International Benchmarking of US Mathematics Research (1997)

Mathematics is the most formal and rigorous of the sciences, but there is no universally accepted definition of mathematics (and the panel has not attempted to formulate one). In this report, to conform with the 1992 report Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States, published by the NRC Board on Mathematical Sciences, mathematics broadly includes pure mathematics, applied mathematics, statistics and probability, operations research, and scientific computing.

Mathematics has several properties that complicate its assessment:

• Mathematical research generally has a particularly long “shelf life”: large parts of it do not become obsolete. Not infrequently, a much earlier result or insight—even from a previous century—is suddenly the key to solving a modern problem.

• Seemingly diverse subfields of mathematics often form unexpected links.

• Throughout science and engineering, mathematics provides a universal language, tools for analysis, abstractions to guide understanding, and methods for solving problems. Consequently, mathematical research has a tightly coupled, two-way connection with other fields: mathematical discoveries influence research in other fields, and developments in other fields provide new problems for mathematicians to study. However, the contributions of mathematical research are often not labeled explicitly as such.

• Mathematical training is a central part of the education of all US citizens, from kindergarten through high school and college.

Because of the first two properties, we have not defined and separately assessed subfields of mathematics, but rather have focused on mathematics as a whole. Because of the third and fourth properties, our evaluation reaches beyond mathematics to fields and activities where mathematics research has a direct and visible impact. We also recognize that important mathematical research is conducted by people whose affiliations and titles do not explicitly identify them as mathematicians; some of the difficulties associated with this phenomenon are described in SIAM (1995).

US mathematics research was defined by the panel as research in the mathematical sciences conducted by residents of the United States working in US institutions.

The panel would like to mention 5 key caveats with respect to its analysis. First, following its charge from COSEPUP, this report is based on the qualitative judgments of the panel members informed by both their own knowledge and the sparse quantitative data available. The panel has attempted to be as fair and impartial as possible, balancing the points of view of US academic mathematical researchers with views of leading mathematicians from outside the United States, nonmathematical US researchers, and industrial researchers. In addition, the panel

was specifically charged not to make recommendations. With more time and effort, additional opinions and data could have been collected, but such efforts would have been expensive relative to the additional guidance obtained.

Second, given the diversity of mathematical sciences, no panel could represent all the subfields of mathematics. The panel has done its best to review all subfields, but some are undoubtedly better analyzed than others. The findings and conclusions here do not apply uniformly to all areas of the mathematical sciences For example, statistics has enjoyed much stronger employment prospects than most other areas. Thus, some variation in interpretation of these results is required when focusing on specific areas.

Third, many statements in this report are not based on numerical data, mainly because of the paucity of statistics that allow meaningful comparisons among countries. For example, undergraduate and PhD degrees in the United States are not directly comparable with all similarly labeled degrees in other industrialized countries. Even when quantitative information is available, sometimes it contains so many ambiguities that we were reluctant to rely on it. To understand the position in other countries when suitable data were unavailable or unclear, the panel relied on the informed judgments of panel members and colleagues from outside the United States.

Fourth, a substantial amount of mathematics is carried out by people bearing other labels: physicists, chemists, electrical engineers, economists, computer scientists, and statisticians. Some Nobel prizes in these fields have been awarded for mathematical work.

Fifth, had this report been written 7 years ago, the Soviet Union would have loomed large as a competitor of the United States. The collapse of the Soviet Union changed that; Russia is in disarray on all fronts, infrastructure is collapsing, and many of the best former-Soviet mathematicians have found employment abroad, particularly in the United States. At present, mathematics in Russia and Ukraine is a shadow of its former self.

