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c) Recommendations Our recommendations aim at enabling the community of mathe- matical sciences to do all it can toward meeting national needs. Even if all these recommendations are implemented, inherent limi- tations on the rate of growth of research and teaching activities in the mathematical sciences will make it impossible to meet all these needs fully. It is therefore necessary to move as vigorously as possible toward the recommended goals. The needs, outlined in the preceding chapter and throughout our report, are an inevitable consequence of the growing complexity of our society and its increasing dependence on a variety of science- based technologies. These technologies, and the physical, biological, and behavioral sciences that support them, are becoming more and more mathematized. They utilize the ideas and techniques of core mathematics and the methods and results of physical mathematics, statistics, and other applied mathematical sciences. Most of these technologies depend, often in a crucial way, on effective use of electronic computers. Accordingly, society needs a rising level of mathematical literacy and competence at all levels, a sufficient sup- ply of mathematically trained people in many sciences and profes- sions, and a sufficient number of qualified mathematical scientists. # Added in proof: All our estimates and predictions about future numbers of PhD's were formulated before the February 1968 issuance of new Selective Service rules affecting graduate students. If these rules should result in a serious deple- tion of the graduate student population, this would, of course, intensify the predicted shortage of PhD's in the mathematical sciences. 13

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14 Summary Research in the mathematical sciences serves a twofold social function. The research workers create, develop, refine, and adapt the mathematical tools needed now and in the future for the mani- fold applications of mathematics and for the growth of the mathe- matical sciences themselves. The same people, because of the insight and stimulation acquired through active participation in research, are the leaders of the whole educational effort of the mathematical community. A thoughtful policy concerning support of research has to be responsive to both aspects. In these recommendations, as in the report as a whole, we have treated research and education together. Research apprenticeship, as carried out by postdoctoral fellows and research instructors and by graduate students writing theses, is inseparable from research. Education at the undergraduate and early graduate levels is so intimately related on a long time scale- to research, that to con- sider the continuing good of one without the other would be fool- ish. From a national point of view, these are parts of one problem, a problem that must be faced in its entirety. Our recommendations, to be stated and discussed below, fall under six main heads: Improvement in the quality of education in the mathematical sciences at the undergraduate level through expanded federal sup- port, especially at key points. There is a growing shortage calf college teachers in the mathematical sciences. ~ Faculty improvement is essential; and specific kinds of support of early graduate work can avoid losses, both of people and of opportunities for sound train- ing. (See Recommendations 14 through 17 and the report) of our Panel on Undergraduate Education.) Maintenance of momentum in research, research apprenticeship, and graduate education. This will require continuing growth in federal support of these activities. Even if this is provided, the mathematical community will riot grow fast enough to meet national needs. Accordingly, recent slackening in federal support) are a cause of deep concern to the mathematical community and should, we {eel, be a matter of general concern. (See Recommenda- tions 1 and 11 through 13.) Support for the explosive :,rowth~ of computer science, especially ~ This is documented and discussed in Chapter 7 under Quality and Distribution of Mathematical-Science Faculty (see page 127) as well as in the report) of our Panel on Undergraduate Education. -I- See reference 2, Volume XVI, Appendix C.

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Recommendations 15 as a field of research in its own right. (See Recommendations 2, 3, and 19.) Support of research and education in the applied mathematical sciences as such, and not merely in connection with either mathe- matics or the particular sciences that use mathematics. Such sup- port will not be expensive, but a failure to seize today's oppor- tunity would be costly. (See Recommendations 4, 18, 21, and 22.) Agencies and mechanisms for federal support of research and research apprenticeship in the mathematical sciences. (See Recom- mendations 5 through 10.) A continuing program of information-gathering about research and education in the mathematical sciences. (See Recommendation 20.) RESEARCH The encouragement and support that the research effort in the mathematical sciences has received during the last 20 years in the United States have made this effort eminently successful by any test. The record shows numerous and great intellectual achie~re- ments as well as substantial material and social benefits resulting directly from the endeavors thus set in motion. The United States now holds a position of leadership in all mathematical sciences. To prepare for future needs, the momentum of research should be preserved. Growth and Level Federal support of research and research apprenticeship in the mathematical sciences has developed and nurtured a process of growth limited more by natural abilities and by individual prefer- ences among fields than by available funds. Even if this momentum continues, the mathematical community will not grow fast enough to meet national needs. 1. We recommend that, as a national policy, federal support for basic research and research apprenticeship in the mathematical sciences and in each of their major subdivisions including the areas of core mathematics-continue to grow in proportion to the number of appropriately qualif ed investigators and graduate students.

