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/ - Th . BY THE 10 lVui~ ~ SAC Mathematical '^ sciences. . A Report COMMITTEE OIL SUPPORT OF RESEARCH IN THE MATHEMATICAL SCIEN CES OF THE NATIONAL RESEARCH COUNCIL FOR THE COMMITTEE ON SCIENCE AND PUBLIC POLICY NATIONAI, ACADEMY OF SCIENCES PUBLICATION 1681 National Academy of Sciences WASHINGTON, D.C. 1968 . - ~ a\ '/

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Available from Printing and Publishing Once National Academy of Sciences 2101 Constitution Avenue Washington, D.C. 20418 Library of Congress Catalog Card Number 6841402

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February 2, 1968 Dr. Fred crick Seitz, Presid ent Na t lo na l A cad emy o f Scien ces 2101 Constitution Avenue Washington, D.C. 20418 Dear Dr. Seitz: The Committee on Science and Public Policy takes pleasure in forwarding to you the report of the Committee on Support of Research in the Mathematical Sciences. Our committee has reviewed this report at several stages in its development, and we feel that it presents a clear and cogent case for the support of tile mathematical sciences and a good picture of the present state of both research and ed ucation in mathematics and related d isciplines. The extraordinary rate at which classical mathematical theorems have fallen before the onslaught of young American mathematicians in recent years is impressive, as is the leading position acquired by the United States in recent times. The report calls attention to the penetration of mathematics and mathematical modes of thought into many new areas of scholarship and the resultant great increase in the relative enrollment in undergraduate mathematics courses, especially at the upper levels. It also brings out the increasing importance of mathematical methods in industry and government, the rapid growth in employment of mathematical scientists outside the universities, and the pace of growth of mathematical sciences relative to that of all other disciplines. The report calls attention to the special problem of computer science in universities, brought about by its high cost relative to the rest of mathematical research and education and by the fact that it is both a scientific and an engineering discipline, which has made it difficult to fit into the academic structure. Meaningful research in computer science often requires rather large-scale support for complex projects of a partly developmental nature that do not always lend themselves to the project-grant type of support typical in pure mathematics and some other scientific disciplines. The development of computer science only as a . · .

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by-product of the application of computer techniques in other fields often results in failure to develop a distinctive body of theory and technique in computer science in its own right. The report identifies special problems concerning mathematical statistics and physical (i.e., classical applied) mathematics. In physical mathematics the problem is not the need for rapid growth of financial support, as in computer science, but for a place i'n the academic structure that does not tie it too closely to a particular set of applications. The authors of this report feel that there are too few distinguished applied-mathematics groups in American universities, and' that special encouragement is needed for building such groups. Whatever place applied mathematical sciences find in the academic structure, however, it is essential to maintain a strong and continuing interaction between these sciences and pure mathematics an interaction in which ideas and people must move in both directions. An outstanding feature of this report in comparison with some of the earlier disciplinary reports is the attention it devotes to questions of education, not only for mathematical scientists but also for the many users of mathematics. The Committee on Support of Research in Mathematical Sciences is to be congratulated not only for its impressive work in the preparation of the present volume but also for the valuable contributions of two supplementary volumes produced under its aegis. One of these, the detailed report of the Committee's Panel on Und ergrad uate Ed ucation, analyzes the uniquely d emand ing special problems of American undergraduate education in the mathematical sciences and makes many suggestions for dealing with these problems. The other, a collection of twenty-two essays by distinguished mathematical scientists, brings together, in a form accessible to a wide range of readers, expositions of the achievements, traditions, and prospects in a rich variety of mathematical fields. Sincerely, HARVEY BROOKS, Chairman Committee on Science and Public Policy 1V

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THIS REPORT on the mathematical sciences is another in the series of comprehensive survey reports on major fields of science pre- pared under the aegis of the Academy's Committee on Science and Public Policy. With the growing importance of the mathematical sciences, both in their own right and through their penetration of other sciences and of technology, it has become clear that a report on mathe- matics should be added to the survey reports. Thus the Committee on Support of Research in the Mathematical Sciences was appointed to undertake the preparation of such a report under the chairman- ship of Professor Lipman Bers. I am pleased to express gratitude to the Committee for its dedi- cated work. I trust the findings and recommendations set forth here will prove useful to those with responsibilities in the continuing support of American science. FREDERICK SEITZ, President National Academy of Sciences v