Our first means of evaluation was an ad hoc survey. The panel divided the mathematical sciences into 19 subfields corresponding roughly to the classification used by the International Mathematical Union, and each panel member was assigned a set of subfields in which he or she was knowledgeable. For every subfield, the assigned panel members identified a reasonable number (between 5 and 10) of internationally recognized leaders in several countries. These experts were asked to draw up a list of about 10 speakers (10 is the average number of speakers on a given subfield at an international congress on mathematics), without regard to nationality, for a hypothetical international mathematics congress. Speakers were to be selected because their research was at the leading edge and driving the subfield. Our intent was not to identify the most-famous or best-established people, but rather those whose research was the most important at the time. The demographic results of this informal process were remarkably uniform in each subfield and across subfields. In 17 of the 19 subfields, at least half those named are citizens or permanent residents of the United States. In the other two, about 40% of those named work in the United States.

A second means of evaluating research leadership was to examine the lists of winners of major prizes in mathematics. This measure can be criticized on several grounds: far fewer prizes are awarded in mathematics than in many other sciences, prizes do not uniformly cover all subfields of mathematics, and prizes awarded on an international basis are invariably subject to (unstated) requirements that they not be dominated by a single country. Nonetheless, the panel felt that the clear leadership of US mathematics was demonstrated by looking at two of the most prestigious prizes in mathematics: the Fields medal and the Wolf prize. The Fields medal is presented every 4 years at the International Congress of Mathematicians (ICM) and is by tradition given to mathematicians younger than 40. Of the 38 Fields medals awarded so far, 14 (about 37%) went to people in the United States; more than 40% of Fields medal winners are now working in the United States. The Wolf prizes have been given annually since 1978 for outstanding achievement in physics, chemistry, medicine, agriculture, the arts, and mathematics. More than half (15 of 28) of the recipients of the Wolf prize in mathematics, which is not restricted by age of the recipient, now live in the United States. Moreover, that although there is no Nobel prize in mathematics, the 1994 Nobel prize in economics was shared by a US mathematician, John F. Nash.

A third indicator is the US representation among the plenary speakers at two large and prestigious international mathematics meetings. The ICM is typically attended by about 4,000

mathematical researchers. ICM speakers are chosen by distinguished committees with some attention to balanced geographic distribution among the speakers. At the last ICM, in 1994, 8 of 16 (50%) of the 1-hour plenary speakers were from the United States. In 1990 (Kyoto), 9 of 15 (60%) were American, and in 1986 (Berkeley), 8 of 16 (50%). For the International Congress on Industrial and Applied Mathematics (ICIAM), attended by about 2,500 mathematical researchers, plenary speakers are chosen by a committee representing mathematics societies from 12 countries or regions, and substantial attention is paid to balancing the plenary speakers among those countries. In the 1995 ICIAM, 7 of 20 (35%) of the invited plenary speakers were from the United States, and in 1991, 6 of 20 (30%).

The three indicators just described reflect clear US leadership based on the accomplishments of a relatively small number of stellar mathematicians. We felt that we should also assess the more-robust measure of depth in research leadership; US leadership would be fragile if it depended on the location of a few individuals. We believe that the United States has substantial depth in all subfields of mathematics, on the basis of the following observations:

• In the United States, 183 institutions award PhDs in the mathematical sciences. The 20 or so top-ranked mathematics departments in this group are comparable in research excellence with those at the best universities anywhere in the world.

• As shown in figure 1, US mathematicians consistently produced about 40% of the total research publications in mathematics from 1981 to 1993.

• The American Mathematical Society (AMS), an organization of researchers in the mathematical sciences, has 30,000 members, including 22,000 US members. Non-US citizens join AMS through reciprocity agreements. The Society for Industrial and Applied Mathematics (SIAM), an organization of researchers in applied mathematics and scientific computing, has 6,400 US members in a total membership of 9,000. Annual attendance by US residents at the joint research conferences held by AMS and the Mathematical Association of America (MAA) is about 3,500. Annual attendance by US residents at SIAM research conferences is about 2,300. The meetings and publications of these societies play a major role in disseminating mathematical ideas. Membership in the societies is a rough measure of sustained interest.

As mentioned in section 2, the quality of mathematical research can partly be measured by its effects on closely related activities. We consider four: scientific and engineering research, industry, government, and education. It is difficult to carry out this analysis, because it is hard to document which mathematics-related activities are conducted by mathematicians and which by people trained in other scientific and engineering fields.