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16 Summary DISCUSSION An analysis of the present level of support for research in the mathematical sciences is presented in Chapter 10. An extraor- dinary development of mathematical competence and prestige has accompanied modest expenditures: a relatively small (approxi- mately $15 million in 1966) annual investment by the National Science Foundation in academic mathematical research, a larger total investment in basic mathematical research at universities (approximately $35 million in 1966) by all federal agencies, the much larger government investment in both basic and applied research in the mathematical sciences (approximately $125 million in 1966~. Vastly larger investments (e.g., $2 billion estimated for the purchase and utilization of computers by the government in 1967) are importantly affected in their effective use by research in the mathematical sciences. The ratio of the national investment in basic research to the investment in fields of application is so small, and the benefits from basic research so large, that the goal of programs for the support of basic mathematical research, in core mathematics and in the applied mathematical sciences, should be limited only by the availability of high-quality investigators. We estimate that at present about one out of every six PhD's in the mathematical sciences is consistently active in research. Over the past five years the number of PhD's has been growing at an average rate of 18 percent per year; a rate of at least 10 percent is projected for the next five years (see Chapter 8~. We believe that throughout the next decade the increase of qualified investigators in the mathematical sciences will lag substantially behind society's need. We consider that the level and rate of growth of support was adequate a few years ago at least within the core areas. (In the other mathematical sciences there has been a shortage of funds for basic research not tied to specific applications. In computer science the support for unrestricted basic research in the software area has been quite inadequate.) At present the failure of current and pro- jected budgets to provide expansion adequate to take account of larger numbers of qualified mathematicians, advancing costs of research, and advancing overhead rates is a matter of serious con- cern. The above recommendation can be thought of as urging the * For 1966 the $35 million spent in basic academic mathematical research was 2.4 percent of the total federal research expenditures in that year.

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Recommendations 17 maintenance of at least a natural rate of growth throughout the mathematical sciences. The Special Situation of Computer Science National needs and developments in contemporary technology require the stimulation of higher than natural rates of progress in some areas ot the mathematical sciences. Of particular urgency at this time are the requirements of computer science. 2. We recommend that at the national level special priority be given to support of the expansion of research and graduate study in computer science. Appropriate actions would include: special support for developing and updating courses, support for research during the academic year when needed, grants to departments to cover costs of computer usage in research, special attention to needs for space, and expansion of numbers of research assistantships and traineeships to stretch the capacity of all departments of high quality. DISCUSSION The proliferation of high-speed electronic data-process- ing equipment, combined with the rapidly expanding art of its use, constitutes one of the newest and most dynamic forces affecting the mathematical sciences. There is a critical shortage of research leaders in computer science, and urgent steps are required to over- come it as fast as possible. Electronic computers have evolved so rapidly that in many areas they have become an integral part of operations before there has been time for the research needed to determine the best, or even fairly good, ways of using them. The vast expenditures for computing in the operations of the federal government alone mean that even modest improvements achievable from research at relatively low cost will almost surely pay off in large cost reductions and improvements in service within a very few ... . . years. Despite the large sums available to finance computing service on campus, the money available for research in computer science has been, with the exception of a few spectacular projects, seriously inadequate. ~# The role of computers in higher education across the board has recently been studied in the Pierce reports of the President's Sci

