Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.

Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 39

3 Research Progress and Prospects
A survey of modern science and technology shows the mathematical
sciences supporting crucial advances and giving rise to a wealth of
creative and productive ideas. The mathematical achievements of this
century, among the most profound in the history of the discipline,
have been fundamental to the development of our technological age.
And beyond their practical applications, the development of these
mathematical sciences can be counted among the great intellectual
achievements of humankind.
The mathematical sciences constitute a discipline that combines rigor,
logic, and precision with creativity and imagination. The field has
been described as the science of patterns. Its purpose is to reveal the
structures and symmetries observed both in nature and in the abstract
world of mathematics itself. Whether motivated by the practical prob-
lem of blood flow in the heart or by the abstraction of aspects of
number theory, the mathematical scientist seeks patterns in order to
describe them, relate them, and extrapolate from them. In part, the
quest of mathematics is a quest for simplicity, for distilling patterns to
their essence.
Of course, the nonmathematician who tries to read a mathematics
research paper is bound to see the terms as anything but simple. The
field has developed a highly technical language peculiar to its own
needs. Nonetheless, the language of mathematics has turned out to be
eminently suited to asking and answering scientific Questions.
v
Research in the mathematical sciences is directed toward one of two
objectives, and in some cases toward both: (1) to build on and expand
39

OCR for page 39

RENEWING U.S.MATHEMATICS
the core areas of the discipline and (2) to solve problems or create
problem-solution techniques for the increasingly numerous areas of
science and technology where mathematics finds applications. Thus
mathematical sciences research spans a spectrum from the examina-
tion of fundamentals to the application-driven solution of particular
problems. This chapter surveys selected recent research achievements
across this spectrum and mentions a sampling of current research
opportunities that build on and promise to extend recent progress.
These opportunities for progress are real and exist in every major
branch of the mathematical sciences. What is unusual about the mathe-
matical sciences at this time is that, collectively, they are poised to
make striking contributions across the whole spectrum of science and
· ~
engineering.
THE MATHEMATICAL SCIENCES YESTERDAY
This potential for progress is the result of the remarkable growth of
the mathematical sciences along three more or less parallel paths,
stemming from a branching point in the 1930s, when mathematics as a
"pure" discipline entered a new era characterized by reexamination of
its foundations and exploitation of powerful new tools of abstraction.
The result was an extraordinary flourishing of the discipline and an
acceleration of the development of its major branches through the
post-World War II years. The pace was so rapid that specialization
increased, and for a while it seemed that topologists, algebraic geome-
ters, analysts, and other groups of mathematical scientists could barely
speak to one another. Each was inventing powerful new mathematical
structures to unify previously disparate ideas and shed new light on
classical problems.
These developments were accompanied by independent spectacular
advances in applied mathematics and statistics during the same pe-
riod. A major stimulus for these efforts was World War II, which
presented an array of scientific and technological challenges. In
communication, control, management, design, and experimentation,
the power of mathematical concepts and methods was felt in the post-
war years as never before.
A third line of inquiry rooted in the 1930s resulted in the evolution of
the computer. The original work of a handful of mathematicians and
electrical engineers some 50 years ago gave rise to a new cliscipline-
~0

