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Experiments in International Benchmarking of US Research Fields
ATTACHMENT 1
INTERNATIONAL BENCHMARKING OF US MATHEMATICS RESEARCH
Panel on International Benchmarking of US Mathematics Research
Committee on Science, Engineering, and Public Policy

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Experiments in International Benchmarking of US Research Fields
International Benchmarking of US Mathematics Research
Panel Members
PETER D. LAX (Chair), Professor of Mathematics and Director, Courant Mathematics and Computing Laboratory, New York University, New York, NY
MICHAEL F. ATIYAH, Master, Trinity College, Cambridge, England
SPENCER J. BLOCH, Professor, Department of Mathematics, University of Chicago, Chicago, IL
JOSEPH B. KELLER, Professor, Departments of Mathematics and Mechanical Engineering, Stanford University, Stanford, CA
JACQUES-LOUIS LIONS, President, French Academy of Sciences and Professor, College de France, Paris, France
YURI I. MANIN, Director, Max Planck Institut fur Mathematik, Bonn, Germany
RUDOLPH A. MARCUS, A.A. Noyes Professor of Chemistry, California Institute of Technology, Pasadena, CA
GARY C. McDONALD, Head, Operations Research Department, GM Research and Development Center, Warren, MI
CATHLEEN S. MORAWETZ, Professor Emeritus, Courant Institute of Mathematical Sciences, New York University, New York, NY
PETER SARNAK, Chair, Department of Mathematics, Princeton University, Princeton, NJ
I.M. SINGER, Institute Professor, Massachusetts Institute of Technology, Cambridge, MA
MARGARET H. WRIGHT, Distinguished Member of Technical Staff, Bell Laboratories, Lucent Technologies, Murray Hill, NJ
Project Staff
DEBORAH D. STINE, Study Director
PATRICK P. SEVCIK, Research Associate
JOHN R. TUCKER, Senior Program Officer
NORMAN GROSSBLATT, Editor

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Mathematics Benchmarking Guidance Group
MARYE ANNE FOX (Chair), Vice President for Research, University of Texas at Austin, Austin TX
PHILLIP A. GRIFFITHS, Director, Institute for Advanced Study, Princeton, NJ
DANIEL KLEPPNER, Professor of Physics, Massachusetts Institute of Technology, Cambridge, MA
AVNER FRIEDMAN, Director, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN
SAMUEL WINOGRAD, IBM Fellow, International Business Machines Corporation, Thomas J. Watson Research Center, Yorktown Heights, NY
ROBERT MacPHERSON, Professor, School of Mathematics, Institute for Advanced Study, Princeton, NJ
Staff:
LAWRENCE E. McCRAY, Executive Director, COSEPUP
DEBORAH D. STINE, Associate Director, COSEPUP
JOHN R. TUCKER, Director, Board on Mathematical Sciences
NORMAN METZGER, Executive Director, Commission on Physical Sciences, Mathematics, and Applications

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Experiments in International Benchmarking of US Research Fields
CONTENTS
EXECUTIVE SUMMARY
67
1 BACKGROUND
69
2 SCOPE AND NATURE OF THE PANEL'S EVALUATION
71
3 RELATIVE POSITION OF US RESEARCH IN MATHEMATICS
74
3.1 The Discipline
74
3.1.1 Leadership
74
3.1.2 Depth
75
3.2 Mathematics in a Broader Context
76
3.2.1 Science and Engineering
76
3.2.2 Industry
78
3.2.3 Government Laboratories and Agencies
80
3.2.4 Mathematics Education
81
4 FACTORS THAT INFLUENCED PAST US PERFORMANCE IN MATHEMATICS
83
4.1 Attractiveness to Talent from Outside the United States
83
4.2 Quality and Structure of Graduate Education in Mathematics
84
4.3 Diversity of the US Research Enterprise
84
4.4 Adequate Funding
86