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18 Summary ence Advisory Committee. The detailed impact of projections made by this and similar studies on the requirements for research and education in computer science itself is, however, most inadequately understood. This matter obviously demands urgent attention. 3. We recommend a thorough study of the implications for research and advanced education in computer science of an ade- quate implementation of the recommend ations of the Pierce report.3 Basic Research in Applied Mathematics Research of sufficiently high quality is wisely supported as research for its own sake. In at least two areas of applied mathematics- physical mathematics (sometimes called classical applied mathe- matics) and the mathematics underlying operations research and modern economics there are growing communities of mathemati- cal scientists whose efforts meet this criterion. Their basic research should thus be supported partly for its own sake and as a field in its own right, rather than solely because of its immediately per- ceived contributions to particular fields of application. 4. We recommend that f ed eral su pport f or research and research apprenticeship in high-quality basic applied mathematics be given on the basis of intellectual worth (recognizing, of course, the over- all importance of progress in applied mathematics to many sciences). DISCUSSION There is a significant distinction between such applied mathematical sciences as physical mathematics and the mathe- matics underlying operations research and such partly mathemati- cal sciences as statistics and computer science. Mathematics is applied in all these disciplines, as it is in so many others. The partly mathematical sciences gain their identity and their only partly mathematical nature-from the existence of problems, not initially mathematical, that run broadly through most fields of science and technology. The applied mathematical sciences, like physical mathe- matics and operations research, have had the en ect of uniting mathematics with specific areas of application- an effect that will not disappear. However, they have not developed a sufficiently strong identity of their own. Much can be gained from the develop

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Recommendations 19 ment of such identities, the foundations for which already exist in intellectually worthwhile research of high quality. Recent work, characterized by the evolution of ever more appropriate mathe- matical models, together with the evolution of mathematical tech- niques,~ display clearly the comprehensive nature of the discipline of applied mathematics. Support of such work for its own sake is now clearly justified and need not interfere with other work in these areas supported because of its more immediate usefulness. Sources of Support The major federal support of research and higher education in the mathematical sciences comes from a variety of agencies, the most important being the National Science Foundation and certain of the mission-oriented agencies, as indicated in more detail in Chapters 10 and 11. We believe that activities in the mathematical sciences will continue to be relevant to the tasks of all these agencies, and that all of them should continue to share in the future support of these sciences. 5. We recommend a level of growth that will enable the National Science Foundation to continue effectively in its central role in support of basic research and higher education in the mathematical sciences. DISCUSSION The National Science Foundation is the agency of the federal government whose direct mission is the promotion and support of basic research and education in the sciences. In carrying out this mission it has evolved a versatile and highly effective array of programs. A vital ingredient in the success of these programs has been the system of peer-evaluation for ensuring high quality in the research and educational activities supported. The National Science Foundation has been a very important source of support for the mathematical sciences, furnishing in recent years close to one third of the total federal support of mathe- matical activities in basic research and approximately half of the federal support specifically allotted to mathematical activities in higher education. As the sum total of our discussion in various parts # In studies, for example, of the dynamics of the ocean, the structure of galaxies, the physics of low-density gases, and the optimal use of water-supply systems.

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20 Summary of the present report indicates, we feel that the whole range of National Science Foundation programs in the mathematical sci- ences has been valuable and well conceived. We urge a natural rate of growth in most of these programs and a more rapid growth in several. We also suggest a few new programs. AL ~ Several mission-oriented agencies of the government rely on ad- vanced mathematical techniques in accomplishing their tasks. For such agencies it is important to maintain close contact with con- temporary activity and competence in the mathematical sciences. 6. We recommend that mission-oriented agencies that expect to derive significant benefits from the use of mathematical sciences continue and expand their partnership with the community of mathematicians by: (:aJ participating in the sponsorship, not only of research that promises predictable returns in applications, but also of basic inves- tigations that enlarge the intellectual foundations of the field, and (bJ evolving organized plans for bringing their unsolved scientific problems to the attention of the mathematical-sciences community and for provid ing the opportunity to qualified research mathe- maticians to further, at times and in the depth of their choosing,, the mathematization of major realms of scientific and technical eff ort of national concern. DISCUSSION The past record of sponsorship of mathematics research by the mission-oriented agencies shows its effectiveness. The pos- sibility of rapid adaptation and ollow-through in connection with newly developed techniques of mathematical analysis has greatly assisted the mission-oriented agencies; and familiarity with some of the difficult scientific and technological obstacles that must be overcome has been instrumental in stimulating fruitful funda- mental research endeavors. Time may be lost and effort wasted in the achievement of tech- nology-dependent objectives, and delays may occur in the progress of mathematical techniques for the advancement of other sciences, if we fail to develop and to maintain continuing channels of com- munication between the mathematicians and the heavy users of mathematical sciences. Thus we believe that it is vital to continue,

