I. INTRODUCTION
For the mathematical sciences, the past fifty years have been a golden era of
great discoveries and new developments spanning the entire spectrum from basic
theory to realworld applications. Accomplishments in basic theory have been
wide ranging, including the development of sympletic geometry, mirror symmetry,
and quantum groups; the discovery of solitons; the proof of Fermat’s
theorem some 350 years after it was stated; the classification of all simple
finite groups (the building blocks for groups generally). Subfields, once
viewed as quite disjoint, are now seen as part of a whole. Striking examples of
applications also occurred, such as; the designing of the Boeing 777 airliner,
which relied on mathematical theory, computation modeling, and powerful
simulation techniques to replace physical testing and to speed the design
process; the use of wavelets as a fundamental tool for fingerprint analysis; the
multipole algorithm in electromagnetic computation; and neural network
algorithms in pattern matching applications. U.S. mathematicians have been at
the forefront of these and other world developments in mathematics.
The mathematical sciences  and all other sciences  are performed in a
world that is changing rapidly. The exploding importance of information to all
sectors of society and the pervasive role of technology in maintaining the
security and prosperity of the nation have placed the mathematical sciences in a
position of central importance. Mathematics provides the context for
communication and discovery in many other disciplines. Competitive pressures
throughout business and government, coupled with the broad expansion of computer
analysis and data management, have extended the applications of mathematics to
every domain of human activity.
The fundamental changes taking place in many areas of science and
technology–especially biology, communications, and computation – are
accompanied by important new problems that cannot be solved without new
mathematics. The modern desires to improve decisionmaking (for example, to
make realtime stock market or hedging decisions) and to understand very complex
problems (for example, to model the impact of human activities on the
environment) will require original mathematical techniques. These deep
challenges, which are vitally important to the nation, offer novel opportunities
for research in mathematics. Without increased openness by mathematicians to
problems of other disciplines, mathematics may miss opportunities to contribute
to and gain from these developments.
The
missed opportunities would extend beyond mathematics. Excellent mathematical
ideas developed in biology would probably not be helpful to financial modeling,
say, because the jargon used in the two fields is very different. Mathematics
can help standardize developments in one field for use by others. Two examples
include finite element methods, developed by structural engineers, and sparse
matrix methods, developed by power systems engineers and economists. In both
cases, standardization and generalization of the results by mathematicians have
led to applications in many other fields. The following graphics further
illustrate this point.
In an ideal world, mathematics has a clean flow from its core out to
applications and from applications back to the core. This flow facilitates the
adaptation of mathematical concepts from one field, such as physics, to
economics, and vice versa.
When the mathematical sciences retreat from such multidisciplinary
involvements, mathematics suffers from lack of enrichment by the ideas and
challenges of other disciplines. The other disciplines also suffer, for two
reasons:
In 1993, there were 22,820 doctoral mathematical scientists employed in the
United States. It was estimated(see Endnote 1) that
of these mathematical scientists, 14,670
(64.3%) were employed at universities and 4year colleges (6,427 at doctorate
granting universities), 5,160 (22.6%) in industry, and 960 (4.2%) by the Federal
government. Of those active in research, 35% reported receiving Federal
funding(see Endnote 2). It is worthwhile to compare with other sciences?
1993 Fulltime Academic Doctorate Faculty 


Total Number 
Active in Research 
Receiving Fed Support 
% Active with Fed Support 
Biological Sciences 
68194 
51767 
36143 
69% 
Physical Sciences 
28644 
20029 
13463 
67% 
Mathematical Sciences 
15475 
9517 
3250 
35% 
The mathematical sciences are divided into two largely independent groups:
1) academic mathematicians, and 2) users of mathematics, both inside and outside
the university community. The weak coupling between these two groups is a
central problem for mathematics worldwide (as it is for some other areas of
science and technology). To enhance the vigor of both groups, it is vital that
the creators and users of mathematics be more strongly connected. Excellent
mathematics, however abstract, leads to practical applications. In turn, hard
problems in nature stimulate the invention of new mathematics.
Traditionally, abstract mathematicians follow natural paths of inquiry toward
the development of new concepts and new theories. They are often influenced by
problems arising outside of mathematics, but as often, perhaps more often, they
are driven by the inherent beauty and inner consistency of the results. It
might be years or decades before such concepts find application — if they
ever do. The physicist Eugene Wigner marveled as to "the unreasonable
effectiveness of [abstract] mathematics in the natural sciences;" nowadays,
one would add finance and management. Arthur Jaffe (David I
(see Endnote 3), p. 120) explains this as follows:
"Mathematical ideas do not spring full grown from the minds of researchers.
Mathematics often takes its inspirations from patterns in nature. Lessons
distilled from one encounter with nature continue to serve as well when we
explore other natural phenomena." Even if one values mathematics only for
its role in applications, one must value these basic abstract investigations
because they provide the foundation upon which applied and computational
mathematics, as well as statistics and computer science, are based.