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5 CURRENT TRENDS
87
5.1 Vitality of the Mathematical Sciences
87
5.2 Interdisciplinary Research
88
5.3 Employment Prospects for New PhDs
89
5.3.1 Academic Jobs
89
5.3.2 Industrial Jobs
91
5.4 Foreign Graduate Students
92
5.5 Graduate Education
94
5.6 Support
95
6 LIKELY FUTURE RELATIVE POSITION OF US MATHEMATICS
99
6.1 Intellectual Quality
99
6.2 Interdisciplinary Research
99
6.3 US Graduate Education in Mathematics
100
6.4 Support for Mathematical Research
100
7 REFERENCES
101
APPENDIX A: Panel and Staff Biographical Information
103
APPENDIX B: Statistical Data on the Field of Mathematics
109
Figures and Tables
Report
Figure 1:
Percentage of Mathematics-Research Papers Published by US Authors
76
Figure 2:
Percentage Unemployment Among New US PhDs in Mathematics, Autumn of Year Shown
89
Figure 3:
Number of PhDs Produced by US Mathematics Departments, Spring of Year Shown
90
Figure 4:
Employment Status of PhD Mathematicians in the US
91
Figure 5:
Doctoral Recipients: Total Number and US and Non-US Citizens
92
Figure 6:
Percentage of Foreign Natural-Sciences Doctoral Students in Various Countries
93
Figure 7:
Stay Rates—Percentages of Foreign Doctoral Students Who plan to Remain in the United States, Averaged Over 1988-1992
93
Figure 8:
Total Full-Time PhD Students in Mathematical Sciences
94
Table 1:
Decrease in Applications to PhD Programs in Mathematics, 1994 to 1996
95
Figure 9:
Median Salaries in 1993 of US PhDs Who Received Their Degrees in 1985-1990, by Field
96

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Figure 10:
Percentages of Academic Scientists with Federal Support, 1993
97
Figure 11:
Percentage Increase in Federal R&D Expenditures at Universities and Colleges, by Field
97
Appendix B
Figure B-1:
Number of US Institutions Awarding PhDs in Mathematics, 1920-1995
110
Figure B-2:
Number of PhDs Awarded in Mathematics in the United States, 1920-1995
110
Figure B-3:
Median Time to PhD and Age at Receipt of PhD in Mathematics in the United States
111
Figure B-4:
Doctoral Recipients: Total Number of US and Non-US Citizens
111
Figure B-5:
Number of First Degrees in Mathematics and Computer Science
112
Figure B-6:
Doctoral Degrees in Natural-Sciences, 1992
113
Figure B-7:
Number of PhD Mathematicians Employed in the United States
113
Table B-1:
Employment Status of PhD Mathematicians in the United States
114
Table B-2:
Occupation Status of PhD Mathematicians in the United States
115
Figure B-8:
Median Salaries in 1993 of US PhDs Who Received Their Degree in 1985-1990, by Field
116
Figure B-9:
Citizenship of Full-Time Mathematics Faculty with PhDs Hired During 1991-1992 in the United States
117
Figure B-10:
Source of PhDs of Full-Time Mathematics Faculty Hired During 1991-1992 in the United States
117
Figure B-11:
Percentage of Unemployed New US Mathematics PhDs
118
Figure B-12:
Median Nine-and Twelve-Month Salaries of New US PhDs for Teaching or Teaching and Research in 1995 Dollars
118
Table B-3:
Federal Support for the Mathematical Sciences, Fiscal Year 1995-1998, in Millions, Current Dollars; and in Millions, Constant 1992 Dollars
120
Figure B-13:
Percentages of US Academic Scientists with Federal Support, 1993
121
Figure B-14a:
Federal Funding of US Mathematical Research—Academic, 1993-1995 Average
122
Figure B-14b:
Federal Funding of US Mathematical Research—All R&D
122
Figure B-15:
Percentage of Mathematical Research Papers Published by US authors
124
Figure B-16:
Number of Mathematical Research Papers by US and EC Authors, 1981-1996
124

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EXECUTIVE SUMMARY
The United States is clearly preeminent in mathematics today. The field is thriving in terms of both quality and opportunities. Not only are there stellar researchers in all fields at American institutions, but they are backed by a broad and active research community. Mathematical research in the United States has many links with science, engineering, and technology and is broadening its contacts with education at all levels. But this position of eminence is fragile. Increasing demands are placing a strain on the mathematics community.
In making judgments about mathematics, the International Benchmarking of US Mathematics Research Panel kept these points in mind:
Mathematics is the language and tool of most of the sciences.
Mathematical results often have a long life.
Mathematical research is conducted on a very broad front, and seemingly disjointed branches often turn out to be intimately related.
Ideas of abstract mathematics often are crucial ingredients in practical applications.
Mathematics is one of the pillars of education in kindergarten, elementary school, high-school, and college.
The present strength in US mathematics is due to:
Continued attractiveness of the United States to talented people around the world.