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Recommendations 21 and to strengthen where it is already established this pattern of cooperation between mission-oriented agencies and the mathemati- cal community, and to extend it further as our national commit- ments venture into areas in which the role played by science and technology becomes ever more intricate. Areas of possible expansion include housing and urban development, transportation, manage- ment of this country's natural resources, and the guidance of its educational efforts. Of the federally sponsored research in the mathematical sciences, reported as basic by the supporting agencies, 60 percent is con- ducted in academic institutions. Of this fraction, a little less than one half is supported by the National Science Foundation. Other agencies, mainly the Department of Defense, the Atomic Energy Commission, and the National Aeronautics and Space Administra- tion, account for the remainder of the university research and for the bulk of the basic work conducted under government sponsorship at nonacademic establishments, amounting to just under three fourths of the total federal commitment to basic research in the mathe mat~cal sciences. 7. We recommend that the Department of Defense, the Atomic Energy Commission, the National Aeronautics and Space Admin- istration, and the National Institutes of Health continue programs for the sponsorship of basic research in the mathematical sciences, and especially in physical and engineering mathematics, statistics, computer science, and operations research and management science. This support should increase at rates that will enable these agencies to share responsibly in maintaining at least the natural growth rate and that provide for higher rates of expansion in areas with long-term relevance to the agency's mission. DISCUSSION At issue here for the most part is basic research, con- ducted at universities, government laboratories, and industrial establishments in the areas of applied mathematics and statistics, computer science, and operations research and management science. The Department of Defense has the longest history of cooperation with the community of mathematics, and there can be no question that over the years defense technology has benefited in many and

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22 Summary vital ways therefrom (e.g., in computer and communications tech- nology, quality control arid life testing, and programming of mas- sive supply operations). The agencies in question bear an important share in the steward- ship of one of this country's vital resources its research potential in the mathematical sciences. Moreover, the multiple character of the support has itself contributed greatly to the vigorous state of American mathematics. Forms of Supports Since World War II, the overwhelming bulk of federal support of research in mathematical sciences has beers support of individual or group projects. Two decades of experience have demonstrated the effectiveness of the project system. S. We recommend that federal agencies sponsoring basic academic research in the mathematical sciences continue to use the project system as the primary mechanism for support. DISCUSSION The project system has proved compatible with almost every pattern of departmental university organization; it has also proved flexible in adjustment to the tasks required and eRective in linking the problems of sponsoring agencies with relevant contem- porary mathematical research. A more detailed discussion and evaluation of the project system is given in Chapter 10. ~ik ~ We believe, however, that project grants and contracts are not always best suited for fulfilling several necessary tasks: assisting departments of quality and promise to become truly outstanding, developing new centers of leadership in the applied mathematical sciences, and providing centers of research and graduate education in geographical regions so far deprived of them. # This committee is aware that authoritative voices have proposed very radical revisions of the whole federal system for supporting academic research and uni- versity education, abandoning the present forms of support in favor of direct federal subsidy to universities. We feel that a discussion of this problem lies out- side our competence. The fact that we do not mention these possibilities in our report, however, should not be taken as evidence that we oppose them. It is self- evident that in any thorough discussion of such radical changes the special prob- lems of the mathematical sciences would have to be taken into account.

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Recommendations 23 9. We recommend increased exploitation of departmental grants to supplement the traditional project grants and the recently estab- lished university science development awards. DISCUSSION The departmental Cant provides a mechanism for support of the mathematical sciences with desirable flexibility in meeting the diverse needs and capacities for academic mathematical research. The university science development grants have proved useful for the support of research and graduate educational activ- ities that correlate well with those of other departments. In the mathematical sciences these have usually tended to be better suited to the needs of computing, statistics, and traditional applied mathe- matics than to those of core mathematics. The departmental grants should extend such opportunities to core mathematics, as well as permitting other areas of the mathematical sciences to develop their identities and programs. Peer Evaluation Federal support of basic academic research has relied on judgment by qualified investigators in the evaluation of proposals for research grants as well as in processing applications for fellowships. This practice is doubtless one reason why such support has worked as well as it has. 10. We recommend that the principle of peer judgment continue to be used with respect to all forms of support for basic academic research and research education. An essential part of peer judgment should be representation of specialized areas, especially applied mathematical sciences, on evaluation groups that are likely to deal with applications from these areas. DISCUSSION The purpose of such representation is to ensure that proposals from an area of specialization receive a fair evaluation by people familiar with the aims and standards of the field. (The range of judgment required in peer evaluation is indicated by the # We are aware of the twin dangers of "gimmickitis" and "hit and run" financing so eloquently described by George Pake t"Basic Research and Financial Crisis in the Universities," Science, 157, 517-520 (Aug. 4, 1967) i. \Ve hope that pro- grams of departmental grants (and indeed all grants) will be so administered as to minimize these dangers.

