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Renewing ~ T C! Mathematics A PLAN FOR THE l990s Committee on the Mathematical Sciences: Status and Future Directions Board on Mathematical Sciences Corrosion on Physical Sciences, Mathematics, and Applications National Research Council NATIONAL ACADEMY PRESS Washington, D.C. 1990

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NOTICE: The project that is the subject of this report was approved by the Governing Board of the National Research Council, whose members are drawn from the councils of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine. The members of the committee responsible for the report were chosen for their special competences and with regard for appropriate balance. This report has been reviewed by a group other than the authors according to procedures approved by a Report Review Committee consisting of members of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine. The National Academy of Sciences ~ a private, nonprofit, self-perpetuating society of distinguished scholars engaged in scientific and engineering research, dedicated to the furtherance of science and technology and to their use for the general welfare. Upon the authority of the charter granted to it by the Congress in 1863, the Academy has a mandate that requires it to advise the federal ~overn- , cat , _ , ment on scientific and technical matters. Dr. Frank Press is president of the National Academy of Sciences. The National Academy of Engineering was established in 1964, under the charter of the National Academy of Sciences, as a parallel organization of outstanding engineers. It is autonomous in its administration and in the selection of its members, sharing with the National Academy of Sciences the responsibility for advising the federal government. The National Academy of Engineering also sponsors engineering programs aimed at meeting national needs, encourages education and re- search, and recognizes the superior achievements of engineers. Dr. Robert M. White is president of the National Academy of Engineering. ~ _ O The Institute of Medicine was established in 1970 by the National Academy of Sciences to secure the services of eminent members of appropriate professions in the examination of policy matters per- taining to the health of the public. The Institute acts under the responsibility given to the National Academy of Sciences by its congressional charter to be an adviser to the federal government and, upon its own initiative, to identify issues of medical care, research, and education. Dr. Samuel O. Thier is president of the Institute of Medicine. The National Research Council was organized by the National Academy of Sciences in 1916 to associate the broad community of science and technology with the Academy's purposes of further- ing knowledge and advising the federal government. Functioning in accordance with general policies determined by the Academy, the Council has become the principal operating agency of both the National Academy of Sciences and the National Academy of Engineering in providing services to the government, the public, and the scientific and engineering communities. The Council is administered jointly by both Academies and the Institute of Medicine. Dr. Frank Press and Dr. Robert M. White are chairman and vice chairman, respectively, of the National Research Council. Support for this project was provided by the Air Force Office of Scientific Research, the Army Research Office, the Department of Energy, the National Science Foundation, the National Security Agency, and the Office of Naval Research under contract number DMS-8821296. Library of Congress Catalog Card Number 90-60044 International Standard Book Number 0-309-04228-3 Available from National Academy Press 2101 Constitution Avenue, NW Washington, DC 20418 S109 Printed in the United States of America Summaries of this report are available from the Board on Mathematical Sciences, National Research Council, 2101 Constitution Avenue, NW, Washington, DC 20418.

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N AT I O N AL RE S E ARC H C O U N C I L 2101 CONSTIT~ION AVENUE OFFICE OF THE CHAIRMAN WASHINGTON, D. C. 20418 This report, Renewing U.S. Mathematics: A Plan for the lotus, updates the 1984 "David Report," which recommended a national plan to renew and ensure the health of the U.S. mathematical sciences enterprise. The new report, presenting the committee's assessment of progress made since 1984, communicates a sense of promise and achievement, but also the conviction that further corrective action is urgently needed to ensure the vitality of U.S. mathematics. Substantial progress can be seen in increased federal support for graduate education and postdoctoral researchers, as well as in stronger leadership and improved cohesiveness within the mathematical sciences community. Yet major problems remain: the continuing inadequacies in support for mathematical sciences research, especially for principal investigators; the slow response on the part of many members of the mathematical sciences community to the serious issues of renewal; and the absence of a concerted response by universities to problems clearly described six years ago. The high drop-out rate from mathematical sciences career paths warns that U.S. mathematical research faculties, institutions, and education at all levels must be renewed. At the same time, this report's presentation of some of the exciting recent achievements in mathematical sciences research, as well as the wealth of opportunities for future research and applications. Points . . ~ . , to the promise of what is achievable in the mathematical sciences and by extension in U.S. science and technology. We have an opportunity now to deal with the issues confronting the mathematical sciences community and the nation, especially the challenge of attracting and educating tomorrow's professors and researchers. This report recommends specific actions to address those issues. I commend it to your attention. j~'6~ Frank Press Chairman National Research Council . ~ fit