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A strong system of graduate education.
Diversity and flexibility of the US research enterprise.
Sustained funding for research from universities and the federal government.
The United States continues to attract some of the best graduate students and postdoctoral fellows from all over the world; a substantial portion of active research mathematicians now in the United States come from outside the United States. But we are in danger of losing our preeminent position if we do not face some critical issues and challenges. Some critical issues and challenges must be faced:
US leadership in mathematics rests on the health of research universities, which today are experiencing severe financial pressure and conflicting demands.
The United States is not taking sufficient advantage of its native mathematical talent: while graduate enrollment from abroad thrives, the number of American students applying to graduate school in mathematics is diminishing.
Serious thought is needed about how to make better connections between mathematics and other fields, because mathematics is crucial in much interdisciplinary research.
US industry has reduced its commitment to long-range research in mathematics.

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1
BACKGROUND
In 1993, the Committee on Science, Engineering, and Public Policy (COSEPUP) of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine issued the report Science, Technology, and the Federal Government: National Goals for a New Era. This report recommended that the United States be among the world leaders in all major fields of science and maintain clear leadership in selected fields. A similar recommendation was made in a later National Research Council (NRC) report, Allocating Federal Funds for Science and Technology, published in 1995—that the United States ''strive for clear leadership in the most promising areas of science and technology and those deemed most important to our national goals."
Both reports stated that quantitative measures, such as dollars spent and number of scientists supported, were inadequate indicators of leadership and that policy decisions about programmatic issues or resource allocation would be better informed by comparative international assessments. Independent field-specific panels were suggested as the best means for obtaining such evaluations. Each panel would consist of researchers in the particular field, researchers in closely related fields, and research users who follow the field, and each panel would include researchers from outside the United States.
In late 1996, COSEPUP began an experimental study of the effectiveness and outcome of such panels. The present report—an evaluation of US research in mathematics—was prepared by the first panel and will be followed by studies in materials science and immunology. Each panel has been asked to address the following questions:

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at different demographic and employment characteristics and by taking different cohorts. This provides for both longitudinal and time-series analyses, as shown here.
Of course, differentiating between research and teaching in determining the type of work for faculty is difficult. However it is fruitful to think about the nonresearch and teaching positions that mathematicians are obtaining and how they are changed over time.
Figure 4 in section 5.3.2 shows some of this information graphically. Note how the percentage of mathematicians employed as tenured and tenure-track faculty has declined while the percentage of mathematicians employed in industry has increased. The percentage in government employment has remained stable.
Figure B-8 shows the median salaries for PhD mathematicians and PhD holders in several related fields.
Figure B-9 shows the citizenship of faculty hired in 1991-1992 and figure B-10 the source of their PhDs.
Of particular concern is the unemployment status of new PhDs. Figure B-11 shows the change in unemployment rate for new mathematics PhDs from 1989 to 1996. The salaries of the new PhDs who attained academic employment are shown in figure B-12; the 9-month salaries included data on 102 men and 38 women, and the 12-month salaries included data on 20 men and 7 women.
FIGURE B-8 Median salaries in 1993 of US PhDs who received their degree in 1985-1990, by field. Source: NSF 1996a, appendix table 5-27.

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FIGURE B-9 Citizenship of full-time mathematics faculty with PhDs hired during 1991-1992 in the United States. Source: AMS 1992, pp. 314-315.
FIGURE B-10 Source of PhDs of full-time mathematics faculty hired during 1991-1992 in the United States. Source: AMS 1992, pp. 314-315.

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FIGURE B-11 Percentage of unemployed new US mathematics PhDs. Source: AMS 1996, 1997c.
FIGURE B-12 Median nine- and twelve-month salaries of new US PhDs for teaching or teaching and research in 1995 dollars. Source: AMS 1996.