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32 Summary for the U.S. Government. As its needs for mathematical techniques have grown, however, the government has come increasingly to rely on outside contractors rather than on the expansion of mathemati- cal resources within its own research and development establish- ments. 21. We recommend that a broadly qualified ad hoc group be con- vened to study the desirability and feasibility of creating research units within one or a few of the government's key research and development establishments whose mission would be the develop- ment and imaginative application of mathematical-science results and techniques in contexts pertinent to federal efforts. New National Goals Growing up alongside the national programs that call for physics and heavy engineering, there are now programs that, with increasing frequency, receive at least equal priority and that are designed to ameliorate the lives of individuals or to develop beneficial social organizations, and hence involve the problems of environment and people. The use of mathematical techniques in these contexts can be very substantial and must be expected to depend largely on the applications that are made of electronic data-processing facilities as well as the constructions of systems analysis, operations research, and the management sciences. 22. We recommend that recently established agencies of the fed- eral government, whose missions strongly depend on science and technology, cooperate with the National Science Foundation in a thorough review of those activities in the mathematical sciences that deserve attention in the context of their missions. IMPLICATIONS FOR PROGRAM AND RESOURCES PLANNING Most of the recommendations of the previous sections make budget- ing and policy demands on the conduct of the federally sponsored research and advanced education programs in the mathematical sciences. Relevant information and data have been developed

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Recommendations 33 throughout the report. For the convenience of program planners and managers, the key elements are summarized here in direct juxtaposition with our key recommendations. The baseline for our projections is constituted by the allocations of fiscal year 1966, the last complete year for which data became available while the present survey was in progress. For that year, the agencies of the federal government reported research and devel- opment obligations totaling $125 million for research in the mathe- matical sciences. Of this amount a conservative estimate identifies at least $45 million as having been spent on basic research. The remainder of $80 million has served to fund applied research directly supporting the missions of the sponsoring agencies, as well as a few major projects principally under the aegis of the Ad- vanced Research Projects Agency of the Department of Defense- in which it proved impossible to separate basic from applied com- ponents. No long-term rates of growth have been projected for this remainder item; its future level will be established by the needs and opportunities as they are identified by the individual agencies. Returning to the allocation of $45 million to basic research in 1966, we have estimated that about $35 million of this went for the support of academic research, leaving a remainder of roughly $10 million for the conduct of basic mathematical-sciences research under programs administered at the local level by the major fed- erally sponsored research and development centers. Again, no attempt has been made to project the growth of this fraction over the next few years. Within the $35 million for academic research, we have identified slightly over $90 million as allocated to project support, the remainder accounting for other than project-type sup- port, such as portions of interdisciplinary efforts, departmental and institutional grants, and conference activities. In addition to the $125 million in research and development obligations in base year 1966, approximately $10 million of other federal funds were allocated in that year to the support of graduate study in the mathematical sciences and approximately $5 million to programs of further faculty training and various activities in undergraduate educational improvement in these fields. Our recom- mendations dealing with levels and forms of support are addressed principally to the budget for academic research ($35 million in 1966) and for these closely related items in higher education ($15 million in 1966~.

