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Appendix A
Executive Summary of the 1984
Reports
I. BACKGROUND
The Ad Hoc Committee on Resources for the Mathematical Sciences
was established in June 1981 by the National Research Council's As-
sembly of Mathematical and Physical Sciences to review the health
and support of mathematical research in the United States. Prelimi-
nary evidence presented to the Assembly by its Office of Mathematical
Sciences had suggested that in the nation's major universities external
support for mathematics had lagged considerably behind correspond-
ing support in other fields of science. The evidence was sufficiently
dramatic that the charge to the Committee contained more emphasis
on financial support than is usual for a review of the health of a
scientific fielcl. Committee members with a range of scientific inter-
ests and experience were chosen to ensure that this review Woolf] be
carried out with a broad perspective.
Early in our Committee's deliberations, we came to three important
realizations:
· Mathematics is increasingly vital to science, technology, and
society itself.
Paradoxically, while mathematical applications have literally
Reprinted from Renewing U.S. Mathematics: Critical Resource for the Future (Na-
tional Academy Press, Washington, D.C., 1984), pp. 1-10.
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78
APPENDIX A
exploded over the past few decades, there has been declining atten-
tion to support of the seminal research which generates such benefits.
.
Opportunities for achievement in mathematical research are at
an all-time high, but capitalizing on these will require major new
programs for support of graduate students, young investigators, and
faculty research time.
These perceptions guided the activities of our Committee as we pur-
sued our charge.
II. THE MATHEMATICAL SCIENCES
A. Strengths and Opportunities
The period since World War II has been one of dazzling accomplish-
ments in mathematics. The flourishing of the discipline has run hand-
in-hand with burgeoning applications, which today permeate the theo-
retical fabrics of other disciplines and constitute important parts of
the intellectual tool kits of working scientists, engineers, social scien-
tists, and managers. These developments were nurtured by coopera-
tion between the universities and the federal government, and fueled
by a national commitment to strengthening scientific research and
education. The injection of federal funds into universities, combined
with a pervasive sense of the importance of research, attracted num-
bers of the best young minds in the country into science and mathe-
matics and propelled the United States into world leadership in the
mathematical sciences.
The field expanded and diversified enormously during this period.
Mathematical statistics matured. Operations research was born.
Mathematics in engineering flowered with prediction theory, filter-
ing, control, and optimization. Applied mathematics extended its
reach and power, and the discipline of mathematics grew at a breath-
taking pace.2
Since World War II, the impact of mathematics on technology and
engineering has been more direct and more profound than in any
historical period of which we are aware. When we entered the era of
high technology, we entered the era of mathematical technology.
Historically, the work of Wiener and Shannon in communication and
information theory highlights the change. The mathematical under-

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APPENDIX A
pinnings of the computer revolution, from van Neumann onwarcl, and
the sophisticated mathematical design of the fuel-efficient Boeing 767
and European Airbus airfoils further exemplify the increased impact
of applied mathematics.
The discipline of mathematics also aclvanced rapidly and contributed
to the solution of problems in other fields of science. Fundamental
questions in algebra, geometry, and analysis were addressed with
ever-increasing conceptual generality and abstraction; new interac-
tions between parts of contemporary mathematics and physics, as in
gauge field theory, remind us of the payoff of mathematics for other
sciences. Indeed, in the span of little more than the past two years we
have seen four Nobel Prizes awarded to U.S. scientists for largely
mathematical work, much of it employing mathematical structures
and tools developed over the last few decades: Chan~lrasekhar in
astrophysics, Cormack in medicine (tomography), Debreu in econom-
ics, and Wilson in physics.
Major research opportunities for the future exist in the study of non-
linear phenomena, discrete mathematics, probabilistic analysis, the
mathematics of computation, the geometry of three- and four-dimen-
sional manifolds, and many other areas.3 The infusion of mathematics
into society will continue and accelerate, creating further opportuni-
ties and increased demand for mathematical scientists.
B. Prospects for the Future
There are reasons to be quite concerned about the future, in spite of
current vitality and past achievements. In mathematics, the country is
still reaping the harvest of the investment of human and dollar re-
sources made in the mid-to-late 1960s. Investments since that time
have not been adequate to assure renewal of the field, to provide the
seminal work supporting expanded applications, or to pursue the
remarkable opportunities in prospect.