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Funding
The information provided in this section, unless otherwise indicated, is from an analysis conducted by the Joint Policy Board for Mathematics for the American Association for the Advancement of Science. It produces an annual analysis of federal budget data on the field of mathematics.
There are 7 dedicated programs in mathematical sciences at 3 agencies: the Department of Defense (DOD), the Department of Energy (DOE), and the National Science Foundation (NSF). NSF focuses on fundamental research and its vitality, DOD looks on mathematical sciences as a problem-solving technology that can reduce costs in the development and deployment of hardware and software, and DOE and other agencies—such as the Department of Transportation, the Environmental Protection Agency, the National Aeronautics and Space Administration, the National Institutes of Health, and the National Institute of Standards and Technology—maintain mostly-applied mathematics and statistics activities to enable progress in fields related to their missions. All other agencies use applied mathematics and statistics.
Table B-3 shows federal support for academic mathematical-sciences research. Figure B-13 compares the percentage of academic mathematical scientists who have federal support to the percentages in other fields. Federal support for all mathematical research (basic, applied, and development) is shown in figure B-14.
The NSF Department of Mathematical Sciences (DMS) supports development of mathematical and statistical ideas and techniques, encourages the integration of mathematics with other disciplines, and encourages the diffusion of mathematics into technology. Grants are provided to individual investigators, research institutes, and centers for shared computing equipment, postdoctoral fellowships, research conferences, and undergraduate programs such as curriculum development.
NSF supports three mathematics institutes—the Institute for Mathematics and its Applications (IMA) at the University of Minneapolis was supported at $1,900,000 and the Mathematical Sciences Research Institute (MSRI) at the University of California, Berkeley was supported at $3,110,000 in FY1996. The IMA nearly matches the NSF support with funds from industry, sponsoring institutions, other agencies, and the University of Minnesota. The MSRI has limited additional support outside the NSF award. In 1998, there will be a recompetition for the location of the institutes in the mathematical sciences. The MSRI and the IMA are under review for ''bridging" awards until the new national institutes are established as a result of the recompetition. Since its inception in 1989, the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) at Rutgers University and its staff have received a total of $74 million in science and technology center (STC)

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TABLE B-3 Federal Support for the Mathematical Sciences, Fiscal Year 1995-1998, in Millions, Current Dollars
Actual FY 95
Actual FY 96
Estimate FY 97
Percent Changea FY 96-97
Budget Request FY 98
Percent Change FY 97-98
National Science Foundation
87.69
91.70
98.22
7.11%
102.00
3.8%
DMS*
85.29
87.70
93.22
6.29%
97.00
4.1%
Other MPS
2.40
4.00
5.00
25.00%
5.00
0.0%
Department of Defenseb
77.40
77.30
67.80
–14.01%
73.60
8.5%
AFOSR
17.50
16.70
17.10
2.39%
17.10
0.0%
ARO
15.00
15.00
13.00
–15.38%
15.00
15.4%
DARPA
21.00
22.90
19.50
–17.43%
22.40
14.8%
NSA
2.50
2.50
2.10
–19.05%
2.10
0.0%
ONR
21.40
20.20
16.10
–25.47%
17.00
5.6%
Department of Energy
15.70
16.00
16.00
0.00%
16.00
0.0%
University support
6.20
5.50
5.00
–10.00%
5.00
0.0%
National laboratories
9.50
10.50
11.00
4.76%
11.00
0.0%
TOTAL, All Agencies
180.79
185.00
182.02
–1.61%
191.60
5.3%
Federal Support for the Mathematical Sciences, Fiscal year 1995-1998, in Millions, Constant 1992 Dollars
Actual FY 95
Actual FY 96
Estimate FY 97
Percent Changea FY 96-97
Budget Request FY 98
Percent Change FY 97-98
National Science Foundation
81.48
83.44
87.20
4.51%
88.26
1.2%
DMS*
79.25
79.80
82.76
3.71%
83.93
1.4%
Other MPS
2.23
3.64
4.44
1.99%
4.33
–2.5%
Department of Defenseb
71.92
70.34
60.19
–16.86%
63.68
5.8%
AFOSR
16.26
15.20
15.18
–0.13%
14.80
–2.5%
ARO
13.94
13.65
11.54
–18.28%
12.98
12.5%
DARPA
19.51
20.84
17.31
–20.39%
19.38
12.0%
NSA
2.32
2.27
1.86
–22.04%
1.82
–2.5%
ONR
19.88
18.38
14.29
–28.62%
14.00
2.9%
Department of Energy
14.59
14.56
14.20
–2.54%
13.84
–2.5%
University support
5.76
5.00
4.44
–12.61%
4.33
–2.5%
National laboratories
8.83
9.55
9.77
2.30%
9.52
–2.5%
TOTAL, All Agencies
167.99
168.34
161.59
–4.18%
165.78
2.6%
a Column added by authors of this report.
b The FY1998 budgets for DOD's mathematical programs are estimates based on DOD's overall budget request for basic research.
* MPS = Directorate for Math and Physical Sciences.
Source: AAAS Report XII: Research and Development, FY 1998, Chapter 20