The United States excels in abstract mathematical research. To make the best
use of this strength, there is a need for a faster flow of knowledge between the
creators and users of mathematics. The progress of science deteriorates when
mathematicians or the users of mathematics must develop new knowledge "on
demand." Only when the doors of communication are open wide can the
mathematical enterprise function at full potential. Users benefit from quick
access to known mathematics, and mathematicians are challenged by new
formulations and questions from users.
Strengthening the connections between the creators and the users of
mathematics, while maintaining historical proficiency in pure mathematics, is
the most important opportunity now open to the National Science Foundation in
its support of the field. It is imperative to find new ways to speed the flow of
mathematical discoveries between academic mathematical scientists and those who
use their results, and between different fields of the mathematical
sciences.
The third community of the mathematical enterprise consists of the
students. During this time of rapid change, the way students are educated must
keep pace with quickly shifting realities of vocation and employment. There is
a need to broaden and extend the curriculum for future mathematical scientists,
as well as for the users of that science, to make them more flexible and ensure
that the two groups are equipped to interact effectively in the interests of
their disciplines and of society. It will require ingenuity on the part of the
mathematical scientists and of other disciplinarians to do so, without losing
depth.
The importance of the mathematical sciences to society dictates that we adapt
the way we prepare the next generation of mathematical scientists to face new
realities, which include increasingly multidisciplinary work and the extension
of the mathematical sciences into other fields. Tomorrow’s mathematical
scientists must be educated in new ways if they are to contribute to the
mathematical enterprise and to society across the full range of employment
opportunities and professional challenges.
(see Endnote 4)
The Purpose of This Report
This report is part of the National Science Foundation’s response to
comply with the Government Performance and Results Act (GPRA). The act
requires an evaluation of how well the Foundation has met its strategic goals,
which are:
In March 1997, the Division of Mathematical Sciences (DMS) of the National
Science Foundation (NSF) convened a Senior Assessment Panel and charged it to
undertake an assessment of the Mathematical Sciences in the United States. The
Panel was asked to undertake the following tasks: to assess the health and
position of leadership of the United States mathematical sciences; to evaluate
the connections of mathematics with the other sciences, technology, education,
commerce, and industry; to appraise the performance of mathematics in the
education and training of professional mathematical scientists; and to make
recommendations for action. This report describes the results of the
Panel’s work.
The principal strategy used by NSF to achieve its objectives is to support
research and education in universities. The report therefore focuses on NSF's
performance in supporting the performance and teaching of academic mathematical
sciences and in encouraging interactions between academic mathematicians and the
users of mathematics.
The Process Used by the Panel
The Panel consisted of leading mathematicians drawn largely from outside
the United States and individuals from important U.S. stakeholder communities
that are strongly dependent on mathematics (science, technology, education,
government, and finance). None had received recent NSF funding in mathematics.
Members who were mathematicians brought to the Panel their expertise in the
various subdisciplines, the progress of international research, and the means
used by other nations to support mathematical research. Members of the
stakeholder communities provided judgments on their mathematical needs, on
opportunities for mathematicians in these communities, and on the effectiveness
of mathematical knowledge in service to society.
The Panel enjoyed staff support from the Division of Mathematical Sciences,
which provided data, analysis of data, and a wealth of reports. Members of the
Panel met four times (March 2022, June 57, September 5, and September 22,
1997) at the National Science Foundation Headquarters in Arlington, Virginia to
study data and reports, to discuss appropriate criteria for making assessment,
and to formulate recommendations of a qualitative nature.
The Panel also discussed, at considerable length, the vital importance of
mathematics in K12 education in the United States. Not without regret, the
Panel concluded that, given its composition and expertise, it should refrain
from making assessments or recommendations in this area. However, the Panel
wishes to emphasize its sense of the essential importance of K12 mathematics to
the wellbeing of the United States, to underscore the assessments and
recommendations made in previous reports, and to affirm that much needs to be
done in this area. The education of present, and, more importantly, future,
teachers, will be the key to the improvement of K12 mathematics. Because the
education of teachers is the task of current and future university and college
mathematicians, the quality of graduate students in mathematics and the
education they receive will be crucial to the improvement of K12
mathematics.
The Structure of the Report
This report includes a benchmarking comparison of U.S. mathematics
with mathematics in Western Europe and the Pacific Rim. The report combines
detailed benchmarking of the individual fields of the mathematical sciences with
a strategic analysis of the role of mathematics in building and maintaining U.S.
strength across the range of fields that comprise and use mathematics. This
range extends from fundamental discoveries in mathematics to the application of
mathematics in other scientific and engineering disciplines and in such
"user" areas as government, finance, and manufacturing.
The methodological sections of the report describe the
procedure followed by the Panel (Chapter III) and the means and data used for
making benchmarking comparisons (Chapter IV and Appendix 2).
The substantive results of the Panel’s work are presented as
follows:
The report also includes (in Appendices) data which support the findings and
materials that address many elements of the report in greater detail.