The Panel examined numerous categories of data in trying to benchmark the activity and health of U.S. mathematical sciences in relation to those of other countries. It found a number of indicators (see Appendices) and qualitative observations to be significant, as discussed below.


In comparing the United States with Europe, the Panel found the United States mathematics produce more research papers. Pacific Rim nations trail that of the United States and Europe, but have increased their output significantly since the early 1980s (not shown) (see Endnote 11).

 Publications in the Mathematical Sciences

1989 1991 1992
United States 39.9% 42.1% 38.9% United Kingdom 5.7% 6.3% 5.9% Germany 6.6% 7.0% 6.5% France 7.5% 4.6% 8.5% Other Western Europe 12.7% 12.6% 13.1% Japan 4.3% 4.6% 3.6% Other Pacific Rim 3.9% 3.7% 4.4%

In a separate study by CHI Research, Inc. of ISI journals with emphasis on research publications in pure mathematics, the proportion of papers published by various nations remained stable, with U.S. mathematicians authoring 40-50%. The study noted that international coauthorship was increasing.

Data on the number of Ph.D.s in the mathematical sciences is difficult to determine by geographic region. More generally, for the natural sciences, in 1992 there were 6593 doctorates in Asia, 18,951 doctorates in Europe, and 13,344 doctorates in North America (with the United States producing 12,555 doctorates). However, the report noted, "a declining pool of college-age students in Europe has not resulted in declining numbers of natural science and engineering degrees, as has occurred in the United States." (see Endnote 12)

International Congress Participation

Data on invited speakers at international conferences vary somewhat. In 1994, 50% of invited one-hour speakers at the quadrennial International Congress of Mathematicians were from the United States. In the forthcoming 1998 Congress, 38% of the invited one-hour speakers are from the United States and 48% are from Europe; on the other hand, 48% of the 45-minute speakers are from the United States (1/3 of them having non-U.S. origins) and 36% are from Europe. U.S. speakers accounted for 35% of the plenary speakers at the 1995 International Congress of Industrial and Applied Mathematics.


A tally of the leading awards in mathematics provides another useful benchmark. Of the 16 mathematicians awarded the Fields Medal between 1970 and 1990, eight resided in the United States at the time of the award and 11 currently do. Four of the eight medalists honored in the 1990s reside in the United States, but only one was born in the United States. (The Nobel Prize is not awarded in mathematics; one U.S. mathematician has received the Nobel Prize in economics.) Of the mathematicians who have received the Wolf Prize, an award for distinguished scientists, 14 of 27 are U.S. mathematicians and two others have spent substantial time in the United States.

Subdiscipline comparisons

The Panel, on the basis of its own expertise, undertook a qualitative benchmarking of subdisciplines, which appears in full as Appendix 2. The Panel concluded that the United States has strengths in all subdisciplines, but that it is not the major contributor in some. In several fundamental subdisciplines, including Foundations, Symbolic Computation, and Ordinary Differential Equations, foreign contributions outweigh those of the United States. Overall, however, the U.S. mathematical research enterprise is judged capable of responding to advances occurring anywhere in the world, an ability enhanced by the very high level of interaction among mathematicians worldwide.

Budgetary comparisons

The Panel was unable to provide meaningful country-by-country comparisons of research funding. One reason for this is the wide variations in budgetary and institutional relations between governments and universities. For example, in Canada, Europe, and Japan, faculty are paid on a 12-month basis for both teaching and research. This means that all faculty in these countries have funding for summer research; in the United States, only 35% of active researchers do. This "have" and "have-not" situation is very destructive to the fabric of the U.S. mathematical sciences community and decidedly discouraging to young U.S. researchers. Funding agencies in Europe and Japan play differing roles from that of NSF or the DoD agencies, often using different means to promote the health of science and mathematics within their countries. For example, the Japanese Society for the Promotion of Science spends most of its money to support university science libraries and visits by distinguished foreign scientists.

A few generalizations are possible, however. In Europe, bursaries for students admitted to graduate school meet their full living costs and do not need to be supplemented by money earned through teaching. The docent programs in Germany and Eastern Europe support students completing the doctorate and intending to seek a post in a research university, as do the collegiate fellowships in England. Recently, France has introduced postdoctoral fellowships, as has the European Union. By contrast, the Panel finds that in the United States, there are few research assistantships for the mathematical sciences. The United States is overly dependent on teaching funds to provide support for graduate students in the mathematical sciences, a custom that prolongs the time to degree and makes the field of mathematics less attractive to U.S. students. Also, except for a few NSF funded postdoctoral research fellowships and a few research instructorships funded by universities, postdoctoral research opportunities for mathematical scientists in the United States are exceedingly scarce.