Figure 1: Percentage of mathematics-research papers published by US authors

Source: NSB 1996, appendix table 5-31.

Numerous studies have documented in great detail, the strong connections of mathematical sciences research with the physical, biologic, and social sciences, engineering, and medicine. We list a small subset of diverse recent instances in which US mathematical research is closely linked with other fields. That a broad spectrum of mathematics contributed to these examples emphasizes the unity of the mathematical sciences (see section 2).

• Physics has been the science closest to mathematics for the longest time, and their closeness continues today. Many of history's most famous scientists worked in both physics and mathematics (from Newton, Euler, Gauss, Lagrange, Poisson, Kelvin, Maxwell, Poincaré, and Rayleigh to Einstein, Weyl, von Neumann, and Witten). Mathematical physics is extremely active in the United States; many questions of common interest in mathematics and physics arise from quantum mechanics and field theory, general relativity, fluid and multiphase flow, electromagnetic theory, and materials science. Semiconductor modeling, thin films, and signal transmission in optical fibers are three special areas of high mathematical content (see, for example, NRC 1993).

• Biology, physiology, and theoretical and computational chemistry are adopting mathematical approaches, and some eminent US researchers in these fields actively collaborate with mathematicians (NRC 1995a, b). Mathematical Challenges from Theoretical/ Computational Chemistry (NRC 1995a) describes recent successes, such as the development of commercial products from quantitative structure-activity relationships and the insights into molecular structure gained from group theory and topology. Materials scientists and mathematicians have increasingly been forming research partnerships. A 1995 Minerals, Metals, and Materials Society-SIAM workshop on modeling microstructural evolution (Chen and others 1996) produced 70 papers by US mathematician, physicist, and materials scientist coauthors. A joint 1996 initiative of the National Science Foundation (NSF) and the Defense Advanced Research Projects Agency (DARPA) in materials science attracted numerous proposals involving research from all subfields of mathematics.

• Computer science traditionally relies on particular branches of mathematics, certainly logic but also combinatorics and number theory. Recently, the astonishing growth of traffic on the Internet has led to new applications in queuing theory, discrete mathematics, combinatorial optimization, and protocol verification. The report Cryptography's Role in Securing the Information Society (NRC 1996a) discusses the need for mathematical research in, for example, number theory and logic to develop and analyze cryptographic techniques guaranteed to remain secure when faced with continuing gains in raw computing power.

• Imaging has relied on and inspired new mathematics for over 20 years. The basis of the CAT-Scan, the Radon transform, was first described nearly 80 years ago. Mathematical research today in, for example, deblurring and real-time detection of anomalies links directly with medical applications. The report Mathematics and Physics of Emerging Biomedical Imaging (NRC 1996b) states that “many of the envisioned innovations in medical imaging are fundamentally dependent on the mathematical sciences.” The mathematics of imaging is also important in astronomy, biology, geosciences, weather, and cartography.

• Engineering, in all its fields, uses sophisticated mathematics to formulate, analyze, and solve problems, particularly those in which early prototyping and experimentation are too expensive or too risky. The 1995 symposium on “Frontiers of Engineering” held at the National Academy of Engineering highlighted four topics of research in engineering—biotechnology, design and manufacturing, environmental engineering, and information technology; US mathematicians have been active contributors to all four (NAE 1996; SIAM 1995).

• In meteorology, biotechnology, and other “grand-challenge” problems, mathematicians and scientists in other fields have had major successes. The nature of such interdisciplinary collaborations is discussed in two studies—1 by COSEPUP (1996) and the other by the National Academy of Public Administration (NAPA 1995).

That list demonstrates the success of US mathematics research in taking part in research in science, engineering, and medicine. Much of it has been achieved through computing, both locally and through supercomputer centers. There was no feasible way for the panel to determine the interactions of mathematics with science and engineering in other countries. Panel members from outside the United States and our sampling of reports from other countries confirm that all industrialized nations are vigorously encouraging interdisciplinary research in which mathematics plays a part (see section 5.2).