During the past few years, concern about the future of mathematics
has been reflected in an unprecedented probing and searching within
and by the mathematical sciences community. The state of mathemat-
ics, its applications, and its future promise have been assessed in:
· the report of the COSEPUP Research Briefing Panel on Mathe-
matics presented to OSTP and NSF
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APPENDIX A
.
its supplementary report to DOD and the DOD~University Forum
· reports to the NSF Advisory Committee for the Mathematical
Sciences by J. Glimm, on the future of mathematics, and I. Olkin and
D. Moore, on statistics
· the G. Nemhauser/G. Dantzig report on research directions in
· ~
operations science
· the report of the NSF/DOD Pane] on Large-Scale Computing
in Science and Engineering
.
reports of the NRC Committees on Applied and Theoretical
Statistics and on the Applications of Mathematics.
In all of these the theme recurs: in mathematics itself and in its capa-
bilities for application there is a multitude of major opportunities, but
the resources, people, and money are not available to capitalize on
them.
Our Committee has found the support situation in mathematics to be
worse than the preliminary evidence suggested:
Since the [ate 1960s, support for mathematical sciences research in the
United States has declined substantially in constant dollars, and has come to
be markedly out of balance with support for related scientific and technologi-
cal efforts. Because of the growing reliance of these efforts on mathematics,
strong action must be taken by the Administration, Congress, universities,
and the mathematical sciences community to bring the support back into
balance and provide for the future of the field.
III. THE WEAKENING OF FEDERAL SUPPORT
A. How It Happened
In many ways, the history of support for mathematical research re-
sembles that of other sciences: a rapid buildup of both federal and
university support through the 1950s; some unsettling changes in the
early-to-mid-1960s; then a slackening of federal support in the late
1960s and early 1970s, because of increased mission orientation of
federal R&D and reductions in federal fellowships; and finally, more
than a decade of slow growth.
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APPENDIX A
However, mathematics faced special problems, owing to its concentra-
tion at academic institutions and its dependence for federal support
on two agencies: the National Science Foundation (NSF) and the
Department of Defense (DOD).4 In the mid-1960s, increased focus on
mission-oriented research (a change accelerated by the 1969 Mansfield
Amendment) caused DOD to drop nearly all of its support of pure
mathematical research and parts of basic applied work as well. Then
dramatic reductions in federal fellowships beginning in 1971 removed
virtually all federal support of mathematics graduate students and
postdoctorals. Compensation for these two types of losses could only
be made at NSF, but at NSF constant dollar support of mathematical
research decreased steadily after 1967. We estimate the [ass in federal
mathematical funding to have been over 33% in constant dollars in the
period 1968-73 atone; it was followed by nearly a decade of zero real growth,
so that by FY 1982 federal support for mathematical sciences research stood
at less than two-thirds its FY 1968 [eve! in constant dollars.5
While federal support for related sciences also dipped in 1969-70,
these sciences received (constant dollar) increases in NSF funding in
the years 1970-72 and thereafter, as well as support from other agen-
cies; mathematics did not.6 This resulted in the present imbalance
between support for mathematics and related sciences:
Comparisons of Federal Support in Institutions of Higher
Education for Three Fields of Science, 1980
Mathematical
Chemistry Physics
Sclences
Doctoral scientists in R&D
Faculty with primary or secondary
activity in R&D
Faculty in R&D federally supported
Approximate annual Ph.D. production
Graduate research assistants
federally supported
Postdoctorals federally supported
9,800
7,600
3,300
1,500
3,700
2,500
9,200
6,000
3,300
800
2,900
1,200
9,100
8,400
2,300
800
200
50
SOURCES: NRC Survey of Doctoral Recipients, National Science Board Status
of Science Review.
B. Why It Escaped Notice
Three things made it difficult for mathematicians and policy-makers
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APPENDIX A
to quickly grasp the full extent of the weakening of support for mathe-
matics:
· After the sharp decline of 1968-73, universities increased their
own support for many things which earlier would have been carried
by research grants. It was only after financial problems hit the univer-
sities in the mid-1970s that the severe lack of resources became evi-
clent.