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FIGURE B-13 Percentages of US academic scientists with federal support, 1993. Source: NSB 1996, appendix table 5-27.
and individual-investigator grants, of which NSF support has accounted for 50%. In 1995, total funding was $9.9 million. The STC program is nearing its end, and DIMACS will need to decide soon whether it will recompete for NSF STC funds. Other large projects supported by NSF include the Institute for Advanced Studies at $1,333,000 and the National Institute for Statistical Science at $1,068,000 in FY1996.
In DOD, the Air Force Office of Scientific Research supports research in subjects such as optimization, signal-processing, probability and statistics, computational mathematics, and dynamics and control. The Army Research Office focuses on the mathematics of materials science, high-performance computing, stochastic methods in image analysis, and mathematical and computational issues in intelligent manufacturing. The Office of Naval Research supports research in the mathematical subfields of applied analysis, discrete mathematics, numerical analysis, operations research, and probability and statistics. The Defense Advanced Research Projects Agency supports research that facilitates the development of technologies needed to meet future military needs. Of particular interest recently have been mathematical aspects of signal-and image-processing, electromagnetics, modeling and simulation of manufacturing processes, and optimized portable application libraries.
The National Security Agency is the nation's largest employer of mathematical scientists. It has a competitive grants program that

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FIGURE B-14a Federal funding of US mathematical research—academic, 1993-1995 average.
FIGURE B-14b Federal funding of US mathematical research—all R&D. Key: NSF = National Science Foundation; DOD = Department of Defense; DOE = Department of Energy; HHS = Department of Health and Human Services; NASA = National Aeronautics and Space Administration; USDA = Department of Agriculture; DOC = Department of Commerce; DOT = Department of Transportation; DOI = Department of the Interior. Source: NSB 1996.

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supports unclassified academic research in discrete mathematics, algebra, number theory, probability, statistics, and cryptology.
The DOE focuses its R&D support on applied computer and computational mathematics, science and technology.
Papers and Citations
Two recent reports—one from Australia and the other from the United Kingdom—have analyzed scientific performance on a comparative basis using research-paper production and citation data. As noted in the Australian Bureau of Industry Economics report Australian Science: Performance from Published Papers (1996), there are a number of problems in using such data, including a bias toward roman script and English-language journals; the greater attention paid to papers by renowned authors than to high-quality papers by less-known authors, technical papers, review articles, and recipes with little frontier science; and self-citation and citation circles.
Other problems occur because journal prestige and variation among disciplines is not considered. Time lag is a problem. There can be differential counting or miscounting due to multiple authorship, multiple field allocation, limits on the number of citations by journal, and changes in the number of journals in the field over time. And authors might use the same material with slight elaborations or break up a major article into several minor ones.
Papers "ahead of their time" and research communicated in nonjournal form (such as working papers, scientific equipment, computer programs, and seminar papers) might not be cited. Other outputs (such as teaching, advice to government, commercial research, and scientific services) are not included in bibliometric analyses.
Thus, citation rates measure visibility but not inaccessible work and not necessarily quality.
Figure B-15 shows the percentage of mathematical-research papers published by US authors relative to authors in 4 other countries that have strong mathematics programs. Figure B-16 compares the number of papers produced by US mathematicians with those produced in the European Community.
The UK report The Quality of the UK Science Base (1997) identifies the following as the top countries according to share of world's citations in mathematics:
United States.
United Kingdom.
Germany.
France.
Japan.

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FIGURE B-15 Percentage of mathematical-research papers published by US authors. Source: NSB 1996, appendix table 5-31.
FIGURE B-16 Number of mathematical-research papers by US and EC authors, 1981-1996. Source: Institute for Scientific Information, National Science Indicators on Diskette, 1981-1996. Philadelphia, PA.

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Another measure that was used in the UK report is the relative citation impact. The relative citation impact for a country in a particular field is defined as the country's share of the world's citations in the field divided by its share of world publications in the field. It can be thought of as a comparison of a country's citation rate for a particular field with the world's citation rate for the field. A relative citation impact (or rate) higher than 1 shows that the country's citation for the field is higher than the world's. According to the UK report, it is a measure of both the impact and the visibility of a country's research (as disseminated through publications) and gives some indication of the quality of the average paper.
The top countries in mathematics according to the relative citation impact index are:
Denmark.
Norway.
UK.
US.
Netherlands.

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