established in June 1981 by the National Research Council's As- sembly of Mathematical and Physical Sciences to review the health and support of mathematical research in the United States. Prelimi- nary evidence presented to the Assembly by its Office of Mathematical Sciences had suggested that in the nation's major universities external support for mathematics had lagged considerably behind correspond- ing support in other fields of science. The evidence was sufficiently dramatic that the charge to the Committee contained more emphasis on financial support than is usual for a review of the health of a scientific fielcl. Committee members with a range of scientific inter- ests and experience were chosen to ensure that this review Woolf] be carried out with a broad perspective. Early in our Committee's deliberations, we came to three important realizations: Mathematics is increasingly vital to science, technology, and society itself. Paradoxically, while mathematical applications have literally Reprinted from Renewing U.S. Mathematics: Critical Resource for the Future (Na- tional Academy Press, Washington, D.C., 1984), pp. 1-10. 77

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78 APPENDIX A exploded over the past few decades, there has been declining atten- tion to support of the seminal research which generates such benefits. . Opportunities for achievement in mathematical research are at an all-time high, but capitalizing on these will require major new programs for support of graduate students, young investigators, and faculty research time. These perceptions guided the activities of our Committee as we pur- sued our charge. II. THE MATHEMATICAL SCIENCES A. Strengths and Opportunities The period since World War II has been one of dazzling accomplish- ments in mathematics. The flourishing of the discipline has run hand- in-hand with burgeoning applications, which today permeate the theo- retical fabrics of other disciplines and constitute important parts of the intellectual tool kits of working scientists, engineers, social scien- tists, and managers. These developments were nurtured by coopera- tion between the universities and the federal government, and fueled by a national commitment to strengthening scientific research and education. The injection of federal funds into universities, combined with a pervasive sense of the importance of research, attracted num- bers of the best young minds in the country into science and mathe- matics and propelled the United States into world leadership in the mathematical sciences. The field expanded and diversified enormously during this period. Mathematical statistics matured. Operations research was born. Mathematics in engineering flowered with prediction theory, filter- ing, control, and optimization. Applied mathematics extended its reach and power, and the discipline of mathematics grew at a breath- taking pace.2 Since World War II, the impact of mathematics on technology and engineering has been more direct and more profound than in any historical period of which we are aware. When we entered the era of high technology, we entered the era of mathematical technology. Historically, the work of Wiener and Shannon in communication and information theory highlights the change. The mathematical under-

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APPENDIX A pinnings of the computer revolution, from van Neumann onwarcl, and the sophisticated mathematical design of the fuel-efficient Boeing 767 and European Airbus airfoils further exemplify the increased impact of applied mathematics. The discipline of mathematics also aclvanced rapidly and contributed to the solution of problems in other fields of science. Fundamental questions in algebra, geometry, and analysis were addressed with ever-increasing conceptual generality and abstraction; new interac- tions between parts of contemporary mathematics and physics, as in gauge field theory, remind us of the payoff of mathematics for other sciences. Indeed, in the span of little more than the past two years we have seen four Nobel Prizes awarded to U.S. scientists for largely mathematical work, much of it employing mathematical structures and tools developed over the last few decades: Chan~lrasekhar in astrophysics, Cormack in medicine (tomography), Debreu in econom- ics, and Wilson in physics. Major research opportunities for the future exist in the study of non- linear phenomena, discrete mathematics, probabilistic analysis, the mathematics of computation, the geometry of three- and four-dimen- sional manifolds, and many other areas.3 The infusion of mathematics into society will continue and accelerate, creating further opportuni- ties and increased demand for mathematical scientists. B. Prospects for the Future There are reasons to be quite concerned about the future, in spite of current vitality and past achievements. In mathematics, the country is still reaping the harvest of the investment of human and dollar re- sources made in the mid-to-late 1960s. Investments since that time have not been adequate to assure renewal of the field, to provide the seminal work supporting expanded applications, or to pursue the remarkable opportunities in prospect. During the past few years, concern about the future of mathematics has been reflected in an unprecedented probing and searching within and by the mathematical sciences community. The state of mathemat- ics, its applications, and its future promise have been assessed in: the report of the COSEPUP Research Briefing Panel on Mathe- matics presented to OSTP and NSF 79