The growth of computer science support masked the decline in
mathematics support because of the federal budget practice of carry-
ing "mathematics and computer science" as a line item until 1976.
· The explosion of the uses of mathematics caused funding to
flow into applications of known mathematical methods to other fields.
These were often labelled "mathematical research" in federal support
data. The category grew rapidly, masking the fact that support for
fundamental research in the mathematical sciences shrank.
C. Its Consequences
The absence of resources to support the research enterprises in the
country's major mathematical science departments is all too apparent.
In most of them, the university is picking up virtually the total tab for
postdoctoral support, research associates, and secretarial and operat-
ing support; as a result, the amounts are very small. Graduate stu-
dents are supported predominantly through teaching assistantships,
and (like faculty) have been overloaded because of demands for under-
graduate mathematics instruction, which have increased 60% in the
last eight years. The number of established mathematical scientists
with research support, already small in comparison with related fields,
has decreased 15% in the last three years. Morale is declining. Prom-
ising young people considering careers in mathematics are being put
off.
Ph.D.'s awarded to U.S. citizens declined by half over the last decade.
A gap has been created between demand for faculty and supply of
new Ph.D.'s. It may well widen as retirements increase in the l990s.
There is the prospect of a further 12% increase in demand for doctoral
mathematical scientists needed for sophisticated utilization of super-
computers in academia, industry, and government.
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APPENDIX A
The most serious consequence has been delayed. In a theoretical
branch of science with a relatively secure base in the universities,
sharp reduction in federal support does not leave large numbers of
scientists totally unable to do their research, as might be the case in an
experimental science. There is a considerable time lag before there is
a marked slowing down of research output. The establishecl research-
ers and the young people who were in the pipeline when reduction
began carry the effort forward! for 15 or 20 years, adjusting to in-
creased teaching loads, to decreased income or extra summer work,
and to simply doing with fewer of most things. If the number of first-
rate minds in the field is large at the onset of the funding reduction, an
effort of very high quality can be sustained for quite some time.
This is what has been happening in the mathematical sciences in the
United States for over a decade. The situation must be corrected.
IV. FUTURE SUPPORT
A. The Needs of Research Mathematical Scientists
The research community in the mathematical sciences is concentrated
heavily at academic institutions spread throughout the country. Over
90% of productive research mathematicians are on the faculties of the
nation's universities and colleges. Their numbers equal those of physics
or chemistry, some 9,000-10,000.
To pursue research effectively, mathematical scientists need:
1. research time
2. graduate students, postdoctorals, and young investigators of
high quality
3. research associates (visiting faculty)
4. support staff (mostly secretarial)
5. computers and computer time
6. publications, travel, conferences, etc.
During the fifties and sixties, these needs were effectively met by the
injection of federal funds for research into universities. That spurred
remarkable growth and propelled the United States into world leader-
ship in the mathematical sciences. The erosion of support since the
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APPENDIX A
late 1960s has slowed momentum and decreased the rate of influx of
outstanding young people into the mathematical sciences.
B. A Plan for Renewal
What has been describer! makes it evident that realization of the po-
tential for mathematics and its applications requires a substantial
increase in extra-university support. Because there is often an indirect
relation between mathematical developments and their applications,
significant support from industry will not be forthcoming. Thus, the
role of government is crucial.
Incremental budgetary increases of the usual sort cannot deal with the
severe inadequacy of support. We estimate that the federal support
needed to strengthen mathematical research and graduate education
is about $100 million more per year than the FY 1984 level of $78
million. Significant additional resources are needled in each of the six
basic categories we identified earlier. The resources will:
· allow mathematical scientists to capitalize on the future oppor-
tunities provided by the dramatic intellectual developments now oc-
curring
· provide for the attraction and support of young people to help
renew the field
· sustain the work of established researchers.