Many large companies in the United States that rely on technical innovation support research and development laboratories that employ PhD mathematicians. In 1995, 21% of doctoral mathematical scientists in the US workforce were employed in private industry —a proportion that has steadily increased over the years (see table B-l). In 1975, 11% of PhD mathematicians worked for industry; in 1985, 19%. That trend might indicate that the flexibility of mathematics PhD programs is increasing. A few US industrial research laboratories —Bell Labs, IBM Research, and General Electric—have been world-famous for their research in mathematics and in other sciences. Major US companies, such as AT&T, Boeing, and General Motors, have maintained active, high-quality groups of research mathematicians. However, it is often difficult to identify mathematical research in medium- or small-scale industrial settings because the organizational structures are cross-disciplinary.

In many widely publicized instances, mathematical research has made substantial contributions to US industry successes. The aerodynamic design of the Boeing 777 was accomplished by computing airflow, pressure, temperature in the exterior of the proposed design; research on numerical methods, adaptive grid generation, and optimization was crucial. The visualization system allowed thousands of scientists, engineers, and customers to work together (Council on Competitiveness 1996). Mathematics was also central to the remarkable animation in the 1995 Disney film Toy Story (SIAM 1996). Another example is the soliton, discovered around 1965 by the mathematicians N.J. Kruskal and M.D. Zabusky, that is now poised to play a key role in the transmission of signals in optical fibers. The mathematical concept of wavelets is an increasingly important tool for the storage and recovery of information, for instance, fingerprints.

Mathematical research has played well-documented roles in many other fields of industry that have strengthened the economic position of the United States, particularly in the automotive, pharmaceutical, communication, and computer industries (NRC 1991). Within the last few years, there has been a great deal of interest by banks and investment houses in employing mathematicians to use mathematical techniques to model and analyze financial trends (COSEPUP 1995); financial mathematics depends on a number of recent discoveries and techniques found in mathematics. Attempts to assess precisely the contributions of mathematical research to industrial problems are hindered by the blurring of disciplinary boundaries throughout most of industry (SIAM 1995).

Especially since the 1980s, US academic mathematicians have created programs and organizations, partially funded by industry, that are designed to involve mathematics research with industrial problems. Examples include the Institute for Mathematics and its Applications at the University of Minnesota and several programs at public and private universities.

Outside the United States, strong connections exist between research mathematics and industry. In France, a rapid development immediately followed World War II in the fields of applied mathematics and scientific computing; many strong researchers took jobs in industry. In Germany, recently created institutes combine academic and industrial partners, for example, in aerospace and automotive manufacturing. Several institutions in the UK support research interactions between mathematicians and industry. Examples are the Rolls-Royce Readership in

Computational Fluid Dynamics at Oxford, the Newton Institute at Cambridge, and the Basic Research Institute in the Mathematical Sciences at Bristol (founded with partial funding from the UK branch of Hewlett-Packard).

During and after World War II, the US government established laboratories in which research in mathematics (and many other disciplines) was supported. In 1993, government laboratories employed 4.5% of the mathematical-sciences PhDs in the United States (NSF 1996a, Table 20). Even with the cuts in defense spending that began in the early 1990s, US government labs have maintained substantial investments in mathematical research. Mathematical research is an integral part of the mission of federal laboratories, such as those of the Department of Energy at Los Alamos, Livermore, Berkeley, Argonne, and Oak Ridge; several Department of Defense laboratories; of the National Security Agency; of the National Institute of Standards and Technology; of the National Center for Atmospheric Research; of the National Aeronautics and Space Administration; of the National Oceanic and Atmospheric Administration; and other agencies such as the Bureau of the Census of the Department of Commerce.

The position of mathematics research conducted in government laboratories varies in other industrialized countries. In the United Kingdom, for example, until the 1980s, the missions of several government laboratories (such as the National Physical Laboratory and AERE Harwell) included basic research, but this has now been de-emphasized or eliminated. There apparently remain a large number of research mathematicians in the Defense Research Agency, but their numbers are classified. Major decreases in defense spending in the UK have led to concomitant reductions in mathematical research in the associated laboratories. Nonmilitary research in government laboratories has been cut; several nondefense laboratories that employed research mathematicians have been privatized and have moved toward commercial, short-term activities, rather than long-term research.