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NATIONAL RESEARCH COUNCIL COMMISSION ON PHYSICAL SCIENCES, MATHEMATICS, AND APPLICATIONS 2101 Constitution Avenue Washington, D.C. 20418 BOARD ON MATHEMATICAL SCIENCES Dr. Frank Press Chairman, National Research Council Dear Frank: April 1990 In submitting to you the report of the Committee on the Mathematical Sciences: Status and Future Directions ~ - ~ . . . . . ~ _ let me include some comments and observations which are unusual for a letter of transmittal. I do so in hopes that their message will be promulgated and heard, not only in the National Academies but more widely. Nine years ago you asked me to chair the Ad Hoc Committee on Resources for the Mathematical Sciences? to review the intellectual health of mathematical research in the United States and do an in-depth analysis of federal support for the field. As a communications engineer who could trace his roots back through his thesis advisor Jerry Wiesner to the great mathematician Norbert Wiener, and as a science administrator who had seen first-hand the enormous impact of mathematics and mathematicians, first at Bell Laboratories and later at Exxon Research and Engineering, I was pleased to accept. You gave me a superb committee to do the work. You are as familiar as I am with what we found: a field brimming with intellectual vitality, preeminent in the world, and poised to make even greater contributions to science and technology, yet a field in which the research infrastructure had eroded. in Dart because federal support had been allowed to deteriorate. , . . . In 1984 the ad hoc committee recommended a coordinated set of actions to be taken by government, universities, and the mathematical community over five to ten years to rebuild the infrastructure and enable mathematics to renew itself. recommendations. Halfway through that decade you asked me to chair a different but equally distinguished committee, on behalf of which I am now reporting. We were to assess progress made in implementing the 1984 to assess progress made in implementing _ I was happy to accept because I remain vitally concerned with the health of U.S. mathematics. Our report tells what has happened in the five years since we published Renewing U.S. Mathematics: Critical Resource for the Future. and this second retort recommends what to do now. Our message is in one sense very simple: balance between support for the mathematical sciences and support for related fields must be restored; stay the course, see it through. We do suggest, however, a modification of the original plan for renewal, tying it more closely to human resource issues and concentrating attention on the pipeline which develops mathematical and scientific V

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talent--issues which loom much larger in the minds of all of us today than they did five years ago. This modification may require some policy changes in federal agencies, drawing their research and educational missions closer together. It will certainly require strong commitment and bold action by the mathematical sciences community, working in conjunction with the research universities. Let me end with a personal perspective. Overall, I am both pleased and puzzled by progress since 1984--pleased because strong leadership by individuals in government and the mathematical sciences community has brought substantial progress, creating movement toward renewal, yet puzzled by three matters: 1. 2. 3. the general membership of the mathematical sciences community, unlike its leadership, seems only~beginning to grasp either the nature or seriousness of its renewal problems, problems that are being compounded by the greatly increased need to attend to revitalization of mathematics education: the lack of concerted action by the research universities or their leaders, either in calling attention to the problems of mathematical research funding or in bolstering their mathematical sciences departments; and the inability of the science policy mechanisms of government to deal decisively with a funding problem as easily soluble and vitally important as the one we pointed out back in 1984. On the first point, I plan to continue the work I began five years ago, of encouraging the mathematical sciences community to act vigorously, and am happy to see the intense dialogue which is shaping up for 1990 within the community. As former directors of the Office of Science and Technology Policy, we are not very surprised by points 2 and 3. But, for my own part, I remain somewhat distressed by them. Perhaps we can join together in carrying a message to government and the universities. If the mechanisms of science policy cannot solve this critical problem in mathematics, it is doubtful whether they can solve any problem at all. Sincerely, ^0 ~ :~ /6~' 4. Edward E. David, Jr. Chairman, Committee on the Mathematical Sciences At

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COMMITTEE ON THE MATHEMATICAL SCIENCES: STATUS AND FUTURE DIRECTIONS EDWARD E. DAVID, JR., FED, Inc., Chair CALVIN C. MOORE, University of California at Berkeley, Vice-Chair KENNETH J. ARROW, Stanford University PETER J. BICKEL, University of California at Berkeley RICHARD D. DELAUER, Fairchild Space and Defense Corp. GERALD P. DINNEEN, National Academy of Engineering CHARLES L. FEFFERMAN, Princeton University MICHAEL E. FISHER, University of Maryland KENNETH M. HOFFMAN, Massachusetts Institute of Technology CATHLEEN S. MORAWETZ, New York University DAVID A. SANCHEZ, Lehigh University MORRIS TANENBAUM, AT&T ROBERT E. TARJAN, Princeton University Ex-officio Member PHILLIP A. GRIFFITHS, Duke University Board or' Mathematical Sciences Liaison GUIDO L. WEISS, Washington University Staff LAWRENCE FI. COX, Director DONNA CARTER, Senior Secretary JO NEVILLE, Administrative Secretary RUTH E. O'BRIEN, Staff Associate WILLIAM G. ROSEN, Consultant JOHN TUCKER, Staff Officer SCOTT T. WEIDMAN, Senior Staff Officer vii

