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.