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.
(Russias 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.
Numbers of Full-time Graduate Students and Source of Support (see Endnote 16)