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 15

1
Introduction
STUDY OVERVIEW
The opening years of the twenty-first century have been remarkable
ones for the mathematical sciences. Major breakthroughs have been made
on fundamental research problems. The ongoing trend for the mathemati-
cal sciences to play an essential role in the physical and biological sciences,
engineering, medicine, economics, finance, and social science has expanded
dramatically. The mathematical sciences have become integral to many
emerging industries, and the increasing technological sophistication of our
armed forces has made the mathematical sciences central to national de-
fense. A striking feature of this expansion in the uses of the mathematical
sciences has been a parallel expansion in the kinds of mathematical science
ideas that are being used.
There is a need to build on and solidify these gains. Too many math-
ematical scientists remain unaware of the expanding role for their field, and
this in turn will limit the community’s ability to produce broadly trained
students and to attract larger numbers of students. A community-wide
effort to rethink the mathematical sciences curriculum at universities is
needed. Mechanisms to connect researchers outside the mathematical sci-
ences with appropriate mathematical scientists need to be improved. The
number of students now being attracted to the field is inadequate to meet
the needs of the future.
A more difficult period is foreseen for the mathematical sciences be-
cause the business model for universities is entering a period of rapid
15

OCR for page 15

16 THE MATHEMATICAL SCIENCES IN 2025
change. Because of their extensive role in teaching service courses, the
mathematical sciences will be disproportionately affected by these changes.
These conclusions were reached by the Committee on the Mathemati-
cal Sciences in 2025 of the National Research Council (NRC), which con-
ducted the study that led to this report. The study was commissioned by the
Division of Mathematical Sciences (DMS) of the National Science Founda-
tion (NSF). DMS is the primary federal office that supports mathematical
sciences research and the health of the mathematical sciences community.
In recent years, it has provided nearly 45 percent of federal funding for
mathematical sciences research and the large majority of support for re-
search in the core areas of the discipline. Other major federal funders of the
mathematical sciences include the Department of Defense, the Department
of Energy, and the National Institutes of Health. Details of federal funding
are given in Appendix C.
While the mathematical sciences community and its sponsors regularly
hold meetings and workshops to explore emerging research areas and assess
progress in more mature areas, there has been no comprehensive strategic
study of the discipline since the so-called Odom study1 in the late 1990s.
During 2008, DMS Director Peter March, with encouragement from the
NSF associate director for mathematical and physical sciences, Tony Chan,
worked with the NRC’s Board on Mathematical Sciences and Their Appli-
cations (BMSA) to define the goals of a new strategic study of the discipline.
For the study that produced this report, DMS and BMSA chose a time
horizon of 2025. It was felt that a strategic assessment of the mathematical
sciences needed a target date and that the date should be sufficiently far
in the future to enable thinking about changes that might correspond to a
generational shift. Such changes might, for example, depend on changes in
graduate education that may not yet be implemented.
The specific charge for this study reads as follows:
The study will produce a forward-looking assessment of the current state
of the mathematical sciences and of emerging trends that will affect the
discipline and its stakeholders as they look ahead to the quarter-century
mark. Specifically, the study will assess:
— he vitality of research in the mathematical sciences, looking at such
T
aspects as the unity and coherence of research, significance of recent
developments, rate of progress at the frontiers, and emerging trends;
— he impact of research and training in the mathematical sciences on
T
science and engineering; on industry and technology; on innovation
and economic competitiveness; on national security; and other areas of
national interest.
1 NationalScience Foundation, 1998, Report of the Senior Assessment Panel for the Inter-
national Assessment of the U.S. Mathematical Sciences. NSF, Arlington, Va.