OCR for page 39

RESEARCH PROGRESS AND PROSPECTS
computer science and a new tool, more powerful than any in history,
for storing, processing, and analyzing information. Few people today
need to be told of the impact computers are having on society. But
many people need to have it pointed out that the computer is very
much a mathematical tool that extends the reach and power of mathe-
matics. The computer has already had an enormous impact on ap-
plied mathematics and statistics, and more generally on science and
engineering (see section below, "Computers in the Mathematical Sci-
ences"~.
THE MATHEMATICAL SCIENCES TODAY
In more recent years several dramatic changes have begun to occur
within the cliscipline. Ever more general and more powerful methods
and structures developed within pure mathematics have begun to
reunify its various branches. The gap between pure and applied mathe-
matics has also begun to close as more of the new methods are used in
other fields for example, in biology, medicine, and finance, as well
as in fields usually thought of as mathematical. And the computer
continues to stimulate the need for new mathematics while opening
unprecedented new directions and methods for mathematical explora-
tion per se. The immensity and richness of the methods and ideas
developer] by pure and applied mathematicians and statisticians over
the last 50 years constitute a huge resource being tapped by the intel-
lectual machine of science and engineering.
It is this image of the mathematical sciences today that one should
have in mind while reading this brief survey of the state of the field as
a whole. This is an unusual time in the history of the discipline. The
simultaneous internal unification and greater awareness of external
applications have brought the mathematical sciences into an era of
potentially greater impact on the world around us.
This chapter is a companion to Appendix B. which contains more-
detailed, brief descriptions of 27 important research areas that have
produced significant accomplishments in recent years and that offer
opportunities for further research. The committee emphasizes that
the achievements anal opportunities discussed in Appendix B are not
intended to be comprehensive, nor are they intended to suggest a
specific agenda for funding research in the mathematical sciences.
The aim is rather to demonstrate by example the vigor and compre-
hensiveness of current mathematical sciences research, and how the
41

OCR for page 39

RENEWING U.S. MATHEMATICS
mathematical sciences are reaching out increasingly into all parts of
science and technology even as the core areas of the mathematical
sciences are expanding significantly. The selection of topics discussed
in Appendix B illustrates the very real progress made across that
spectrum in just the last five years.
The many applications of mathematics which are most readily vis-
ible to the larger scientific community can be realized only if the
discipline as a whole remains strong and vibrant internally. Mathe-
matics has indeed been interacting with other disciplines in healthy
and productive new ways while simultaneously receiving an infusion
of new ideas.2 This process is accelerating, and the accounts in Ap-
pendix B illustrate the extent and import of cross-disciplinary research
today. There is now an ever-increasing interest in applied problems-
an interest that was perhaps not so evident a decade or two ago. At
the same time mathematics has developed a substantially greater sense
of internal unity and has displayed healthy cross-fertilization of ideas
between subdisciplines. The intellectual energy produced by these
two trends looking externally for new problems and unifying inter-
nally represents perhaps the greatest opportunity of all for the mathe-
matical sciences over the next five years.
COMPUTERS IN THE MATHEMATICAL SCIENCES
From the convenience of a hand calculator, to the versatility of a
personal computer, to the power of parallel processors, computers
have ushered in the technological age. But they have also ushered in
a mathematical age, since computers provide one of the main routes
by which mathematics reaches into every realm of science and engi-
neering. Computers have profoundly influenced the mathematical
sciences themselves, not only in facilitating mathematical research,
but also in unearthing challenging new mathematical questions. Many
of the research advances described in this chapter and in Appendix B
would not have been possible without computers and the associated
mathematics that is concurrently being developed.
It is sometimes thought that once computers are powerful enough,
mathematicians will no longer be needed to solve the mathematical
problems arising in science and engineering. In fact, nothing could be
farther from the truth. As computers become increasingly powerful,
mathematicians are needed more than ever to shape scientific prob-
lems into mathematical ones to which computing power can be ap-
42