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34 The Staging of the Colleges; Research and Research Education Summary The current level and rate of growth of the demand for profes- sional competence in the mathematical sciences to conduct research and research education has been estimated in connection with: (a) Projected requirements for the teaching of mathematical sciences at the college level; (b) The funding of applied research in the mathematical sciences by the mission-oriented agencies of the federal government; (c) Manpower demands in certain of the applied mathematical sciences, especially computer science and those intervening in the field of operations research. Conservative estimates anticipate a need of some 8,000 additional full-time college faculty members by the academic year 1970-1971 over the 10,750 in service in 1965-1966. According to these estimates only about 41 percent of these new faculty members will have doctorates, even on the optimistic assumption that in the interven- ing five years no less than 70 percent of the new PhD's will be teach- ing mathematical sciences at universities and four-year colleges. This would represent a lowering of quality in the sense that cur- rently 46 percent of those teaching the mathematical sciences in universities and four-year colleges have PhD degrees in the mathe- matical sciences. (Another 6 percent have PhD degrees in other fields, primarily education.) Government support of applied research in the mathematical sciences has grown at an average rate of 51 percent per year during the period 1960 to 1966, largely because of rapidly increasing commitments In computer research and development and in oper- ations research. Having now reached a level of about $80 million per year, this support shows a slackening growth rate, which is, however, still running well ahead of the annual growth rate for support of basic research. Correspondingly, growing manpower de- mands in the applied mathematical sciences indicate that current shortages will become even more severe in the next few years. Thus, shortages of mathematical manpower, for both teaching and research, are increasing. These are occurring in spite of a rela- tively high rate of growth in PhD production during recent years averaging 18 percent per year over the period 1960-1965 and in .. . .

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Recommendations 35 spite of the fact that not many capable graduate students appear to have been lost because of lack of support during that period. As a consequence, we have had to conclude that even optimal planning and management of sponsored programs in research and professional education will not build up the mathematical-sciences community fast enough to meet national needs. It is therefore im- portant that economic deterrents not retard the replenishment of the group on which the responsibility for innovation, research training, and college education devolves. Hence, our recommenda- tions (Recommendations 1, 8, 9, and 11) call for an expansion in the support of basic academic research and research apprenticeship in the mathematical sciences at least at a rate that will not inter- pose economic barriers to the achievement of competence in re- search and research education. In the absence of much of the necessary information, only rela- tively crude planning factors can be established. Taking Recom- mendations 1 and 11 together because they involve common ele . ~ meets, we estimate that: (a) In the $20 million worth of project research, a total of approximately 920 tenure research investigators (TRI) participated, so that the average expenditure of such funds (investigators' salaries, visitors, research associates and assistants, secretarial sup- port, publication, overhead) amounted to around $22,000 per TRI. (b) One out of six PhD's in the mathematical sciences ends up doing research found worthy of support. Allowing for the elapse of approximately five years between receipt of the doctorate and the acquisition of tenure, the figures on earned doctorates provide an estimate of approximately 1,400 for the group of Trues by 1971. (c) The planning factor of $22,000 per TRY will MOW, because of (i) the increasing cost of research, which is certainly no less than 4 percent per year, (ii) the cost of growing requirements for machine computing, which cannot be reliably estimated at present but, in particular instances, reach magnitudes that dwarf all other costs, and, finally, (iii) the increase in the number of research assistants per TR! called for by Recommendation 11. We arrive, conservatively, at a minimum of $29,000 per All by 1971.~ (d) Comparable growth should be provided in the nonproject forms of academic research support, under the assumption that no ~ In computer science itself the corresponding average annual cost is estimated in the section on Computer Science (page 205) to be approximately $60,000 per TRI.

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36 Summary radical change is made in the balance with project support (Recom- mendation 9~. (e) Graduate-student enrollment will grow from its 1966 level of 9,400 to approximately 18,400 by 1971. With 1,300 covered by re- search assistantships See (b) above], a total of 4,800 will have to be accommodated by fellowship and traineeship programs if one third of them are to be given research apprenticeship support, in accord- ance with Recommendation 11. The corresponding figure for 1966 is 1,834, and the at least 4 percent per year increase in cost of re- search also affects the rate per research apprentice in these programs. Total costs can now be projected for 1971 and converted into an annual percent rate of growth for the period 1966-1971. Specifically, there would be 566 million for academic research, of which at least $38 million would be in the form of project research, and another $30 million in fellowship and traineeship support. The equivalent annual growth rates turn out to be 14 percent for research, 24 percent for research apprenticeship' and 16 percent overall. Of special significance and thus to be emphasized is the relatively greater increase in support for research apprenticeship than for research, in order to prepare for meeting national needs in the mid-1970's. # There is another, simpler, kind of calculation that also leads to this over-all annual growth rate for the period 1966-1971. The Westheimer reports on chem- istry (page 166) tied PhD production to total federal obligations for basic re- search in the field, leading to a figure for "Federal support cost per PhD produced." For 1962, the year reported on in the Westheimer report, the mathe- matical sciences had the lowest such cost, namely approximately $22.6 million _ $55 OOO/PhD 410 PhD's ' For 1966, we find that it was approximately $50 million $65 OOO/PhD indicating that from 1962 to 1966 the cost per PhD had increased by approxi- mately 4 percent per year. Supposing it to continue to increase at this rate, this cost will be approximately $80,000/PhD by 1971. With about 1,300 PhD's pro- jected to be produced in 1971, this gives, for federal support of basic research in 1971, $80,000/PhD X 1,300 PhD's = $104 million. This is slightly more than double the 1966 figure and may be computed to call for growth at an average annual rate of just about 16 percent.

