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~ Background and Introduction THE SITUATION IN 1984 In the early 1980s the Office of Mathematical Sciences of the National Research Council (NRC), chaired by William Browder, presented to the Assembly of Mathematical and Physical Sciences of the NRC star- tling evidence suggesting the deterioration of federal support for mathematical sciences2 research in the United States. Because of the critical dependence of science and technology on continued genera- tion of new mathematical methods and concepts, the Ad Hoc Commit- tee on Resources for the Mathematical Sciences was established by the NRC to review the health and support of the field. This panel of scientists, engineers, and mathematicians was asked in particular to determine whether federal and/or university support had in fact dete- riorated and, if so, how this had come about and what should be Lone about it to provide for the future health of the discipline. After three years of investigation and analysis, the ad hoc committee presented its findings and recommendations in Renewing U.S. Mathe- matics: Critical Resource for the Future (the "David Report"; National Academy Press, Washington, D.C., 1984), referred to herein as the 1984 Report. It told a story that was deeply disturbing to both practi- tioners and policymakers in science: . Federal support for mathematical sciences research had come to be markedly out of balance with support for related fields of science and engi- neering. Discrepancies in support for essential research needs were very large. The 1984 Report summarized that committee's estimate of the 11

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RENEWING U.S. MATHEMATICS number of funded senior investigators needed in the mathematical sciences as follows (p. 64~: Apply to the mathematics faculty the lowest percentage for those with federal support in other fields, 54% [from a 1980 National Science Foundation report]. One obtains 2400 as a base figure for the number of mathematicians to sup- port. The mathematical sciences faculty is 1.3 times the size of that in mathe- matics, suggesting that . . . 3100 is about right for the number of mathematical science faculty members on grants. From this, subtract 400 young investiga- tors (Ph.D. age three to five years), to obtain 2700 as an appropriate number of established investigators.... Guido Weiss of our Committee surveyed chair- men of mathematical science departments nationally, asking them to examine their faculties and judge how many researchers without support were doing research of the quality done by those with support. Extrapolation from the responses led to the estimate 2600-2900 for the total of "supported" plus "equally qualified." . . . we adopt 2600 as the threshold level for the number of estab- lished investigators to support. Goals given in the 1984 Report for other categories of support were estimates of the numbers of young people needed at each stage in the mathematical sciences pipeline in order to replenish this necessary core of 2600 senior researchers at the rate of some 100 per year. This situation had come about through a combination of (1) abrupt losses of support for mathematics in the five-year period from 1968 to 1973 caused by shifts in federal policy e.g., the Mansfield Amendment, fellowship cutbacks), and (2) steady deterioration of support over the decade 1973 to 1983, during which the growth of computer science as a discipline and the practice of lumping this field together with mathematics in aggre- gate federal research data masked the deterioration of funding for the mathematical sciences. . The infrastructure supporting the mathematical research enterprise had beer seriously weakened, especially in university mathematical sci- ences departments, which contained 90% of the mathematical research- ers, with the result that the field was in serious danger of being unable to renew itself. . This weakening had also gone largely unnoticed, for two closely related reasons: (1) the mathematical sciences community did not bring its growing problems to the attention of the broader scientific commu- nity until the early 1980s; and (2) the spectacular performance of Ameri- can mathematics, which had risen to a position of world leadership in the decades immediately following World War II, continued unabated 12

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BACKGROUND AND INTRODUCTION Physical sciences Life sciences - 32% \ 70% 52% 22% \ ~, 72%) _ 13% ~ //^ ~15%~ ~ is/? 6% - 69% i, All fields - 59% 16% ~athemadcal and computer sciences E. . nglneenng - Federally 1//~1 Other I I Non- ~ponsored r'~JI sponsored I I sponsored FIGURE 1.1 Research time in universities, November 1978 to October 1979. SOURCE: From National Science Foundation Report 81-323, reprinted from Na- tional Research Council, Renewing U.S. Mathematics: Critical Resource for the Future (Na- tional Academy Press, Washington, D.C., 1984), p. 32. throughout the 1970s, relying heavily on creative talent developed and incorporated into the field before the deterioration began to take its toll. The conclusions of the 1984 Report were supported by data such as that given in Figures 1.1 and 1.2 and in Table 1.1, which are reprinted here from that report. These data document the imbalances in support experienced at that time by the U.S. university mathematics sector. 13