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APPENDIX A . its supplementary report to DOD and the DOD~University Forum reports to the NSF Advisory Committee for the Mathematical Sciences by J. Glimm, on the future of mathematics, and I. Olkin and D. Moore, on statistics the G. Nemhauser/G. Dantzig report on research directions in ~ operations science the report of the NSF/DOD Pane] on Large-Scale Computing in Science and Engineering . reports of the NRC Committees on Applied and Theoretical Statistics and on the Applications of Mathematics. In all of these the theme recurs: in mathematics itself and in its capa- bilities for application there is a multitude of major opportunities, but the resources, people, and money are not available to capitalize on them. Our Committee has found the support situation in mathematics to be worse than the preliminary evidence suggested: Since the [ate 1960s, support for mathematical sciences research in the United States has declined substantially in constant dollars, and has come to be markedly out of balance with support for related scientific and technologi- cal efforts. Because of the growing reliance of these efforts on mathematics, strong action must be taken by the Administration, Congress, universities, and the mathematical sciences community to bring the support back into balance and provide for the future of the field. III. THE WEAKENING OF FEDERAL SUPPORT A. How It Happened In many ways, the history of support for mathematical research re- sembles that of other sciences: a rapid buildup of both federal and university support through the 1950s; some unsettling changes in the early-to-mid-1960s; then a slackening of federal support in the late 1960s and early 1970s, because of increased mission orientation of federal R&D and reductions in federal fellowships; and finally, more than a decade of slow growth. 80

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APPENDIX A However, mathematics faced special problems, owing to its concentra- tion at academic institutions and its dependence for federal support on two agencies: the National Science Foundation (NSF) and the Department of Defense (DOD).4 In the mid-1960s, increased focus on mission-oriented research (a change accelerated by the 1969 Mansfield Amendment) caused DOD to drop nearly all of its support of pure mathematical research and parts of basic applied work as well. Then dramatic reductions in federal fellowships beginning in 1971 removed virtually all federal support of mathematics graduate students and postdoctorals. Compensation for these two types of losses could only be made at NSF, but at NSF constant dollar support of mathematical research decreased steadily after 1967. We estimate the [ass in federal mathematical funding to have been over 33% in constant dollars in the period 1968-73 atone; it was followed by nearly a decade of zero real growth, so that by FY 1982 federal support for mathematical sciences research stood at less than two-thirds its FY 1968 [eve! in constant dollars.5 While federal support for related sciences also dipped in 1969-70, these sciences received (constant dollar) increases in NSF funding in the years 1970-72 and thereafter, as well as support from other agen- cies; mathematics did not.6 This resulted in the present imbalance between support for mathematics and related sciences: Comparisons of Federal Support in Institutions of Higher Education for Three Fields of Science, 1980 Mathematical Chemistry Physics Sclences Doctoral scientists in R&D Faculty with primary or secondary activity in R&D Faculty in R&D federally supported Approximate annual Ph.D. production Graduate research assistants federally supported Postdoctorals federally supported 9,800 7,600 3,300 1,500 3,700 2,500 9,200 6,000 3,300 800 2,900 1,200 9,100 8,400 2,300 800 200 50 SOURCES: NRC Survey of Doctoral Recipients, National Science Board Status of Science Review. B. Why It Escaped Notice Three things made it difficult for mathematicians and policy-makers 81

