V. FINDINGS

U.S. mathematics has been and remains distinguished. Academic mathematicians in the United States have been very successful in creating new fundamental concepts. This excellence has been clearly and repeatedly recognized in the large number of professional awards received by U.S. mathematical scientists. In addition, U.S. mathematical scientists have been quick to develop and extend new concepts created elsewhere. There is no question that the U.S. academic community has been among the strongest in the world since World War II and remains so today.

The success of mathematicians from U.S. graduate programs in the mathematical sciences attracts students from every country, including Western Europe. Generally, U.S. graduate programs are larger and broader than those available elsewhere, which adds to their appeal.

Although the United States is the strongest national community in the mathematical sciences, this strength is somewhat fragile. If one took into account only home-grown experts, the United States would be weaker than Western Europe. Interest by native-born Americans in the mathematical sciences has been steadily declining. Many of the strongest U.S. mathematicians were trained outside the United States and even more are not native born. A very large number of them emigrated from the former Soviet Union following its collapse. (Russia’s strength in mathematics has been greatly weakened with the disappearance of research funding and the exodus of most of its leading mathematicians.) Western Europe is nearly as strong in mathematics as the United States, and leads in important areas. It has also benefited by the presence of émigré Soviet mathematical scientists.

It is worth noting that prior to World War II, the United States lagged well behind Europe in mathematical research. After the war, the presence of German refugees, growth of federal investment in science, and expansion of the university system all fueled the powerful growth of U.S. mathematical sciences. But federal funding has not kept pace with the growth in the size of the mathematical science community, and the growth of the university system has stopped in all but two or three states. The impetus that led to U.S. leadership in the mathematical sciences no longer prevails.

The U.S. lead in the mathematical sciences is declining in some subfields, which are further endangered by a lack of young people in several areas where U.S. leaders will soon retire. An example is Foundations, which during the past two decades has failed to attract enough young mathematicians to contribute to or respond to advances in other countries. As a result, the average age of the leaders in mathematical logic in the United States is above 50 years (even higher in proof theory), significantly higher than in other fields of mathematics. In symbolic computation, a subarea where Europe is strong, the United States has considerable commercial presence but little academic depth. The separation of computer science from the mathematical sciences in U.S. universities has had a negative impact on combinatorics, discrete mathematics, symbolic computation, and other areas. It has also resulted in the training of computer scientists who have limited mathematical backgrounds.

U.S. strength in mathematics rests heavily on mathematicians who have come from outside the United States. Many distinguished U.S. mathematicians who have received international awards were neither born nor trained in the United States. An increasing number of all U.S. academic mathematicians received their early training outside the United States. A yet-to-be-published study by COSEPUP reports that 21% of tenured faculty and 58% of tenure-track faculty at 10 highly rated mathematics departments received their undergraduate degree outside the United States. This situation is not confined to the highly rated departments. The citizenship of full-time mathematics faculty with Ph.Ds hired during 1991-92 by U.S. universities and colleges were as follows: 37% were U.S. citizens, 16% were Western Europeans, 13% were Eastern Europeans, 22% were Asiatics and 12% were citizens of other countries (see Endnote 15). Of these hires, 26% came directly from overseas. U.S. industry constantly seeks to recruit mathematical scientists outside of the United States and sends abroad much work which requires mathematical skills. Although mathematics is a very international field, this trend suggests that U.S. academic mathematics is not as robust as suggested by its high level of academic recognition. Unless the United States can make mathematics more attractive as a career to U.S. citizens, several developments threaten to push the supply of trained mathematicians below that needed by academia, let alone by industry: (i) the collapse of the Soviet Union as a producer of highly trained mathematicians; (ii) the pressure on U.S. graduate students who are Chinese citizens to return to China after completing their studies; (iii) worldwide decline of student interest in mathematics; and (iv) competition by Western Europe to retain first-rank European-trained mathematicians.

Lack of financial support thwarts the careers of many young mathematical scientists. Not only is there a lack of sufficient postdoctoral fellowships for new doctorates in the mathematical sciences, but few young researchers are successful in obtaining research grants. With only 35% of academic research mathematical scientists receiving such grants, it is exceedingly difficult for young researchers to pursue careers in research. This lack of support, especially when compared with support for young researchers in the physical, biological, and engineering sciences, discourages young mathematicians, many of whom have left academia for Wall Street and other nonacademic fields. This loss of young researchers has the potential to undermine future U.S. strength in the mathematical sciences.