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BOARD ON MATHEMATICAL SCIENCES PHILLIP A. GRIFFITHS, Duke University, Chair LAWRENCE D. BROWN, Cornell University RONALD DOUGLAS, State University of New York-Stony Brook DAVID EI)DY, Duke University FREDERICK W. GEHRING, University of Michigan JAMES GLIMM, State University of New York-Stony Brook WILLIAM JACO, American Mathematical Society JOSEPH KADANE, Carnegie-Mellon University GERALD J. LIEBERMAN, Stanford University ALAN NEWELL, University of Arizona JEROME SACKS, University of Illinois GUIDO L. WEISS, Washington University SHMUEL WINOGRAD, IBM TV. Watson Research Center Ex-officio Member MORRIS DEGROOT, Carnegie-Mellon University (deceased) Staff LAWRENCE H. COX, Director DONNA CARTER, Senior Secretary JO NEVILLE, Administrative Secretary RUTH E. O'BRIEN, Staff Associate WILLIAM G. ROSEN, Consultant SEYMOUR SELIG, Staff Officer JOHN TUCKER, Staff Officer SCOTT T. WEIDMAN, Senior Staff Officer Vt''

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COMMISSION ON PHYSICAL SCIENCES, MATHEMATICS, AND APPLICATIONS NORMAN HACKERMAN, Robert A. Welch Foundation, Chair GEORGE F. CARRIER, Harvard University HERBERT D. DOAN, The Dow Chemical Company (retired) DEAN E. EASTMAN, IBM T.~. Watson Research Center MARYE ANNE FOX, University of Texas GERHART FRIEDLANDER, Brookhaven National Laboratory PHILLIP A. GRIFFITHS, Duke University NEAL F. LANE, Rice University CHRISTOPHER F. MCKEE, University of California at Berkeley RICHARD S. NICHOLSON, American Association for the Advancement of Science JEREMIAH P. OSTRIKER, Princeton University Observatory ROY F. SCHWITTERS, Superconducting Super Collider Laboratory LARRY L. SMARR, University of Illinois at Urbana-Champaign NORMAN METZGER, Executive Director ax l

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Acknowledgments Many individuals in and beyond the mathematical sciences commu- nity contributed to this report. The interdisciplinary Committee on the Mathematical Sciences: Status and Future Directions, which devel- oped the ideas that form the writing and core of this report, actively and willingly participated in the creation of it, as is reflected in its finished form. Their names are listed on a previous page. I particu- larly acknowledge Calvin C. Moore, vice-chair of the committee, for seeking out, selecting among, organizing, and presenting well a di- verse collection of research opportunities in the mathematical sci- ences. For a field that offers so many rich opportunities for the future, this was a challenging task. Thanks are due also to some 40 mathe- matical scientists (listed in Appendix B) for providing written expert summaries. These summaries provided much of the substance for the section titled "Accomplishments and Opportunities." The committee had before it a great deal of information that influ- enced its conclusions. Much of that information is not contained in the report itself since it will appear elsewhere. The committee thanks those who brought it forward, often at considerable effort. Among them are the directors, deputy directors, and staff of the ICEMAP federal agencies, and William G. Rosen and other members of the staff of the Board on Mathematical Sciences. The chairpersons of some 25 university mathematical sciences departments furnished detailed data on their departments as compared to physics, chemistry, biology, and engineering departments. The committee also wishes to thank four chairpersons in particular who, through thoughtful essays, provided valuable glimpses of conditions for fulfilling the research and educa- tion goals of the mathematical sciences. At

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ACKNOWLEDGMENTS Thanks are due to the National Science Foundation and the other ICEMAP federal agencies for commissioning this study ant! to the Board on Mathematical Sciences, National Research Council (NRC), for organizing it. The board supplied valuable information and ad- vice concerning both the research opportunities and wider matters, through its chairperson, Phillip A. Griffiths, and participants Guido L. Weiss, Peter J. Bickel, and Cathleen S. Morawetz. Conscientious and effective support was given to the committee by the board's staff, particularly by its Executive Director, Lawrence H. Cox, and Senior Staff Officer Scott T. Weidman. The attention to detail yet quick response of the NRC offices responsible for the review and production of the report' and of the independent reviewers of the report assigned by the NRC, also deserve mention as constructive influences. Edward E. David, Jr. Chairman, Committee on the Mathematical Sciences . Ott

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Contents EXECUTIVE SUMMARY 1 BACKGROUND AND INTRODUCTION The Situation in 1984, 11 Imbalance in Support for Research, 15 The 1984 National Plan, 16 The Current Report, 18 RESPONSE TO THE 1984 NATIONAL PLAN Federal Response, 21 Response of the Universities, 31 Response of the Mathematical Sciences Community, 32 3 RESEARCH PROGRESS AND PROSPECTS.. The Mathematical Sciences Yesterday, 40 The Mathematical Sciences Today, 41 Computers in the Mathematical Sciences, 42 Accomplishments and Opportunities, 44 The Unifying Science, 51 The Production of New Mathematics, 51 4 THE PROBLEM OF RENEWAL ....... Demand for Mathematical Scientists, 55 Shortfall in Supply, 56 Reasons for the Shortfall, 57 Addressing the Shortfall, 61 .... 11 21 39 55 . ~ ~ Xtft

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CONTENTS 5 RECOMMENDATIONS.... Primary Recommendations, 69 Directed Recommendations, 72 APPENDIXES A Executive Summary of the 1984 Report, 77 B Recent Research Accomplishments and Related Opportunities, 87 xv .... . . .. . 69