OCR for page 15

INTRODUCTION 17
The study will make recommendations to NSF’s Division of Mathematical
Sciences on how to adjust its portfolio of activities to improve the vitality
and impact of the discipline.
To carry out this study, the NRC appointed a broad mix of people
with expertise across the mathematical sciences, extending into related
fields that rely strongly on mathematics and statistics. Biographical sketches
of the committee members are included in Appendix F. As was done for
two earlier NRC strategic studies chaired by former Presidential Science
A
dvisor Edward David—Renewing U.S. Mathematics: Critical Resource for
the Future (1984) and Renewing U.S. Mathematics: A Plan for the 1990s
(1990)2—and the aforementioned one led by William Odom, a chair was
sought who is not a mathematical scientist. (Dr. David was trained as an
electrical engineer and General Odom was an expert on the Soviet Union.)
This was done so the report would not veer into advocacy and also so it
would be steered by someone with a broad view of how the mathematical
sciences fit within broader academic and research endeavors. The breadth
of the current committee—only half of the members sit in academic depart-
ments of mathematics or statistics—enabled the study to assess the actual
and potential effects of the mathematical sciences on the broader science
and engineering enterprise.
To inform its deliberations, the committee interacted with a wide range
of invited speakers, as shown in Appendix B. At its first meeting, the focus
was on learning about other NRC strategic studies of particular disciplines:
how they were carried out and what kinds of results were produced. At
that meeting, the committee also engaged in discussion with a pair of ex-
perienced university administrators, one a mathematician, to explore the
changing setting for academic research and what might be on the horizon.
In addition, the committee examined a large number of relevant reports and
community inputs. The second meeting featured discussions with a range
of individuals who employ people with mathematical skills, to explore the
kinds of skills (in emerging industries, especially) that are needed, and
the adequacy of the existing pipelines. Inputs from these first two meetings
influenced Chapters 3 and 6 in particular.
To gather inputs from the mathematical sciences community, the com-
mittee established a Web site for input and requested comments through
a mass e-mail to department heads and other community leaders, using a
list maintained by DMS. It also produced announcements that were pub-
lished in the May 2011 issues of the Notices of the American Mathematical
S
ociety (AMS) and the AMSTAT News of the American Statistical Associa-
2 These are also known colloquially as the “David I” and “David II” reports. Both were
published by the National Academy Press, Washington, D.C.

OCR for page 15

18 THE MATHEMATICAL SCIENCES IN 2025
tion (ASA). Eight inputs were received through this route. The committee
sent specific requests for comments to the leaders of selected committees
of the AMS, the Society for Industrial and Applied Mathematics (SIAM),
ASA, and the Mathematical Association of America. At its third meeting,
which was held in Chicago, the committee organized a panel discussion
with representatives from eight mathematical sciences departments in the
vicinity of Chicago. That discussion focused on challenges and oppor-
tunities facing departments and the profession and on how to respond.
Similar questions were discussed with several dozen community members
at open sessions the committee held at the Joint Mathematics Meetings in
New Orleans, January 2011; the International Congress of Industrial and
A
pplied Mathematics in Vancouver, July 2011; and the Joint Statistical
Meetings in Miami, August 2011. In addition, helpful discussions were held
in March 2011 with the AMS Committee on Science Policy, in April 2011
with the SIAM Science Policy Committee, and in October 2011 and April
2012 with the Joint Policy Board for Mathematics.
A mechanism that proved particularly valuable was a series of 11
conference calls that members of the committee held in March-May 2011
with selected experts across the mathematical sciences. Salient observations
raised by these experts (who are listed, as already mentioned, in Appen-
dix B) are reflected in Chapters 3 and 4.
Coincidently, the current study overlapped analogous examinations
in the United Kingdom and Canada. A member of the study committee
chaired the U.K. assessment, and two members of the committee served on
the advisory board for the Canadian assessment; in addition, the commit-
tee was briefed on the Canadian study by that study’s executive director.
Through these links and examination of specific materials, this study was
informed by the U.K. and Canadian work.3
As part of this study, the committee also produced an interim product
titled Fueling Innovation and Discovery: The Mathematical Sciences in the
21st Century. That short report highlights a dozen illustrations of research
progress of recent years in a format that is accessible to the educated public
and conveys the excitement of the discipline. It recounts how research in the
mathematical sciences have led to Google’s search algorithm, advances in
medical imaging, progress in theoretical physics, technologies that contribute
to national defense, methods for genomic analysis, and many other capabili-
ties of importance to all people. But that report merely skims the surface,
because the mathematical sciences nowadays touch all of us in so many ways.
3 Engineering and Physical Sciences Research Council (EPSRC), 2010, International Review
of Mathematical Science. EPSRC, Swindon, U.K.; Natural Sciences and Engineering Research
Council (NSERC), 2012, Solutions for a Complex Age: Long Range Plan for Mathematical
and Statistical Sciences Research in Canada 2013–2018. NSERC, Ottawa, Canada.