Finding 2: Interactions with Users of Mathematics

Academic mathematics is insufficiently connected to mathematics outside the university. One of the greatest — and most difficult -- opportunities for academic mathematics is to build closer connections to industry. The poor communication between the university and industry cannot be blamed exclusively on either party. Academic mathematics is an intense, focused, and sometimes solitary intellectual activity. By contrast, mathematical scientists in industry tend to work in teams, usually addressing analytical challenges rather than developing new concepts. A further difficulty is that most companies do not have a separate division devoted to mathematics or, indeed, the job classifications of "mathematician" or "statistician." This situation, which evolved in an era when mathematics was much less pervasive in industry and less central to economic competitiveness than it is today, makes it difficult for academic mathematicians to contact their industrial counterparts.

It is clear that both industrial and academic mathematics must reach out to one another if the two are to interact effectively. Industry could enhance communication by organizing its mathematicians so they can be easily identified and contacted by their university colleagues. Academic mathematicians will have a larger perspective of their discipline if liaisons can be developed between industry and academics, as exists in chemistry, pharmacology, and engineering. Good models exist, at Boeing, Lucent, IBM, AT&T, the applied mathematics groups in the pharmaceutical companies, and the financial industry, where mathematical scientists are easy to identify, work on well defined and sophisticated mathematical problems, and welcome faculty consultants and student interns. Effective interactions like these are creating new specialties in applied mathematics, such as financial engineering and computational drug design.

Academic mathematics could interact fruitfully with other disciplines in ways which are often obscured by the inward focus of mathematics and science departments. We believe that mathematics is a field of almost unlimited opportunity -- provided that it looks outward toward its interfaces with other fields. The opportunities at disciplinary interfaces -- for example, in bioinformatics, communications networks, and global climate modeling -- are not only important in a practical sense, but they are also intellectually challenging. By tradition, however, academic mathematicians are reluctant to seek such interactions — as are members of other science and engineering disciplines. This reluctance means foregoing much professional stimulation and precludes the solution of problems that require new concepts and techniques in mathematics. This is less the case with statisticians, who have always worked with others.

A narrow vision of mathematics in academic departments translates into a narrow education for graduate students, most of whom are oriented toward careers only in academic mathematics. Although it may be appropriate for some departments to maintain a "pure" academic focus, a higher level of interaction with other disciplines is essential for the mathematical enterprise as a whole as it is for other disciplines.

The structure of universities mitigates against multidisciplinary research. While the above finding criticizes mathematical scientists for not collaborating more actively with other scientists and engineers, part of the fault lies with the organization and culture of universities, here and abroad, which restrains collaboration across scientific boundaries. The academic award system does not encourage collaboration; in fact, individuals who straddle fields reduce their chances of tenure. Given the growing need for multidisciplinary research, forward-looking universities must find ways to break down the disciplinary walls that inhibit collaboration.

Scientific problems of the future will be extremely complex and will require collaborative mathematical modeling, simulation, and visualization. Mathematical modeling and experimental observation go hand in hand. Modeling, which is built on both observation and theory, leads to further experiment and more precise measurements. Good modeling demands the most relevant mathematical theory. It is nearly impossible for a single researcher to maintain sufficient expertise in both mathematics/computational science and a scientific discipline to model complex problems alone. A well defined model requires multidisciplinary teams that include both mathematical and disciplinary scientists. Each member of such teams will need to understand the expertise of the other members well enough to recognize their competencies and limitations. Developing this degree of breadth takes time and commitment from all members. Funding agencies need to provide financial support that recognizes and rewards multidisciplinary activities and to recognize the long time required to become competent in such work.

The existence of physically separate departments of "applied mathematics" and "pure mathematics" has often perpetuated a narrow view of what mathematics can or should be applied. Historically "applied mathematics" has meant the application of the subarea "analysis" to problems in the physical sciences and engineering. This view of applied mathematics has greatly limited the application of all of mathematics to real world problems. With the burgeoning opportunities now available, the view must be that every area of mathematics can contribute and benefit from interactions with other disciplines and with industry and commerce. The division into "pure" and "applied" has been highly destructive to the discipline and must be healed.

Finding 3: Educating the Next Generation

U.S. graduate programs in the mathematical sciences, especially the top 25, are considered to be among the very best in the world, attracting many students from other nations. For the last decade, more than 50% of Ph.D. degree recipients in the mathematical sciences from U.S. graduate schools received their undergraduate degrees from outside the United States. The graduates of the U.S. graduate problems have excelled at what they have been educated to do. Their publications are deeper and more numerous than those of earlier generations.