In France, government research laboratories were established in 1939, and mathematics plays a prominent role in several of them (for example, Institut National de Recherche en Informatique et en Automatique (INRIA) and Institute de Hauts Etudes (IHES)). We found no data specifically about mathematics; national laboratories account for 22% of all research in France, and half of French scientists work full-time on research in laboratories run by government agencies. About 20% of all scientists and engineers work in government laboratories (NSF 1996c, p. 36).

International comparisons are difficult because of differences in organizational structures; for example, many government laboratories in France are integrated within universities.

Of all the sciences, mathematics is most closely scrutinized for its function in education, largely because mathematical skills are seen as a key indicator of the scientific and technologic development of the citizenry. The panel stresses the need for a healthy relationship between mathematical research and mathematics education. To bring modern, simpler ways of looking at material and to incorporate appropriate recent research into graduate and undergraduate education in mathematics and other fields, the role of active researchers is crucial. Research

experiences that encourage mathematical skills and innovation are a growing part of US undergraduate mathematics education.

Research mathematicians from all sectors of the US higher-education system, including the most-prestigious mathematics departments, are increasingly involved in improving the teaching of mathematics, at every level of education, to both specialists and nonspecialists. “Service teaching” of undergraduate mathematics to nonmajors is the responsibility of mathematics departments in all US universities, and many leading US academic mathematicians regularly teach elementary courses for mathematics nonmajors. There have always been divided views on this. There was a period when many nonmathematics departments taught these courses; however, when research universities expanded, these departments were happy to shift this responsibility to mathematics departments. Now the reverse is occurring due to the general shrinking of research universities. The best solution appears to be better teaching by mathematicians operating in full cooperation with the departments concerned. New methods based on that idea have led to major improvements in calculus teaching.

Policy-making and professional organizations of US research mathematicians are deeply involved in education. The NRC Board on Mathematical Sciences, the NRC Mathematical Sciences Education Board, AMS, MAA, and SIAM regularly initiate studies and publish reports about various aspects of mathematics education. Collaboration between high schools and research mathematicians is rare but increasing. However, despite the increased efforts of many mathematicians, we have a long way to go, compared to many other countries, in teaching mathematics effectively in kindergarten through high school, as illustrated in the recent evaluation of 8th-grade mathematics instruction in the Third International Mathematics and Science Study (TIMSS) (ED 1996).

It is beyond the scope of this report to discuss the enormous task of improving the mathematical level of the general population. The contribution of research mathematics to this task in only a small though essential part. However, the level of mathematics education at the K-12 level affects the research community in two ways:

• Inadequate high-school education makes it more difficult for college and university instructors to maintain standards and create an intellectually challenging curriculum.

• Because of the lack of interesting exposure to mathematics, fewer students are interested in studying mathematics or pursuing mathematics as a career.

Mathematics education is a major concern in all other countries of which the panel is aware, but we have only anecdotal data about how research mathematicians elsewhere are involved in education. There is a general concern in Europe (except in France) that the quality of education is going down. Japan, according to a recent report, has an outstanding mathematics curriculum (Askey 1993).

The early recognition and training of the mathematically talented is traditional in Russia, France, Hungary, Romania, and Poland. We are happy to report that this tradition is gaining ground also in the United States, through special high schools, special publications for the young, correspondence courses, statewide contests, national and international olympiads, intense summer programs, and so on.

We have identified four leading influences on the success of US mathematical research: attractiveness to foreign talent, quality and structure of graduate education, diversity of the mathematical research enterprise, and financial support for research and infrastructure.

A policy of welcoming distinguished scientists as citizens or permanent residents has enabled the United States to attract from abroad many of the world's best senior mathematicians and promising young mathematicians. Leading scientists, including mathematicians, fled to the United States from the Nazis during 1933-1945 and were followed by a second flood after World War II, from 1945-1955. This concentration of immigrants raised the level of US mathematics to the top. Substantial increases in the number of outstanding mathematicians immigrating to the United States—for example, from China and the former Soviet Union—have also occurred more recently. The pattern of assimilating top talent from all nations outside the United States has been consistent and striking.