OCR for page 39

RESEARCH PROGRESS AND PROSPECTS
plied. And as science and engineering attempt to solve ever more
ambitious problems involving increasingly large and detailed data
sets and more complicated structures entirely new mathematical ideas
will be needed to organize, synthesize, and interpret.
Computers are fundamentally connected to mathematics, in their
physical design, in the way they organize and process information,
and in their very history. The concept of a machine that could per-
form calculations automatically dates at least to the early nineteenth
century. This concept became practical through the farsightedness of
such mathematicians as Alan Turing, famous for cracking the German
Enigma code during World War II, and John von Neumann, who was
the driving force behind the design and building of the first computer.
Computer scientists continue to draw on theoretical mathematics, since
advances in computing power are dependent upon mathematical ideas.
Faster electronic components are continually appearing, but advances
in hardware alone will not improve computing speed and efficiency,
and most experts agree that the development of efficient software is
not keeping pace with hardware development. Designing and analyz-
ing the efficiency of computer algorithms are largely mathematical
tasks. As machines become faster and computer memory sizes be-
come larger, asymptotic improvements in the efficiency of algorithms
become more and more important in practice.
Recent research in
theoretical computer science has produced significant improvements
in specific algorithms and also new approaches to algorithm design,
such as the use of parallelism and randomization.
The impact of the computer on the mathematical sciences has particu-
larly broadened the domain of the mathematical modeler, who can
now reliably simulate quite complex physical phenomena by com-
puter. Widely used in all sciences and in engineering, and a research
area in its own right) computer modeling plays a major role in the
development of critical technologies such as the fabrication of micro-
electronic circuits and the understanding of fluid flow. Developing
appropriate simulations for a given technology invariably involves a
high degree of scientific knowledge as well as sophisticated mathe-
matical tools to describe and evaluate the model. Validation of these
models may require statistical tests and comparison with an analyti-
cally produced limiting-case solution. Ultimately the model itself has
to be tuned to physical data to confirm or improve its aptness for
representing a physical process or phenomenon.
~3

OCR for page 39

RENEWING U.S.MATHEMATICS
Finally, and by no means least important, the computer is beginning
to have a significant impact on areas of core mathematics through its
use in the visualization of underlying mathematical structures. Its use
in proving theorems is evidencecl by the recent proofs of the four-
color theorem and of the Feigenbaum conjecture.
ACCOMPLISHMENTS AND OPPORTUNITIES
The committee has selected for presentation a collection of specific
recent research achievements (Appendix B) that open up new oppor-
tunities for the future. It is emphasized that this is a partial list only
and that lack of space precludes a fuller and more comprehensive
surrey. These examples calf research progress and opportunities are as
follows:
1. Recent Advances in Partial Differential Equations
2. Vortices in Fluid Flow
3. Aircraft Design
4. Physiology
5. Medical Scanning Techniques
6. Global Change
7. Chaotic Dynamics
X. Wavelet Analysis
9. Number Theory
1 0. Topology
11. Symplectic Geometry
12. Noncommutative Geometry
13. Computer Visualization as a Mathematical Too!
14. Lie Algebras and Phase Transitions
15. String Theory
16. Interacting Particle Systems
17. Spatial Statistics
18. Statistical Methods for Quality and Productivity
19. Graph Minors
20. Mathematical Economics
21. Parallel Algorithms and Architectures
22. Randomized Algorithms
23. The Fast Multipole Algorithm
24. Interior Point Methods for Linear Programming
25. Stochastic Linear Programming
26. Applications of Statistics to DNA Structure
27. Biostatistics and Epidemiology
44

OCR for page 39

RESEARCH PROGRESS AND PROSPECTS
A description in some detail of the specifics of each of these is given in
Appendix B. What follows here is a discussion of how these achieve-
ments and opportunities (referred to by the Appendix B section num-
ber that also corresponds to the numbers in the listing of topics above),
as well as some others not included in Appendix B. fit into the overall
landscape of the mathematical sciences and their many and varied
applications. It is hoped that this brief narrative will convey an appre-
ciation of the breadth, scope, and usefulness of the mathematical sci-
ences ant! how they are changing the contours of science and technol-
ogy. At the same time it is hoped that this discussion will illustrate
the vitality of mathematics as a discipline and show not only how
ideas flow from the core of mathematics out to applications but also
how the applications of mathematics can, in turn, result in ideas flow-
ing to the core areas of the discipline. These interchanges affect al-
most every area of core mathematics.
For instance, developments in such core areas as number theory, alge-
bra, geometry, harmonic analysis, dynamical systems, differential
equations, and graph theory (see, for instance, Sections 1, 7, 8, 9, 10,
11, 12, and 19 in Appendix By not only have significant applications
but also are themselves influenced by developments outside of core
mathematics.
The Living World
The mathematical and the life sciences have a long history of interac-
tion, but in recent years the character of that interaction has seen some
fundamental changes. Development of new mathematics, greater
sophistication of numerical and statistical techniques, the advent of
computers, and the greater precision and power of new instrumenta-
tion technologies have contributed to an explosion of new applica-
tions. Testifying to the impact of mathematics, computers are now
standard equipment in biological and medical laboratories. Several
recent fundamental advances seem to indicate a revolution in the way
these areas interact.
The complexity of biological organisms and systems may be unrav-
eled through the unique capability of mathematics to discern patterns
and organize information. In addition, rendering problems into mathe-
matical language compels scientists to make their assumptions and
interpretations more precise. Conversely, biologists can provide a wealth
of challenging mathematical problems that may even suggest new
45