Despite the excellence of the U.S. graduate programs in the mathematical sciences, the students of these programs are provided substantially less federal funding than are students of the other sciences. They depend almost entirely on teaching assistants stipends and on their own resources. This treatment sends a clear message that the United States does not place high value on the mathematical sciences. This is certainly not the case in Western Europe.

      

      Federal     Institutional      Other      Self Support

      Support        Support        Support

      

      Biological Sciences        16593           20805           3923          6962

      Physical Sciences          10353           14858           2079          1602
      Mathematical                1291            9169            478          2484
      Sciences

      

      

There is a paucity of research assistantships and postdoctoral positions in mathematics. The data shows that in 1995, (see next page), new doctoral recipients in the mathematical sciences were much more likely to move directly into teaching appointments than their counterparts in the physical or biological sciences, and much less likely to obtain research assistantships or postdoctoral fellowships. The "bench science" nature of the physical and biological sciences explains only a part of this disparity. The lack of an adequate number of postdoctoral fellowships slows the professional development of young mathematical scientists and reduces the attractiveness of the field. There is a clear need to decrease significantly the use of TAs in departments of mathematics and to increase research assistant positions as well as postdoctoral fellowships.

Graduate applications in the mathematical sciences have declined. Since 1985, the number of U.S. undergraduate majors has declined, and since the early 1990s, the percentage of U.S. mathematics majors going on to graduate school in the mathematical sciences has dropped. The American Mathematical Society reports(see Endnote 18) that the number of U.S. citizens enrolling in graduate school for the first time dropped 6.9% in fall 1997 from 1996; it dropped 8.9% in subfields other than statistics, applied mathematics, and operations research. Both the number of U.S. and foreign applicants have declined by 1/6 in each of the last three years. Since 1992, many of the best mathematics departments have actually reduced graduate enrollments (6.3% from 1996 to 1997), in part because of lack of graduate student funding and in part because of a tight job market. Even so, a number of these departments failed to meet enrollment targets for first-year students for fall 1997, indicating not only a decline in interest, but also in the quality of the applicant pool.

Annual Percentage Change in Full-time, First year Graduate
Students in Mainline Departments(see Endnote 19)

Good mathematics students can and do excel in many disciplines and are heavily recruited by others. The lack of research assistantships for mathematical science students puts the mathematical science departments at a disadvantage when recruiting.

Careers in mathematics have become less attractive to U.S. students. At least seven factors contribute to this change: (i) Students mistakenly believe that the only jobs available are collegiate teaching jobs, a job market which is saturated (more than 1,100 new Ph.D.s compete for approximately 600 academic tenure-track openings each year); (ii) Academic training in the mathematical sciences tends to be narrow and to leave students poorly prepared for careers outside academia; (iii) Neither students nor faculty understand the kinds of positions available outside academia to those trained in the mathematical sciences; (iv) Within the United States, the number of graduate students on fellowships and research assistantships is significantly lower in the mathematical sciences than it is in the other sciences, so students must depend on teaching assistantships for financial support. For this reason, students perceive that other disciplines are more highly valued by society; (v) Universities depend on graduate students to teach undergraduates; too often, graduate students are exploited as cheap labor rather than nurtured and mentored as students. This custom extends the time to degree, which now averages more than 7 years, and places entering freshmen under the tutelage of novice graduate instructors. The uneven quality of this teaching diminishes the appeal of mathematics as an area of study; (vi) The United States offers limited opportunities for postdoctoral work in the mathematical sciences. In most fields of science in the United States, and in mathematics in Europe, postdoctoral experience is valued as an opportunity to increase one’s breadth of training and to develop independent research. The absence of postdoctoral funding makes mathematics less attractive and decreases opportunities for new Ph.D.s to develop their talent; and (vii) Funding for U.S. research mathematical science faculty is substantially below that for other sciences and engineering fields; sending a very negative signal about the status of the discipline in the United States.

The curriculum in U.S. institutions for undergraduates needs to be strengthened, broadened, and designed for more active participation by students in discovery. The faculty needs to be more involved with students to enable them to experience the joy of mathematical discovery. Also, U.S. students motivated by a desire to apply mathematics often find they must study in departments of economics and engineering, where they may not receive a firm foundation in mathematics.