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RENEWING U.S. MATHEMATICS Then, as now, the academic research communities of mathematics, chemistry, and physics were of comparable size. However, the num- bers of active mathematical scientists and trainees who were sup- ported' and thus encouraged to perform or learn to perform research, were strikingly out of balance with the numbers supported in chemis- try and in physics. The low ratios of graduate research assistants per mathematical researcher and postdoctoral researchers per mathemati- cal researcher pointed to a great many missed opportunities for better training of young people. In fact, there were so few research grants in mathematics that many qualified researchers were without support, while the level of support for graduate students and postdoctoral researchers was so low that the mathematics Ph.D. pipeline suffered both in quality and quantity. There was clear evidence that the field was not renewing itself and therefore legitimate concern that research progress would diminish in the future. The prospect of becoming a professional mathematician had begun to look less and less inviting to students. 4,500 4,000 3,500 3,000 2,500 2,000 ,500 ,000 500 O- l 14 1980 1975 ~ l _ - _ ~ _ Chemistry Physics Materials science ONon-federally supported Supported by NSF 03 Supported by other federal agencies = . ....~.., ., . _ ,,i _~', ~ . .. _ _. _ Computer Mathematics science FIGURE 1.2 Graduate students with research assistantships enrolled full-time in doc- torate-granting institutions, mathematical and physical sciences. SOURCE: From National Science Foundation Report 82-260, reprinted from Na- tional Research Council, Renewing U.S. Mathematics: Critical Resource for the Future (National Academy Press, Washington, D.C., 1984), p. 33.

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BACKGROUND AND INTRODUCTION TABLE 1.1 Postdoctorals in Graduate Institutions, 1981 Federally Non-Federally Total Supported Supported Chemistry 2870 2465 405 Physics 1450 1217 233 Mathematical Sciences. 99 56 43 This number excludes about 75 university-sponsored "research instructors" in mathematics. SOURCE: National Science Foundation; reprinted from National Research Council, Renewing U.S. Mathematics: Critical Resource for the Future (National Academy Press, Washington, D.C., 1984), p. 33. This situation developed during the 1970s, largely due to causes mentioned above. Rather than arguing for increasing Ph.D. produc- tion and hiring, mathematical sciences departments adapted to in- creased teaching responsibilities by expanding the use of graduate teaching assistants, thus freeing up some faculty time for research, but at the expense of providing quality research training for those enter- ing the field. The noteworthy productivity of mathematics researchers in 1984, as now, was at least partly due to a trio of singular events: the emer- gence of mathematics tools from World War II, the post-Sputnik alarms, and the ongoing expansion of the field as computer use widened the demand for mathematics. As the first two of those stimuli fade, the intellectual momentum they sparked is certain to run out. Just as today's mathematicians are prolific due in part to events and support levels of decades ago, tomorrow's mathematics will suffer because of the more recent underfunding and concomitant fading of career op- portunities. IMBALANCE IN SUPPORT FOR RESEARCH The interdisciplinary committee that wrote the 1984 Report quickly realized that problems resulting from insufficient research support for U.S. mathematical sciences were so severe that they threatened the entire scientific enterprise. Due to the enormous disparity in the number of people supported in the mathematical sciences vis-a-vis other sciences and engineering, the overall science and technology base was in poor balance, thus threatening its effectiveness. Revers- 15

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RENEWING U.S. MATHEMATICS ing the underfunding of the mathematical sciences was therefore necessary as much to restore a healthy balance to the overall scientific enterprise as to assure a healthy mathematics research capability. The 1984 committee concludecl that the extreme imbalance in the numbers of junior people involved in research, while partly attributable to the differing needs of laboratory and nonlaboratory fields, nonetheless indicated a distinct shortcoming in mathematical sciences researcher training. Without attempting to define "ideal" balance, that group was able to make recommendations for levels of supported research designed to restore balance. The actions of the mathematical sciences community and the federal funding agencies in response to the 1984 Report make it clear that they have accepted the conclusions that an imbalance existed and needed to be countered. Balance does not necessarily imply funding parity, nor the achieve- ment of equity between fields; rather, it implies supporting each field of science to whatever degree is required to keep it and the totality of science functioning efficiently. The 1984 committee saw the science and engineering disciplines as an ecosystem: while the components have different needs and roles, they must all function in a balanced way for the system as a whole to thrive. The present committee agrees with this analysis and believes that elimination of the imbalance documented clearly in the 1984 Report and still present, as demonstrated by Table B (Executive Summary) and Tables 2.3 and 2.4 (Chapter 2) of this report- is still the most pressing need of the mathematical sciences as a field. Since the argu- ments and underlying premises of the 1984 Report were widely ac- cepted, the present report does not reargue that case but instead refers the reader to the Executive Summary of the 1984 Report, reprinted in this report as Appendix A. THE 1984 NATIONAL PLAN The 1984 Report challenged the Administration and the Congress, the universities, and the mathematical sciences community to implement, through a decade or more of sustained effort, a national plan for re- newing the mathematical research enterprise in the United States. The seven principal elements of this 1984 National Plan for Graduate and Postdoctoral Education in the Mathematical Sciences discussed in Section IV of Appendix A called for the following: 16