OCR for page 39

RENEWING U.S. MATHEMATICS
directions for purely mathematical research. Great differences in ter-
minology and in the cultures of the two fields require a core of re-
searchers with understanding of both areas.
Mathematical techniques for understanding fluid dynamics have made
possible computational models of the kidney, pancreas, ear, and many
other organs (Section 4~. In particular, computer models of the human
heart have led to improved design of artificial heart valves. Mathe-
matical methods were fundamental to the development of medical
imaging techniques, including CAT scans, magnetic resonance imag-
ing, and emission tomography (Section 5~. In the neurosciences the
mathematical simulation of brain functions, especially through com-
puter modeling, has come close enough to reality to be a powerful
guide to experimentation. For instance, mathematical models have
recently helped to elucidate studies in the formation of ocular domi-
nance columns, patches of nerve cells in the visual cortex that respond
to signals from only one eye. In addition, advances in neural network
simulations are starting to have a significant impact on predicting
how groups of neurons behave.
Recently DNA researchers have collaborated with mathematicians to
produce some striking insights. When a new experimental technique
allowed biologists to view the form of DNA under an electron micro-
scope, researchers saw that DNA appeared tangled and knotted.
Understanding the mechanism by which DNA unknots and replicates
itself has led to the application of knot theory (a branch of mathemat-
ics that seeks to classify different kinds of knots) to DNA structure.
At about the same time a breakthrough in knot theory gave biologists
a tool for classifying the knots observed in DNA structure (see Section
10 for details). In addition, researchers are developing three-dimen-
sional mathematical models of DNA and are applying probability theory
and combinatorics to the understanding of DNA sequencing (Section
26).
Computers have brought sophisticated mathematical techniques to
bear on complicated problems in epidemiology. One major effort is
the mathematical modeling of the AIDS epidemic. Analysis of data on
transmission of the human-immunodeficiency virus that causes AIDS
has shown that HIV does not spread like the agents of most other
epidemics. Various mathematical methods have been combined with
statistical techniques to produce a computer model that attempts to
account for the range of factors influencing the spread of the virus.
46

OCR for page 39

RESEARCH PROGRESS AND PROSPECTS
However, because of the complexity and size of the problem, research-
ers are finding current computational power inadequate and are look-
ing for mathematical ways of simplifying the problem (Section 271.
The Physical World
The physical sciences, especially physics itself, have historically pro-
vided a rich source of inspiration for the development of new mathe-
matics. The history of science has many examples of physical scien-
tists hunting for a theoretical framework for their ideas, only to find
that mathematical scientists had already created it, quite in isolation
from any application. For example, Einstein used the mathematical
theory of differential geometry, and, more recently, algebraic geome-
try has been applied to gauge field theories of physics.
Often the mathematical equations of physics cannot be solved pre-
cisely, and so their solutions must be approximated by the methods of
numerical analysis and then solved by computer. Other problems are
so large that only a sample of their solution can be found, with statis-
tical techniques putting this sample in context. For this reason, the
computer has become an indispensable tool for a great many physical
scientists. The computer can act as a microscope and a telescope,
allowing researchers to model and investigate phenomena ranging
from the dynamics of large molecules to gravitational interactions in
space.
The equations of fluid dynamics fit into the broader class of partial
differential equations, which have historically formed the main tie
between mathematics and physics. Global climate change is a topic of
intense debate, and greater quantitative understanding through the
use of mathematical modeling and spatial statistics techniques would
greatly help in assessing the dangers and making reliable predictions
(see Sections 6 and 17~. One of the striking characteristics of today's
applications is the range of mathematical subjects and techniques that
are found to have connections to physical phenomena, from the appli-
cation of symplectic transformations to plasma physics (see Section
11) to the use of topological invariants in quantum mechanics (see
Section 12~. As the various subfields of the mathematical sciences
themselves become increasingly interconnected, new and unexpected
threads tying them to the physical sciences are likely to surface. Over
the past decade the highly theoretical area of Lie algebras has illumi-
nated the physical theory of phase transitions in two dimensions,
~7