Numbers of U.S. bachelor degree recipients in the mathematical sciences increased by 4% from 1989-1995. In that period the number of students receiving conventional mathematics degrees declined by 6.4% and this decline has continued. Several smaller trends moved against this decrease: from 1994-1995, the number of B.S. degrees in mathematics education and statistics doubled, while the number of actuarial majors tripled. (see Endnote 20)

The mathematical sciences play an essential role in precollege education. As mentioned above, the Panel has chosen to defer to other reports on K-12 education, and to underscore the assertion by these reports that mathematics education is crucially important in preparing the workforce of the future. In the United States, the present situation is not acceptable. The U.S. mathematical research community has the opportunity and the obligation to participate in the education of precollege teachers of mathematics, especially those planning to teach at the high school and community college level.

There are mounting pressures to reduce the time allotted to mathematics in the undergraduate education of scientists, engineers, and business majors. At the same time, there is a desire, for pedagogical reasons, to better integrate mathematical concepts and methods with those of the scientific and engineering disciplines. Unless appropriate steps are taken to respond to these felt needs, there is a distinct possibility that "service teaching" in mathematics will be "absorbed" by other disciplines, and that it will no longer be controlled by or provide employment for the mathematical research community. These possibilities are already manifest in some engineering programs. The academic mathematical community must cooperate with faculty from other disciplines in responding to needs for more effective, collaborative instruction. They must respond to a new trend in pedagogy that makes use of scientific problems to motivate mathematical theory and to increase the use of computers, especially with regard to visualization. Students need a sound understanding of mathematics as a basis for life-long learning. The dual nature of mathematics as a theoretical field of its own and as a discipline intrinsically linked to applications must be better reflected in the undergraduate curriculum.

There are exciting mathematical science career opportunities outside the academy. No hard data on career paths for mathematicians outside the academy were available to the Panel (see SIAM Report (op cit)). To gain insight, however, a member of the Panel undertook a survey of financial firms through the International Association of Financial Engineers to assess the use of mathematics in finance, principally by banks and broker-dealers. The survey, which can be found at http.//:www.cmra.com, indicates the high importance of quantitative and mathematical methods to this industry, with 40% of the respondents reporting heavy dependence and at least 75% moderate dependence. One-third of the respondents reported that more than 10% of their professional staffs were quantitative professionals, and more than 80% of the respondents planned to hire additional quantitative professionals. More than 70% of the respondents stated that the proportion of quantitative professionals on their staffs had increased in the past 15 years. Notably, 63% of the respondents stated that academic quantitative research should focus more on industrial applications and that there is not enough communication between theoretical mathematical research in academia and applied mathematical research in the practitioner world.This survey confirmed the Panel’s judgments that (i) there are many exciting opportunities for mathematical careers outside academia, and (ii) many of these opportunities are foregone because of poor communication and academia’s undervaluation of nonacademic employment. With a reorientation of curriculum and of employment expectations, possibilities for a career outside academia are very bright for mathematically talented and well-trained individuals.

Summary of Findings

There is a danger that academic mathematics will be perceived as solely a scholarly endeavor rather than as a full participant in the explosion of scientific and technological advances that began five decades ago. If this danger materializes, mathematics, irrespective of its great beauty, will not compete successfully for resources with disciplines whose goals are more obviously relevant to the solution of societal problems. Were this to occur, mathematics would be relegated to a more limited role, similar to that of some humanistic disciplines. Many intellectual activities of a mathematical nature would be absorbed by the other sciences and engineering because of their importance to applications. As described in Chapter II, this would result in major losses of synergy between academic mathematics and the users of mathematics -- to the detriment of both. It would also reduce our effectiveness in solving critical problems of society. It would do this by distorting the common sequence of events by which a scientific or technological problem is approached. That is, a problem may be generated by theory or experiment, but at some point it moves into mathematical questions; the refinement of these questions into a mathematical problem and the solution of this problem require substantial effort of an exclusively mathematical nature. Conversely, abstract mathematical ideas, which are developed without an application in mind, often prove to be the key to the formulation and solution of scientific and technological problems.

Academic mathematical science must strike a better balance between theory and application. At one extreme, a narrowly inward-looking community will miss both the opportunities that arise outside the mathematical sciences and the opportunities that are part of scientific and technological developments. At the other extreme, an exclusive concern with applications and collaborative research would severely limit the mathematical sciences and deprive the scientific community of the full benefits of mathematical inquiry. At present, the balance is tilted too far toward inwardness.

For U.S. mathematical science to thrive, the discipline must be made more attractive to young Americans with bright and inquisitive minds. The reforms suggested above will enrich the discipline and expand students’ abilities to contribute to their profession and to society. In addition, mathematics students must be funded at a level comparable to students in the other sciences and engineering, and they must be made aware of the range of employment opportunities available outside academia. Young academic researchers must be supported to enable them to reach their full potential.