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BACKGROUND AND INTRODUCTION Restoring a reasonable degree of balance between federal sup- port for mathematical sciences research and support for related fields by increasing mathematics support from $79 million to $180 million ner vear over a five-vear Period (figures in 1984 dollars): r-- -- - ~ i-- r---- ~~~o~~~ ~~~ ~~~~ ~~~~~~~~~ Restructuring the general pattern of use of resources, once they were made available to the mathematical sciences, by moving away from a pattern of small research grants supporting only principal investigators, which was especially prominent at the National Science Foundation (NSF), and toward a grants model consistently supporting graduate students, postdoctorals, and other components of the re- search infrastructure as well.3 Briefly, the 1984 National Plan called for annual summer support for 2600 senior investigators, 24-month research positions for 200 postdoctorals, 15 months plus two summers of research support for 1000 graduate students, and 400 research grants for young investigators. These goals were baser] on the premise that more young mathematical scientists need thorough training with mentors. Reducing the unusual dependency of the mathematical sciences on the NSF and the service agencies of the Department of Defense (DOD) by fostering development of new mathematics programs at other agencies, especially programs concerned with research having long-term payoffs; . Extending the lines of contact and support outward from the mathematical sciences departments to business and industry; Initiating within research universities in-depth reviews of the health of their mathematical sciences departments, focusing on the working circumstances of their faculties, the relationship of federal support to university support, and the widespread university practice of justifying allocations to mathematics departments solely on the basis of the department's instructional role; . Developing within the mathematical sciences community a greater sense of responsibility for its own fate and a greater unity of purpose and action, drawing together professional organizations from across the varied subdisciplines to act in concert in (1) presenting regularly to government and universities the research needs of the field; (2) creating a long-term, coorclinated public information effort 17

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RENEWING U.S. MATHEMATICS aimed at increasing understanding of the roles and the importance of mathematics in science, technology, and culture; and (3) accelerating the efforts of mathematical scientists to attract brilliant young people into their field; and finally Expanding the mathematical sciences community's commitment to and involvement in the revitalization of mathematics education, with special attention to the precollege level. The present committee reconsidered the 1984 National Plan and deter- mined that its goals remain valid and necessary today. Complete implementation of the 1984 National Plan would assure the continued replenishment of the field's personnel base with talented new scientists. This, in turn, would assure continued intellectual production by the discipline. This intellectual output is valuable in itself, contributes substantially to other quantitative fields, provides the environment necessary for training mathematical scientists and educators, and when explained well serves as a beacon to draw students into the mathematical sciences. THE CURRENT REPORT Purpose and Emphasis This report was prepared at the request of the NSF and the Inter- agency Committee for Extramural Mathematics Programs (ICEMAP). Specifically, this committee was charged to (1) update the 1984 Re- port, describing the infrastructure and support for U.S. mathematical sciences research; (2) assess trends and progress over the intervening five years against the recommendations of the 1984 Report; (3) briefly assess the field scientifically and identify significant opportunities for research, including cross-disciplinary collaboration; and (4 ~ make appropriate recommendations designed to ensure that U.S. mathe- matical sciences research will meet national needs in coming years. While recognizing that many critical issues face the mathematical sciencesespecially demographic and educational ones this report focuses on university research departments. These are the intellectual wellsprings of the field and the source of many teachers who set the pace for educational progress. By so focusing, this report comple- 18

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BACKGROUND AND INTRODUCTION meets other recent and ongoing efforts within the mathematical sci- ences community.4 Definition of the Mathematical Sciences The discipline known as the mathematical sciences encompasses core (or pure) and applied mathematics, plus statistics and operations re- search, and extends to highly mathematical areas of other fields such as theoretical computer science. The theoretical branches of many other fields for instance, biology, ecology, engineering, economics- merge seamlessly with the mathematical sciences. This intellectual definition does not correspond exactly to the admin- istrative definitions under which data are collected. Most data in- cluded in this report adhere to the NSF definition of mathematical sciences, which is somewhat more restrictive than the intellectual definition given above. By using these data, the committee sought to maintain continuity with the 1984 Report, being confident that trends and conclusions would not be skewed by such a small mismatch in the basis. The research progress reports included in Appendix B were deliberately chosen to span the broader definition of the field. NOTES tThe Assembly of Mathematical and Physical Sciences at the NRC subsequently evolved into the newly constituted Commission on Physical Sciences, Mathematics, and Applications. 2See below, section headed "Definition of the Mathematical Sciences." 3The specific issue addressed in this part of the 1984 National Plan was not individ- ual investigators versus group research. Rather it was grants that support only the research time of principal investigators versus grants that do that and also support graduate students, postdoctorals, and so on. In the early 1980s the average NSF re- search grant in mathematics supported two months of summer research time for a principal investigator and little else. For example, the National Research Council reports Everybody Counts: A Report to the Nation on the Future of Mathematics Education (National Academy Press, Washington, D.C., 1989) and A Challenge of Numbers: People in the Mathematical Sciences (National Academy Press, Washington, D.C., 1990). 19

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