America has long been viewed as the “promised land” of freedom, wealth, and opportunity. In addition, mathematicians were drawn to the United States for several practical reasons, discussed below—more and better jobs, high salaries, funding for research, and greater mobility than in any other country. On the last issue, for example, European professors tend not to move once they secure a chair. In France, professors migrate toward Paris. In contrast, it is not uncommon for many of the best US professors to change jobs repeatedly.

Because the brightest students want to study with the best people, the presence, described above, of leading mathematicians at universities throughout the United States has been a major factor in the visibility and appeal of US graduate education since the end of World War II, when the “GI Bill” enabled poor but talented students to take advantage of educational opportunities.

US graduate education in the sciences, mathematics, and engineering has been concurrently boosted by two other influences. After the launch of Sputnik in 1957, the United States adopted various national policies that strongly encouraged the study of mathematics, science, and engineering from elementary through graduate school. The large number of “baby boom” undergraduates entering colleges and universities in the 1960s and 1970s led to substantial expansion in mathematics departments and graduate programs throughout the United States.

One structural aspect of US graduate education in mathematics stands out in comparison with other countries: the much lower level of specialization required to enter a graduate program. For example, it is possible to enter a US PhD program in mathematics without an undergraduate degree in mathematics; such late shifts of major are extremely rare in other countries. This “late start” feature increases flexibility and choice for prospective students. By the time US students begin their dissertation research, they are typically as well prepared as their counterparts elsewhere, but possibly older.

Strong US graduate programs in mathematics have been able to attract top-quality students not only in the United States but also from abroad, showing again the great appeal of the US mathematics environment to foreign talent. A well-known example is the enrollment in the 1980s of a large number of brilliant graduate students from China, Taiwan, Hong Kong, and the USSR. Many of them then remained in the United States after receiving the PhD, and some outstanding mathematicians in the United States today belong to this group.

Before World War II, only a handful of US research universities were distinguished in mathematics; today, at least 2 dozen have uniformly high-quality faculty across most subfields of mathematics, and many more have stellar researchers in particular subfields. Rather than being dominated by a few institutions or individuals, this diffuse structure allows a wide range of mathematicians from across the entire country—and with their institutions—to excel. The diffusion of talent is strengthened by a level of professional mobility that is unmatched elsewhere.

Four independent research institutes in the mathematical sciences contribute to the quality of US mathematics. The Institute for Advanced Study, founded in the 1930s, has long been a major force in pure mathematics, drawing talented people from around the world. The Center for Discrete Mathematics and Theoretical Computer Science at Rutgers, the Mathematical Sciences Research Institute at the University of California, Berkeley and the Institute for Mathematics and its Applications at Minneapolis—all funded largely by the National Science Foundation (NSF)— have been created since 1980. A somewhat different model is the Courant Institute which is integrated with the mathematics and computer science departments at New York University. Those four and the National Institute of Statistical Sciences have increased awareness of research accomplishments, brought leading and junior researchers together, provided support for postdoctoral students, and created ties between different subfields of mathematics—for example, geometry and mathematical physics—and between mathematics and industry. There are many analogous institutes abroad (for example, the Max Planck Sonderforschung and the Oberwolfach in Germany, the Euler Institute in Russia, the Mittag-Leffler Institute in Sweden, the Newton Institute at Cambridge, and the IHES in France).

Since World War II, mathematics in US universities has branched out and expanded into new fields. The invention and development of electronic computers provided a stimulus for mathematics throughout the world. The United States pioneered in the use of computers, thanks partly to the leadership of yon Neumann, and to the success of the semiconductor, software, and computer industry. Computing in England began during the war, primarily from the work of Alan Turing. The rest of the industrial world is catching up, but the United States still dominates.

The field of computer science was spawned jointly by mathematics and electrical