OCR for page 39

RENEWING U.S. MATHEMATICS
which has applications to the behavior of thin films (see Section 141.
The study of chaotic dynamics, which employs a range of mathemati-
cal tools, has demonstrated that unpredictable behavior can arise from
even the simplest deterministic systems and has been used to describe
diverse phenomena, such as the interfaces between fluids (see Section
7~. Investigation of quasicrystals, a category of matter combining
properties of crystals and glasses, utilizes the mathematical theory of
tiling, which describes ways of fitting geometrical figures together to
cover space.
Other areas of science and engineering have benefited from the close
connections between the mathematical and physical sciences. Because
of the increased power of instrumentation technology, many phenom-
ena can be observed with a precision that allows questions to be
formulated in terms of mathematical physics. In fact, computational
methods in fluid dynamics have made it possible to model a host of
phenomena in chemistry, astrophysics' polymer physics, materials
science, meteorology, and other areas (see Sections 1, 2, 3, 4, 6, and
23~. The degree of precision achieved by these models usually is
limited partly by the modeling and physical understanding, and partly
by the available computing power. Improvements to the mathemati-
cal model or algorithm can often significantly increase the actual
computing power achievable with given hardware, and hence the degree
of model accuracy.
Theoretical physics has often posed profound challenges to mathemat-
ics and has suggested new directions for purely mathematical research.
One spectacular instance of cross-fertilization came with the advent of
string theory. This theory proposes the intriguing idea that matter is
not made up of particles but rather is composed of extended strings.
Algebraic geometry, a highly abstract area of mathematics previously
thought to have little connection to the physical world, is one of the
ingredients providing a theoretical framework for string theory. In
addition, string theory is supplying mathematicians with a host of
new directions for research in years to come. Section 15 in Appendix
B provides details.
The Computational World
Much of the research on algorithms is highly mathematical and draws
on a broad range of the mathematical sciences, such as combinatorics,
complexity theory, graph theory, and probability theory, all of which
48

OCR for page 39

RESEARCH PROGRESS AND PROSPECTS
are discussed in Appendix B. A' striking and recent algorithmic ad-
vance is the development of interior point algorithms for linear pro-
gramming (Section 24), a mathematical method used in many business
and economics applications. Linear algebra and geometry were used
in the development of these algorithms, which have found many
applications, such as efficient routing of telephone traffic.
In addition to the design of algorithms, the mathematical sciences
pervade almost every aspect of computing: in designing hardware,
software, and computer networks; in planning for allocation of com-
puting resources; in establishing the reliability of software systems; in
ensuring computer security; and in the very foundations of theoretical
computer science. In addition, all kinds of computations are depend-
ent upon the branch of mathematics known as numerical analysis,
which seeks to establish reliable and accurate means of calculation.
For example, a current success in numerical analysis is the application
of wavelet analysis (Section 8), which grew out of a body of theoretical
mathematics, to produce faster signal processing algorithms. Another
area of current research is computational complexity, which mathe-
matically analyzes the efficiency of algorithms.
In statistics, modern computational power has permitted the implem-
entation of data-intensive methods of analysis that were previously
inconceivable. One of these, the resampling method which can be
thought of as a Monte Carlo method in the service of inference—is
finding a wide range of applications in medical science, evolutionary
biology, astronomy, physics, image processing, biology, and econo-
metrics. The subject is still in its infancy, and one can expect more
sophisticated developments to be stimulated by new applications.
Computers and statistics are symbiotic in other ways as well. The
statistics community is becoming involved in the statistical analysis
and design of computer models arising in science and industry. The
use of randomization in algorithms (see Section 22) has proved to be
highly successful in certain kinds of applications and has stimulated
renew research in the properties of pseudorandom number generators.
Meanwhile, other statisticians are assisting computer science by de-
veloping statistical techniques for characterizing and improving soft-
ware reliability.
As the language of computer modeling, mathematics is revolutioniz-
ing the practice of science and engineering. In many instances, com-
puter simulations have replaced costly experiments, for instance in
49