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APPENDIX A to quickly grasp the full extent of the weakening of support for mathe- matics: After the sharp decline of 1968-73, universities increased their own support for many things which earlier would have been carried by research grants. It was only after financial problems hit the univer- sities in the mid-1970s that the severe lack of resources became evi- clent. The growth of computer science support masked the decline in mathematics support because of the federal budget practice of carry- ing "mathematics and computer science" as a line item until 1976. The explosion of the uses of mathematics caused funding to flow into applications of known mathematical methods to other fields. These were often labelled "mathematical research" in federal support data. The category grew rapidly, masking the fact that support for fundamental research in the mathematical sciences shrank. C. Its Consequences The absence of resources to support the research enterprises in the country's major mathematical science departments is all too apparent. In most of them, the university is picking up virtually the total tab for postdoctoral support, research associates, and secretarial and operat- ing support; as a result, the amounts are very small. Graduate stu- dents are supported predominantly through teaching assistantships, and (like faculty) have been overloaded because of demands for under- graduate mathematics instruction, which have increased 60% in the last eight years. The number of established mathematical scientists with research support, already small in comparison with related fields, has decreased 15% in the last three years. Morale is declining. Prom- ising young people considering careers in mathematics are being put off. Ph.D.'s awarded to U.S. citizens declined by half over the last decade. A gap has been created between demand for faculty and supply of new Ph.D.'s. It may well widen as retirements increase in the l990s. There is the prospect of a further 12% increase in demand for doctoral mathematical scientists needed for sophisticated utilization of super- computers in academia, industry, and government. 82

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APPENDIX A The most serious consequence has been delayed. In a theoretical branch of science with a relatively secure base in the universities, sharp reduction in federal support does not leave large numbers of scientists totally unable to do their research, as might be the case in an experimental science. There is a considerable time lag before there is a marked slowing down of research output. The establishecl research- ers and the young people who were in the pipeline when reduction began carry the effort forward! for 15 or 20 years, adjusting to in- creased teaching loads, to decreased income or extra summer work, and to simply doing with fewer of most things. If the number of first- rate minds in the field is large at the onset of the funding reduction, an effort of very high quality can be sustained for quite some time. This is what has been happening in the mathematical sciences in the United States for over a decade. The situation must be corrected. IV. FUTURE SUPPORT A. The Needs of Research Mathematical Scientists The research community in the mathematical sciences is concentrated heavily at academic institutions spread throughout the country. Over 90% of productive research mathematicians are on the faculties of the nation's universities and colleges. Their numbers equal those of physics or chemistry, some 9,000-10,000. To pursue research effectively, mathematical scientists need: 1. research time 2. graduate students, postdoctorals, and young investigators of high quality 3. research associates (visiting faculty) 4. support staff (mostly secretarial) 5. computers and computer time 6. publications, travel, conferences, etc. During the fifties and sixties, these needs were effectively met by the injection of federal funds for research into universities. That spurred remarkable growth and propelled the United States into world leader- ship in the mathematical sciences. The erosion of support since the 83

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APPENDIX A late 1960s has slowed momentum and decreased the rate of influx of outstanding young people into the mathematical sciences. B. A Plan for Renewal What has been describer! makes it evident that realization of the po- tential for mathematics and its applications requires a substantial increase in extra-university support. Because there is often an indirect relation between mathematical developments and their applications, significant support from industry will not be forthcoming. Thus, the role of government is crucial. Incremental budgetary increases of the usual sort cannot deal with the severe inadequacy of support. We estimate that the federal support needed to strengthen mathematical research and graduate education is about $100 million more per year than the FY 1984 level of $78 million. Significant additional resources are needled in each of the six basic categories we identified earlier. The resources will: allow mathematical scientists to capitalize on the future oppor- tunities provided by the dramatic intellectual developments now oc- curring provide for the attraction and support of young people to help renew the field sustain the work of established researchers. As the framework for this, we have determined through analysis the elements of a program to renew U.S. mathematics. This program can be carried out through expansion of support to the $180 million level over the next five years. This National Plan for Graduate and Postdoc- toral Education in the Mathematical Sciences has these features: ~ Each of the approximately 1,000 graduate students per year who reaches the active level of research for a Ph.D. thesis would be provided with 15 months of uninterrupted research time, preceded by two preceding summers of unfettered research time. Two hundred of the 800 Ph.D.'s per year would be provided with postdoctoral positions averaging two years in duration at suit- able research centers. 84