OCR for page 15

INTRODUCTION 19
NATURE OF THE MATHEMATICAL SCIENCES
This report takes an expansive and unified view of the mathematical
sciences. The mathematical sciences encompass areas often labeled as core
and applied mathematics, statistics, operations research, and theoretical
computer science. In the course of the study that led to this report, it be-
came clear both that the discipline is expanding and that the boundaries
within the mathematical sciences are beginning to fade as ideas cross over
between subfields and the discipline becomes increasingly unified. In addi-
tion, the boundaries between the mathematical sciences and other subjects
are also eroding. Many researchers in the natural sciences, social sciences,
life sciences, computer science, and engineering are at home in both their
own field and the mathematical sciences. In fact, the number of such people
is increasing as more and more research areas become deeply mathemati-
cal. It turns out that the expansion of the mathematical sciences is a major
conclusion of this report, one that is discussed in Chapter 4. The discipline
has evolved considerably over the past two decades, and the mathematical
sciences now extend far beyond the definitions implied by the institutions—
academic departments, funding sources, professional societies, and princi-
pal journals—that support the heart of the field.
The mathematical sciences underpin a broad range of science, engi-
neering, and technology, including the technology found in many everyday
products. Many mathematical scientists are motivated by such applications,
and they target their work so as to create particular mathematical and
statistical understanding and capabilities. Such work—for example, the
compressed sensing research highlighted in Chapter 2—usually goes far
beyond routine application of an existing idea, tending to be instead very
innovative and deep. A large fraction of mathematical science work is not
motivated by external applications, and the reader who focused only on
applications would be misled about something central to the culture of the
mathematical sciences: the importance of discovery for its own sake and
the quest for internal coherence, both common drivers of research. But the
words (such as “beauty”) that are often used to describe the motivation
for such research fail to capture the power and value of the work. Whether
externally or internally motivated, mathematical sciences research aims
to understand deep connections and patterns. Researchers are driven to
understand how the world is put together and to find its underlying order
and structure. This leads to concepts with deep interconnections. When a
researcher explores unanswered questions, she or he may catch glimpses
of patterns, of unexpected links. The desire to understand “why” is very
compelling, and this curiosity has a long history of leading to important
new developments. Moreover, when a researcher succeeds in proving that
those glimpses are backed up by precisely characterized connections, the