OCR for page 39

missions.
RENEWING U.S. MATHEMATICS
aircraft design (Section 3~. From visualization of the folding of protein
molecules (see Section 26) to calculation of combustion patterns (see
Sections 2 and 3), mathematical analysis combined with computing
power has produced profound insights. The development of reliable
and accurate simulations requires both understanding of the scientific
problem at hand and knowledge of the mathematical tools to describe
and evaluate the model.
Operations researchers have applied mathematics to a range of indus-
trial problems such as efficient scheduling and optimization of re-
sources, and statistical methods are now commonplace in evaluating
quality and productivity (see Sections 18 and 23~. Control theory an
interdisciplinary field drawing on mathematics, computer science, and
engineering—has applications to such problems as autopilot control
systems, chemical processing, and antilock brake systems on cars.
The advent of communications technologies has depended on mathe-
matical developments. Image processing, acoustical processing, speech
recognition, data compression, and other means of transmitting infor-
mation all require sophisticated mathematics. One surprising example
of the application of theoretical mathematics to such areas came re-
cently from a branch of number theory dealing with elliptic curves. It
turns out that a research result from that area has produced an en-
tirely new approach for efficiently packing spheres. Because commu-
nications signals are sometimes modeled as higher-dimensional spheres,
this result will help in improving the efficiency and quality of trans-
Computer graphics have opened a whole world of visualization tech-
niques that allow mathematicians to see, rotate, manipulate, and in-
vestigate properties of abstract surfaces. In particular, the subject of
the mathematical properties of soap films "minimal surfaces" that
are visible analogues of the solutions to optimization problems in
many fields has witnessed a recent breakthrough and a resurgence
of interest because of computer modeling, as described in Section 13.
Computer visualization techniques have also contributed to under-
standing the mathematics of surface tension in crystalline solids.
Computers are now routinely used in investigating many questions in
number theory, a subject that examines properties of the integers and
often finds application to such areas as computer science and cryptog-
raphy. Symbolic manipulators- computers that perform operations
on mathematical expressions, as opposed to numerical calculations-
50

OCR for page 39

RESEARCH PROGRESS AND PROSPECTS
are powerful new instruments in the too} kit of many mathematical
scientists and are increasingly being used in the teaching of mathe-
matics.
THE UNIFYING SCIENCE
The mathematical sciences have not only served as part of the bedrock
on which science and engineering rests, but have also illuminated
many profound connections among seemingly disparate areas. Prob-
lems that initially seem unrelated are later seen to be different aspects
of the same phenomenon when interpreted in mathematical terms. In
this way mathematics serves to unify and synthesize scientific knowl-
edge to produce deeper insights and a better understanding of our
world.
The mathematical sciences themselves are unifying in profound ways
that could not have been predicted twenty years ago. The computer
has lent unprecedented technological power to the enterprise, but
mathematical sciences research still proceeds largely through individ-
ual creativity and inspiration. As the discipline becomes increasingly
interconnected, progress in mathematics will depend on having many
mathematicians working on many different areas. History has taught
us that the most important future applications are likely to come from
some unexpected corner of mathematics. The discipline must move
forward on all of its many fronts, for its strength lies in its diversity.
THE PRODUCTION OF NEW MATHEMATICS
It is enlightening to note how so many of the research topics men-
tioned here and in Appendix B for instance, the developments in
partial differential equations, vortices, aircraft design, physiology, and
global change—have developed out of the mathematical research
described in the 1984 Report, both in its body and in Arthur Jaffe's
appendix, "Ordering the Universe: The Role of Mathematics." The
recent developments in number theory and geometry follow naturally
from the Mordell conjecture, which was featured prominently in the
1984 Report. One can likewise see in the report of five years ago the
roots of the recent developments in topology and noncommutative
geometry. Although wavelet analysis was not mentioned in the ear-
lier report, its roots in Fourier analysis were discussed. Similarly, the
development of new algorithms was a prominent topic in the 1984
discussion, although the particular algorithms featured in Appendix B
51