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APPENDIX A There would be at least 40() research grants for young investi- gators (Ph.D. age three to five years). At least 2,600 of the established mathematical scientists who, with the young investigators, provide the training for the more than 5,000 total Ph.D. students and the 400 total postdoctorals would have sufficient supported research time not only to conduct their own re- search, but also to Provide the requisite training for these young people. Support would be provided for associated research needs of the investigators. We believe this plan to be consistent with the priorities set by the mathematical sciences research community through several self-stud- ies in the last few years. C. Implementation It will be up to the Administration and Congress to decide what na- tional priority to assign to these needs. We would remind them that what is at stake is the future of a field central to the country's scien- tif~c and technological effort. While the uses of mathematics in other fields have been supported, somehow the needs of fundamental mathe- matics were lost sight of for over a decade. Since there is about a 15- year delay between the entry of young people into the field and their attainment of the expected high level of performance, this decade of neglect alarms us. We urge immediate strong action, in the form of a five-year "ramping up" of federal support for the mathematical sci- ences (18% real growth annually, for five years). An effort to renew mathematics support has already begun at the National Science Foun- dation. This must be continued for five more years, with a parallel effort at the Department of Defense. This will bring support back into balance and allow for renewal, provided Department of Energy re- sources going to the mathematics of computation are significantly increased to sustain the initiative which we recommend in this field. Appropriate utilization of present and future resources requires a well-thought-out and consistent set of priorities in the expenditures of funds. Recommendations of this type have recently been set forth in the COSEPUP Mathematics Briefing Panel Report prepared for OSTP and its companion report specifically for DOD, as well as recent re- ports of the NSF Advisory Committee for the Mathematical Sciences. 85

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APPENDIX A We have built on these community efforts to systematically and con- sistently direct funding trends. The efforts must continue, to ensure the most efficient and fruitful utilization of resources. Success will also depend on action and understanding within the nation's universities. For too long, they have been silent about the fact that the leered of external support for research in their mathemati- cal science departments is markedly out of balance with the general level of support for science and engineering in the country. The disparity is reflected in the working circumstances of their mathemati- cal faculties and graduate students. As added resources become avail- able, they must be used in part to ease the strain on the mathematical science departments, which embody mathematical research in the United States. Still, the group which has the fullest agenda before it is the mathe- matical sciences research community. It is obvious to anyone that if a field gets into the sort of extreme situation we have described, the associated research community must bear much of the responsibility. We urge the mathematical scientists to greatly step up efforts to in- crease public awareness of developments in the mathematical sciences and of the importance of the broad enterprise to the nation; to set their priorities with long-term needs in mind, and to develop mechanisms for effectively presenting their needs to the universities, to the Ad- ministration and to Congress all with a renewed commitment to the unity of the mathematical sciences. NOTES Now the Commission on Physical Sciences, Mathematics, and Resources. tin addition, computer science developed from roots in mathematics and electrical engineering, then spun off to become a separate discipline. It is important in reading this report not to confuse computer science with the mathematical sciences. The rela- tionship of the fields is discussed in Appendix A [of the 1984 Report]. 3These research opportunities are discussed in detail in Chapter II [of the 1984 Report]. 4The two agencies account for 93% of support. Today, the role of the Department of Energy in supporting work at the interface of mathematics and computation is of ever-increasing importance, however. sFY 1968 was not a peak budget year for mathematical research. It is the year in the period 1966-70 for which we have the most accurate data. Chemistry and physics constant dollar budgets at NSF dipped in 1969-70, then increased by over 25% in the years 1970-72, and continued to grow until the late 1970s. 86