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Recommendations 37 The final sense of our recommendations, however, does not lie in these particular figures but in identifying the factors that must serve to determine them. As our knowledge regarding the latter improves, both budget projections and growth rates can be adjusted accordingly. Hence, our recommendations (especially Recommenda- tions 3 and 20) call for continuing efforts of investigation and analysis to develop suitable planning factors and information about research and education in the mathematical sciences so that evolving needs and trends can be appraised more reliably. As the demand for mathematical-science instructors with PhD education will continue to outrun supply in the shorter time frame in any case, their number must be increased by using opportunities that have so far been neglected for one reason or another. Par- ticular programs toward such an end are the subject of Recom- mendations 13 and 14. For the 50 postdoctoral teaching fellowships of Recommendation 12, the cost of supplementary stipends (about 58-9 thousand each) and administration should not exceed $600 thousand per year. The special part-time graduate fellowships for women under Recommendation 13 would constitute about 10 per- cent of all available full-time graduate fellowships in the mathemati- cal sciences, i.e., about 100 initially and perhaps 200 five years hence. Cost, including administration, would range correspondingly from $300 thousand to $600 thousand. In addition to the need for a basic policy that maintains the present momentum of research and research apprenticeship in the mathematical sciences across the board, our recommendations recog- nize certain critical areas in which more than ordinary efforts are needed if the mathematical-sciences community is to render the required services in today's social fabric. These are Recommen- dations 2, 3, and 4 relating to the support of research and research education in computer science and in the applied mathematical sciences as such. Planning factors to gauge the development of research and re- search education in computer science are provided by the size, the cost, and the growth rate of this country's computer establishment. It is clear that it will be some time before the schools and univer- sities will have caught up even approximately with the require- ments that this is generating. Not the least among these is the grow- ing use of computers in the educational process itself, the full-scale expansion of which has been recommended by the President's Sci- ence Advisory Committee in the Pierce report.3 At the same time,

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38 Summary the availability of support for computer science as a field of research in its own right has been minuscule in comparison to its economic and intellectual importance. Under the conditions, there is no choice except to stretch the capacity of high-quality departments as far as this is possible with resources in faculty, space, and computer facilities, potentially available to them, in order to engage in original research and, especially, to create opportunities for research . . apprenticeship. The appropriate level of support for such programs is notoriously difficult to project, one of the more recent proposals suggesting that it be made a flat percentage (e.g., ~ percent) of the $415 million esti- mated by the Pierce report as being required by 1971 to cover the educational use of computers in universities. There are at present a dozen or more departments of computer science that would qual- ify for support under Recommendation 2; five years from now, their number may well have tripled. This suggests a program, starting at $6 million and stabilizing at $15-20 million no later than five years hence. If, however, the doubling time of the number of eligible departments should be two years, rather than the three years esti- mated above, these projections would represent gross underestimates for the latter years. In contrast, the support of a few research and research training programs of exceptional quality in the applied mathematical sci- ences for their own sake will be of low cost. There are today prob- ably no more than half a dozen universities that would qualify for such a program. Program cost would therefore amount to an initial $500,000, growing to $1.~-2 million per year in the course of the next three or four years and stabilizing at that point. Undergraduate anct Early Graduate Education Of the recommendations in these critical areas, three relate to {acuity improvement in undergraduate colleges. Two of these (Recom- mendations 16 and 17) call for appropriately directed expansion of existing programs. Doubling in the course of five years the number of available National Science Foundation Science Faculty Fellow- ships in the mathematical sciences alone would increase the program by only about $1.5 million a year. Planning in this connection, how- ever, will have to take into consideration the Science Faculty Fel- lowship program as a whole in the establishment of proper balances. With respect to summer institutes, as proposed in Recommendation