OCR for page 39

RENEWING U.S. MATHEMATICS
were not available then. These examples dramatically illustrate both
continuity and innovation in mathematics.
It should be noted that much of the work discussed in Appendix B is
the product of individual investigators. Mathematics is still very
much '~small science," with a tradition of individuals pursuing inde-
pendent research. This is a strength because it allows the total enter-
prise to span a great many topics, remain flexible, and be responsive
to the rest of science. However, the breadth required for many inves-
tigations calls for more collaborative work, which is practicable for
mathematicians regardless of distance but is hindered in practice by
the absence of adequate support for the occasional travel that is re-
quirecl. The recent innovations by the NSF (research institutes and
science and technology centers) and the DOD (university research
initiatives) provide valuable alternatives, both for established investi-
gators and postdoctorals.
The continued production of valuable new mathematics requires not
only that the number of individual investigators be increased as rec-
ommended in the 1984 National Plan, but also that the entire field
reach out to the rest of the scientific world. Collaborating with re-
searchers in other fields and making an effort to understand applica-
tions and improve the mathematical sophistication of others help the
mathematical sciences become increasingly robust and valuable. The
potential demand- in terms of the number of possible applications—
for mathematics research is great, but the actual demand may be lim-
ited by failure of the mathematics community to communicate with
others. Mathematics educators must design courses that meet the needs
of the many students from other disciplines. Likewise, mathematical
science researchers must write reviews and textbooks that are acces-
sible to nonmathematicians. More of them need to make the extra
effort to read journals and attend conferences outside their fields, to
learn how to communicate fluently with potential users and collabora-
tors, and to actively seek new opportunities for their work.
Finally, it is important to reemphasize that the research achievements
and opportunities mentioned here and in Appendix B were selected
from among a number of possibilities to illustrate the vigor and. breadth
of the mathematical sciences. This compilation is not intended to be
comprehensive, nor is it intended to be a research agenda for the
future. Many excellent new ideas and proposals will come to the fore
as part of the natural development of the discipline, and these can and
52

OCR for page 39

RESEARCH PROGRESS AND PROSPECT'S
will compete for the attention of active researchers. The prospects are
indeed bright.
NOTES
'See, for example, Mathematical Sciences: So?ne Research Trends, Board on Mathemati-
cal Sciences (National Academy Press, Washington, D.C., 1988) for a very different set
of topics. Other recent, more specialized reports also list significant research opportu-
nities. These include the BMS advisory panel reports (to the Air Force in 1987 and the
Navy in 1987 and 1990) mentioned in Chapter 2; the American Statistical Association
report Challenges for the '90s, listed in Chapter 2; and Operations Research: The Next
Decade ("CONDOR report"), Operations Research, Vol. 36, No. 4 (July-August 1988), pp.
619-637.
2The NRC Board on Mathematical Sciences is producing a series of cross-discipli-
nary reports to foster this trend. The Institute for Mathematical Statistics has also
addressed the trend in its report Cross-Disciplinary Research in the Statistical Sciences
(Institute for Mathematical Statistics, Haywood, Calif., 1988).
53

OCR for page 39