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Committee on Science ant! Public Policy HARVEY BROOKS, Harvard University, Chairman W. O. BAKER, Bell Telephone Laboratories PRESTON E. CLOUD, JR., University of California, Los Angeles HARRY EAGLE, Albert Einstein College of Medicine CARL ECKART, University of California, San Diego HERBERT FRIEDMAN, U.S. Naval Research Laboratory RALPH W. GERARD, University of California, Irvine J. G. HARRAR, The Rockefeller Foundation G. E. HUTCHINSON, Yale University MARK G. INGHRAM, University of Chicago W. D. MCELROY, The Johns Hopkins University GEORGE A. MILLER, Rockefeller University KENNETH B. RAPER, University of Wisconsin HERBERT A. SIMON, Carnegie-Mellon University CYRIL STANLEY SMITH, Massachusetts Institute of Technology ROBERT E. GREEN, National Academy of Sciences, Executive Secretary V1

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Committee on Support of Research in the Mathematical Sciences LIPMAN BERS, Columbia University, Chairman T. W. ANDERSON, Columbia University R. H. BING, University of Wisconsin HENDRIK W. BODE, Bell Telephone Laboratories R. P. DILWORTH, California Institute of Technology GEORGE E. FORSYTHE, Stanford University MARK KAC, Rockefeller University C. C. LIN, Massachusetts Institute of Technology JOHN W. TUKEY, Princeton University F. J. WEY1,, National Academy of Sciences HASSLER WHITNEY, Institute for Advanced Study C. N. YANG, State University of New York at Stony Brook TRUMAN BOTTS, University of Virginia, Executive Director V~l

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Foreworc! The Committee on Support of Research in the Mathematical Sciences (Cosa~Msj was appointed by the Division of Mathematical Sciences of the National Research Council at the instigation of the Committee on Science and Public Policy of the National Academy of Sciences. Our task was to assess the present status and the pro- jected future needs, especially fiscal needs, of the mathematical sciences. It was clear to us from the very beginning of our work that our report would have to differ somewhat in structure from the corresponding reports for other disciplines that had already appeared. Though mathematics provides the common language for all sci- ences, we realize that even scientific readers of our report, let alone nonscientists, may feel that they are not adequately informed about what mathematical research, especially modern mathematical research, consists of. Similarly, even professional mathematicians, or scientists who customarily use mathematics in their work, may be unaware of the manifold applications of mathematics in various sciences and technologies, especially the new applications influenced by the computer revolution. To provide additional background of factual information con- cerning the mathematical sciences, we are supplementing our report with a collection of essays, written by distinguished authors on various topics in mathematics, in the applied mathematical sciences, and in the applications of mathematics. With three exceptions, which are reprints, these essays were written expressly for this col- lection. They are intended not only for the nonmathematical scientist but also for the scientifically oriented layman. 1X

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Mathematics pervades our whole educational system. As a matter of fact, we believe that the mathematical community has no obli- gations more important than those concerned with education, the most critical area being collegiate education. We have, therefore, included in our report questions of policy regarding higher edu- cation. Our Panel on Undergraduate Education has carried out an intensive study of this area; its report is presented in a separate volume of our report. Simultaneously with our activities, the Conference Board of the Mathematical Sciences has been carrying out a survey of research and education in mathematics, and its Survey Committee has agreed to act as a fact-finding agency for our Committee. The Conference Board Survey Committee's report will contain a wealth of factual and statistical material pertaining to the matters discussed in our report. We take this opportunity to express our gratitude to the Survey Committee and to the Ford Fo~n~1~ion which Nor their work. ^ ~-erred ~ The activities of our Committee have been financed mainly by a grant from the National Science Foundation. This has been sup- plemented by smaller grants from the Sloan Foundation, the Con- ference Board of the Mathematical Sciences, the American Mathe- matical Society, the Association for Computing Machinery, the Association for Symbolic Logic, the Institute of Mathematical Statistics, the Mathematical Association of America, the National Council of Teachers of Mathematics, the Operations Research Society of America, and the Society for Industrial and Applied Mathematics. Columbia University has generously provided us with once space and many auxiliary services. To all these organizations we express our thanks. We are deeply indebted to the authors of the essays, to the chair- men and members of our panels, and to the many other individuals who have contributed their time and expertise to our undertaking. January 20, 1968 x LIPMAN BERS Chairman, Committee on Support of Research in the Mathematical Sciences