OCR for page 15

20 THE MATHEMATICAL SCIENCES IN 2025
way the pieces fall into place is indeed beautiful, and researchers are struck
by the “rightness” or inevitability of this new insight. Synonyms for this
driving concept might be “simplicity,” “naturalness,” “power,” and “com-
prehensiveness,” and mathematical scientists of all stripes put a premium
on results with intellectual depth, generality, and an ability to explain many
things at once and to expose previously hidden interconnections (integrat-
ing ideas from disparate areas).
Even when research is internally motivated, it is strikingly common
to find instances in which applications arise in a different discipline and
the necessary mathematics is already available, having been generated by
mathematical scientists for unrelated reasons. As one example, the commit-
tee cites an interchange in the early 1970s between the mathematician Jim
Simons and the theoretical physicist Frank Yang, when Yang was explaining
a theory he was trying to develop to help him understand elementary par-
ticles in physics. Simons—whose background in mathematics later gave him
the foundation for a very successful career shift into finance—said to Yang,
“Stop, don’t do that.” Yang, taken aback, asked, “Why not?” Simons said,
“Because mathematicians already did it more than 30 years ago.” Yang
then asked, “For what reason? Why would they ever do that?” The answer
is of course that they were motivated by the internal, esthetic considerations
of their, at the time, completely theoretic investigations. This is not an
isolated incident, but rather an example of what happens repeatedly in the
mathematical sciences. The prime numbers and their factorization, initially
studied for esthetic reasons, now provide the underpinnings of Internet
commerce. Riemann’s notion of geometry and curvature later became the
basis of Einstein’s general relativity. Quaternions, whose multiplicaion t
table was triumphantly carved into a Dublin bridge by William Hamilton
in 1843, are now used in video games and in tracking satellites. Operators
on Hilbert space provided the natural framework for quantum mechanics.
Eigenvectors are the basis for Google’s famous Page Rank algorithm and
for software that recommends other products to users of services such as
N
etflix. Integral geometry makes possible MRI and PET scans. The list
of such examples is limited only by the space to tell about them. Some
additional examples are given in Chapter 2, where the interplay between
theoretical physics and geometry is described.
A strong core in the mathematical sciences—consisting of basic con-
cepts, results, and continuing exploration that can be applied in diverse
ways—is essential to the overall enterprise because it serves as a common
basis linking the full range of mathematical scientists. Researchers in far-
flung specialties can find common language and link their work back to
common principles. Because of this, there is a coherence and interdepen-
dence across the entire mathematical sciences enterprise, stretching from
the most theoretical to the most applied.

OCR for page 15

INTRODUCTION 21
Robert Zimmer, a mathematical scientist who is president of the Uni-
versity of Chicago, speaks of the mathematical sciences as a fabric: If it
is healthy—strong and connected throughout the whole—then it can be
tailored and woven in many ways; if it has disconnects, then its useful-
ness has limitations. He also argues that, because of this interconnectivity,
there is a degree of inevitability to the ultimate usefulness of mathematical
sciences research. That is, important applications are the rule rather than
the exception.4 Over and over, research that was internally motivated has
become the foundation for applied work and underlies new technologies
and start-ups. And often questions that arise because of our inability to
mathematically represent important phenomena from applications prompt
mathematical scientists to delve back into fundamental questions and create
additional scaffolding of value both to the core and to future applications.
The fabric metaphor accurately captures the interconnectivity of the
various strands of the mathematical sciences; all of the strands are woven
together, each supporting the others, and collectively forming an integrated
whole that is much stronger than the parts separately. The mathematical
sciences function as a complex ecosystem. Ideas and techniques move back
and forth—innovations at the core radiate out into applied areas; flowing
back, new mathematical problems and concepts are drawn forth from prob-
lems arising in applications. The same is true of people—those who choose
to make their careers in applied areas frequently got a significant part of
their training from core mathematical scientists; seeing the uses and power
of mathematics draws some people in to study the core. One never knows
from which part of the mathematical sciences the next applications will
come, and one never knows whether what is needed for a possible applica-
tion is existing knowledge, a variation on what already exists, or something
completely new. To maintain U.S. leadership in the mathematical sciences,
the entire ecosystem must remain healthy.
EVERYONE SHOULD CARE ABOUT THE
MATHEMATICAL SCIENCES
In everyday life, terms that sound mathematical increasingly appear
in a variety of contexts. “Doing the math” is used by politicians to mean
analyzing the gains or losses of doing something, and language such as
“exponential,” “algorithm,” and “in the equation” frequently appears in
business and finance. A positive interpretation of this phenomenon is that
more and more people appreciate the mathematical sciences, but a not-so-
4 For this reason, this report tends to avoid the terms “core mathematics” and “applied
mathematics.” As can be seen in many places in the report, nearly all areas of the mathemati-
cal sciences can have applications.