As the framework for this, we have determined through analysis the
elements of a program to renew U.S. mathematics. This program can
be carried out through expansion of support to the $180 million level
over the next five years. This National Plan for Graduate and Postdoc-
toral Education in the Mathematical Sciences has these features:
~ Each of the approximately 1,000 graduate students per year
who reaches the active level of research for a Ph.D. thesis would be
provided with 15 months of uninterrupted research time, preceded by
two preceding summers of unfettered research time.
· Two hundred of the 800 Ph.D.'s per year would be provided
with postdoctoral positions averaging two years in duration at suit-
able research centers.
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APPENDIX A
· There would be at least 40() research grants for young investi-
gators (Ph.D. age three to five years).
· At least 2,600 of the established mathematical scientists who,
with the young investigators, provide the training for the more than
5,000 total Ph.D. students and the 400 total postdoctorals would have
sufficient supported research time not only to conduct their own re-
search, but also to Provide the requisite training for these young people.
· Support would be provided for associated research needs of
the investigators.
We believe this plan to be consistent with the priorities set by the
mathematical sciences research community through several self-stud-
ies in the last few years.
C. Implementation
It will be up to the Administration and Congress to decide what na-
tional priority to assign to these needs. We would remind them that
what is at stake is the future of a field central to the country's scien-
tif~c and technological effort. While the uses of mathematics in other
fields have been supported, somehow the needs of fundamental mathe-
matics were lost sight of for over a decade. Since there is about a 15-
year delay between the entry of young people into the field and their
attainment of the expected high level of performance, this decade of
neglect alarms us. We urge immediate strong action, in the form of a
five-year "ramping up" of federal support for the mathematical sci-
ences (18% real growth annually, for five years). An effort to renew
mathematics support has already begun at the National Science Foun-
dation. This must be continued for five more years, with a parallel
effort at the Department of Defense. This will bring support back into
balance and allow for renewal, provided Department of Energy re-
sources going to the mathematics of computation are significantly
increased to sustain the initiative which we recommend in this field.
Appropriate utilization of present and future resources requires a
well-thought-out and consistent set of priorities in the expenditures of
funds. Recommendations of this type have recently been set forth in
the COSEPUP Mathematics Briefing Panel Report prepared for OSTP
and its companion report specifically for DOD, as well as recent re-
ports of the NSF Advisory Committee for the Mathematical Sciences.
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APPENDIX A
We have built on these community efforts to systematically and con-
sistently direct funding trends. The efforts must continue, to ensure
the most efficient and fruitful utilization of resources.
Success will also depend on action and understanding within the
nation's universities. For too long, they have been silent about the
fact that the leered of external support for research in their mathemati-
cal science departments is markedly out of balance with the general
level of support for science and engineering in the country. The
disparity is reflected in the working circumstances of their mathemati-
cal faculties and graduate students. As added resources become avail-
able, they must be used in part to ease the strain on the mathematical
science departments, which embody mathematical research in the United
States.
Still, the group which has the fullest agenda before it is the mathe-
matical sciences research community. It is obvious to anyone that if a
field gets into the sort of extreme situation we have described, the
associated research community must bear much of the responsibility.
We urge the mathematical scientists to greatly step up efforts to in-
crease public awareness of developments in the mathematical sciences
and of the importance of the broad enterprise to the nation; to set their
priorities with long-term needs in mind, and to develop mechanisms
for effectively presenting their needs to the universities, to the Ad-
ministration and to Congress all with a renewed commitment to the
unity of the mathematical sciences.
NOTES
Now the Commission on Physical Sciences, Mathematics, and Resources.
tin addition, computer science developed from roots in mathematics and electrical
engineering, then spun off to become a separate discipline. It is important in reading
this report not to confuse computer science with the mathematical sciences. The rela-
tionship of the fields is discussed in Appendix A [of the 1984 Report].
3These research opportunities are discussed in detail in Chapter II [of the 1984
Report].
4The two agencies account for 93% of support. Today, the role of the Department
of Energy in supporting work at the interface of mathematics and computation is of
ever-increasing importance, however.
sFY 1968 was not a peak budget year for mathematical research. It is the year in the
period 1966-70 for which we have the most accurate data.
Chemistry and physics constant dollar budgets at NSF dipped in 1969-70, then
increased by over 25% in the years 1970-72, and continued to grow until the late 1970s.
86