In Asia, overall investment in mathematics is rising. The Asian Tiger countries -- Singapore, Taiwan, Korea, and Hong Kong -- are building strong research universities and research institutes with significant mathematical components. Japan has begun a five-year program to double its funding for basic science. The Chinese National Science Foundation has given the mathematical sciences its highest priority for development.

Research institutes

Institutes and conference centers are important elements in the infrastructure supporting the mathematical sciences — as important to the field as are lab facilities and observatories to physics and astronomers. The Institute for Advanced Study (IAS) in Princeton, N.J. was the first institute to assemble, for short periods (4-12 months), groups of mathematical science researchers. Such institutes have become popular in the mathematical sciences and are viewed as making significant contributions to the advance of the discipline because they enable explorations of new developments, facilitate collaboration among mathematical scientists and assist in the sharing of ideas between mathematical scientists and those of other disciplines. The NSF provides partial funding to IAS primarily for support of young researchers, core funding for three other research institutes (MSRI, IMA, NISS) as well as for DIMACS, a Science and Technology Center that operates much as the other institutes but concentrates on discrete mathematics, algorithms and theoretical computer science. Western Europe has six research institutes plus two conference centers, Canada has three, and the Pacific Rim has several and is planning more. Germany, which is less than a third the size of the United States, has two Max Planck Research Institutes, the Oberwohlfach Conference Center and seven Sonderforschungberich, which are attached to universities and have some aspects of an institute and some of the U.S. Science and Technology Centers. Some institutes have permanent faculty and some, e.g. those whose core funding comes from the NSF, do not. The Centre Nationale des Recherches Scientifiques (CNRS), in France, and the National Academies of Sciences in Russia and Eastern Europe have significant numbers of full-time researchers in the mathematical sciences.

Collaboration with other disciplines

Communication between academic mathematical scientists and other scientists is poor the world over. Many mathematical scientists have a limited vision of their capacity to interact with other scientists. Graduate education is frequently highly specialized. The structure of universities, where decisions on promotions, awards, and salaries are made by disciplinary departments, mitigates against collaboration outside one’s discipline. The difficult and time-consuming task of understanding a second discipline is also an inhibitor. A few U.S. programs, such as the University of Illinois’ Beckman Institute, have been created to advance multidisciplinary research, but few of these efforts have involved mathematical scientists.

Many nations have begun to promote the collaboration of mathematical scientists in multidisciplinary research. England appears to be doing so aggressively, especially under the aegis of the Isaac Newton Institute. In France, there is significant interaction between mathematical scientists and engineers, in part due to the commonality of their secondary and collegiate education. Collaborative initiatives are becoming prominent in countries emphasizing research that drives the economy (e.g., nations of the European Union and those of the Pacific Rim). Many U.S. mathematicians are becoming more involved in multidisciplinary research, probably more than those in other countries, but science and engineering require much more involvement by U.S. mathematicians. Nonetheless, progress is being made. More than 10% of the NSF/DMS budget is invested in projects cofunded with other Divisions and Directorates. The DMS Group Infrastructure Program did provide funding for collaborative programs, and proposals from U.S. mathematicians have been well received by the MPS Office of Multidisciplinary Activities and by the IGERT program. Other recent NSF initiatives promote multidisciplinary research, often with a strong mathematical component, such as the Knowledge and Distributed Intelligence (KDI) Initiative begun this fiscal year.

Interactions between academia and industry

Another trend, most apparent in England and the Netherlands, is to foster interactions between the very different cultures of academic research and the private sector. England has been especially active on this front with its Smith Institute and the OCIAM at Oxford. Activity in the Isaac Newton Institute has led to significant private investment in academic mathematics. Other European countries have begun such programs, but progress has been slow; mixing the cultures of academia and industry is not easy, and there is concern over intellectual property rights and industrial privacy. In Japan, there is little interaction. In the United States, some mathematics departments and institutes (notably the Institute for Mathematics and Its Applications) interact with industrial and financial entities, but they are very much in the minority. Involvement of the U.S. mathematical sciences in the NSF GOALI (Grant Opportunities for Academic Liaison with Industry) program is small compared to that of other sciences, but is beginning to grow. The Division of Mathematical Sciences (DMS) has recently collaborated with DARPA (the Defense Advanced Research Projects Agency) to fund several initiatives that require collaboration by mathematical scientists with other academic and industrial scientists.