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Recommendations 39 17, the underlying current estimates are that, of the roughly 10,000 college teachers in the mathematical sciences, approximately 10 percent should have the opportunity each year of participating in summer institutes. Effective training groups run about 30 students each, which would lead to some 35 institutes per summer, tripling the currently supported number. Costs per institute will range from $70,000 to $100,000, so that initial program totals would lie at around $3 million, rising in future years. Recommendations 14 and 15 propose certain forms of student assistance. Neither of these programs is likely to be very expensive. The support of graduate students in colleges and universities, offer- ing no PhD degree but a high-quality master's degree in the mathe- matical sciences, is meant to be experimental and therefore limited to perhaps 15 to 20 typical such schools. The program of special fellowships or forgivable loans to promising students, emerging from colleges with inadequate departments in the mathematical sciences, would be gauged to make 200 awards per year at a total program cost of $1 million. The development, finally, of undergraduate programs in the applied mathematical sciences is largely an internal decision of university administrations, which might be expedited only periph- erally by the possibility of federal support. If the primary resources exist to offer such programs be they in comprehensive applied mathematics, statistics, computer science, or operations research- the difference of providing the necessary management, housekeep- ing, once and classroom space, and other facilities is more a matter of support of university infrastructure as a whole than of particular fields. The moral pressure of concerned interest, backed up as neces- sary by an occasional subsidy of the right sort, is all that is needed to implement Recommendation 19. Applied Research Applied research in the mathematical sciences, supported by mis- sion-oriented agencies in the context of and for immediate utiliza- tion in specific applications of significance to the sponsoring agency, is of very recent origin. As late as 1960, it amounted to no more than $6.6 million out of total research obligation of $23.6 million- all of 28 percent. By 1966, it had increased to certainly not less than $62.4 million and probably more nearly S78.4 million, amounting to between 53 percent and 63 percent of the total for mathematical

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40 Summary sciences research. Recommendations 6 and 7 essentially call for agencies that traditionally sponsored applied research in the mathematical sciences to continue to do so at levels and in direc- tions in which it has been found useful, and for newly established agencies to contemplate comparable participation in the develop- ment, adaptation, and use of mathematical techniques relevant to their problems. No long-term rates of growth are projected, and levels are expected to be set by needs and opportunities as they are identified. The yearly rate of growth of this portion of the research budget has been decreasing, but the latest figures place it still above the growth rate for academic research in comparable periods. Sources and Forms Continued participation in the support of academic research by the National Science Foundation and by other agencies is called for by Recommendations 5 and 6. No quantitative apportionment of relative shares is proposed, provided the levels of Recommenda- tions 11 and 12 are met. Recommendations 8 and 9 identify the relative functions of project funds and other forms of support for academic research. Again trends rather than absolute quantities are stressed. It is pro- posed that project support remain pre-eminent and, among the various forms of broader support, areas as well as departmental grants be given increased utilization as against interdepartmental and institution-wide grants for the development of quality in the mathematical sciences. Cautionary Remarks The mathematical sciences, perhaps more than any other major discipline of modern science, play a pivotal role in a wide variety of contexts, both in opportunities for application and in require- ments of education. The programs recommended in our report reflect this diversity. Each of these programs is designed to meet needs that in some contexts have emerged as urgent, but no single priority scale applies across the board; hence their adoption as a whole or in part, down to some given level of priority, cannot be made the subject of a single action at the national level. Instead, we expect our recommendations to be implemented, either indi

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Recommendations 41 vidually or jointly, as permitted by the internal priorities of the various agencies involved and by the national emphasis given to the goals of which our objectives are a part. Many of our recom- mended programs may, of course, be completely merged into other similar but broader programs. The same considerations apply to our crude projections of costs, with their wide differences in reliability, amounts involved, and periods spanned. To combine them all into one grand balance sheet would not be very useful. Each of the contexts for implemen- tation of these programs has its own scale of benefits relative to which the programs must be weighed. Since a number of these programs will be funded by agencies that share in their implementation as parts of other, broader activ- ities, often of a more applied nature, a good deal of support calf the mathematical sciences may be termed "implicit." In particular, this implies that the federal government must be alert to the impact on the mathematical sciences of abrupt shifts in criteria for support . . . . Wit tin mlsslon-orlentec agencies.

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