The expectation of an academic career, rather than one in industry, is particularly strong in the United States. Some 75% of new U.S. doctoral mathematical scientists anticipate academic positions. In continental Europe, by contrast, many universities have Diploma programs from which students seek nonacademic positions. The English universities have recently introduced a new degree intermediate between the baccalaureate degree and the Ph.D. which serves that purpose. Many U.S. mathematical science departments have discussed a "professional masters" degree, which would emulate the Diploma programs, but few departments have tested or established them.

Undergraduate Education

Although U.S. doctoral programs in the mathematical sciences are extremely strong, U.S. undergraduate programs offer less exposure to mathematics, at less depth, than do those in Europe and Asia. There are two important reasons for this: (i) U.S. undergraduates arrive at the college level with less knowledge of mathematics: 50% or more are unprepared to begin the calculus; (ii) In other countries, undergraduate mathematical science students concentrate entirely on the mathematical sciences and related subjects, while U.S. students spend at least 50% of their time on unrelated subjects. This has meant that graduates of U.S. undergraduate programs must spend time catching up with their European counterparts, extending time to the doctorate. On the other hand, the U.S. system allows students greater opportunities to explore outside their discipline and to change specialties. For example, of the students with A and A+ in high school who enrolled in U.S. undergraduate schools in the mathematics sciences in 1984, 75% switched to other programs while 61% of those earning a mathematical science degree by 1989 were recruits from other fields.(see Endnote 13)

Graduate Education

Ph.D. recipients from the best universities, whatever the country, are at the same level of achievement and preparedness for research. U.S. graduate departments generally offer a wider range of fields in which to specialize than is the case in Europe or Asia, but recent developments in the European Union mean that students in Europe can easily move to other universities, possibly in another country. This allows them greater variety of specialization. European graduate students also receive funding sufficient to cover their living expenses. In the United States, most mathematical science students need to teach to cover their living expenses.

Anecdotal information suggests that a much larger percentage of students who begin a doctoral degree program in the mathematical sciences in the United States fail to earn that degree than is the case in Western Europe. This is especially true for U.S students. A number of U.S. universities report that by the third year, no U.S. citizen remains in their doctoral program.

Attractiveness of the field to the young

During the last decade, the number of U.S. citizens pursuing degrees in the mathematical sciences has suffered a decided decline. Between 1985 and 1995, U.S. freshmen interested in the mathematical sciences declined by 32%, and by 23% among the top students (see Endnote 14). This situation is mirrored in other nations. The Netherlands, Germany, France, Russia, and Poland all have reported significant losses in mathematics enrollment during the past five years. In the last three years, there has been a steady decline in the numbers of applications to U.S. graduate schools in mathematics by Chinese students, which is probably a sign of diminishing interest in that country.

European trends toward applications and centralization

In the European Union, the European Commission has shifted emphasis in the direction of mathematical applications which enhance wealth creation or the quality of life. In the current Fourth Framework Programme, almost the only support of pure science is budgeted in the Human Capital and Mobility program. This program does carry benefits for mathematics as a whole in supporting the movement of postgraduates and postdocs between countries of the EU, in funding conferences and networks, and in supporting an agenda that covers a larger proportion of the needs of mathematicians than those of laboratory-based scientists. However, the shifting of funds from local to central control and from fundamental research to applications does not have the universal support of the mathematics community.


On the positive side, the U.S. mathematics community leads other nations in a large number of subdisciplines and is judged overall to be capable of responding to breakthroughs occurring elsewhere in any area of mathematics. Individually, U.S. mathematicians have won more than their share of prestigious awards in the field.

At the same time, the mathematics community in the United States shares with other nations significant disciplinary challenges including a condition of isolation from other fields of science and engineering, a decline in the number of young people entering the field, and a low level of interaction with nonacademic fields, especially in the private sector.

The Panel gained the sense that mathematicians in the United States feel themselves disadvantaged in comparison with mathematicians of other countries, most notably in public support. This low morale is not evident in Western Europe or the Pacific Rim. The European Union is expanding opportunities and funding for graduate students and postdoctorates. U.S. students are overly dependent on teaching income, which extends time to degree, decreases the attractiveness of mathematics to younger students, and contributes substantially to the fragility of the U.S. mathematical enterprise.