OCR for page 15

22 THE MATHEMATICAL SCIENCES IN 2025
fortunate consequence is that the average person may not appreciate the
richness of the mathematical sciences.
The mathematical sciences include far more than numbers—they deal
with geometrical figures, logical patterns, networks, randomness, and pre-
dictions from incomplete data, to name only a few topics. And the math-
ematical sciences are part of almost every aspect of everyday life.
Consider a typical man (Bob) and a typical woman (Alice) in a devel-
oped society such as the United States. Whether they know it or not, their
lives depend intimately and deeply on the mathematical sciences; they are
wrapped in an intricate and elegant net woven with strands from the math-
ematical sciences. Here are some examples. A remarkable fact is that these
extremely varied applications depend crucially on the body of mathe atical
m
theory that has been developed over hundreds of years—on ingenious new
uses of theoretical developments from long ago, but also on some very
recent breakthroughs. Some of the pioneers of this body of theory were mo-
tivated by these applications; some by other applications that would seem
completely unconnected with these; and in many cases by the pure desire
to explore the fundamental structures of science and thought.
• Bob is awakened by a radio clock and usually listens to the news.
But he is unlikely to think twice, if at all, about how the radio can
receive signals, remove noises, and produce pleasant sound, yet all
of these tasks involve the mathematical and statistical methods of
signal processing.
• Alice may begin her day by watching the news in a recently pur-
chased high-definition LCD television. To achieve the high-quality
image that Alice takes for granted, many sophisticated steps are
required that depend on the mathematical sciences: compression
of digital signals, conversion from digital to analog and analog to
digital, image analysis and enhancement, and LCD performance
optimization.
• Bob and Alice love movies like Toy Story, Avatar, and Terminator 3.
A growing number of films feature characters and action scenes that
are the result of calculations performed by computers on math-
ematical models of movements, expressions, and actions based
on mathematical models. Obtaining a realistic impression of, say,
the collapse of downtown Los Angeles, requires intricate math-
ematical characterizations of explosions and their aftermath, dis-
played through the application of high-end computational power
to ophisticated mathematical insights about the fundamental equa-
s
tions governing fluids, solids, and heat.
• If Bob’s plans for his day (or the next few days) take into account
weather predictions, he is relying on the numerical solution of

OCR for page 15

INTRODUCTION 23
highly nonlinear, high-dimensional (meaning tens of millions of un-
knowns) equations and on statistical analysis of past observations
integrated with freshly collected information about atmosphere
and ocean conditions.
• To surf the Internet, Bob turns to a search engine, which performs
rapid searches using a sophisticated mathematical algorithm. The
earliest Web search techniques treated the interconnections of the
Web as a matrix (a two-dimensional data array), but modern
search methods have become much more sophisticated, incorporat-
ing protection from hacking and manipulation by outsiders. Effec-
tive Web search relies more than ever on sophisticated strategies
derived from the mathematical sciences.
• Alice is plagued by unwanted e-mails from irrelevant people who
want to cheat her or sell her items she does not want. A common
solution to this problem is a spam filter, which tries to detect un-
wanted or fraudulent e-mail using information and probability
theory. A major underlying tool is machine learning, in which
features of “legitimate” e-mails (as assessed by humans) are used
to train an algorithm that classifies incoming e-mails as legitimate
or as spam.
• When Alice needs to attend a meeting next month in Shanghai,
China, both the schedule of available flights and the price she will
pay for her ticket are almost certain to be determined using opti-
mization (by the airlines).
• Bob uses his cell phone almost constantly—a feature of modern life
enabled, for better or worse, by new developments in mathematical
and statistical information theory that involve wireless signal en-
coding, transmission, and processing, and by some highly ingenious
algorithms that route the calls.
• Alice’s office building consumes energy to run electric lights, tele-
phone landlines, a local computer network, running water, heating,
and cooling. Mathematical optimization and statistical techniques
are used to plan for efficient energy delivery based on information
about expected energy consumption and estimated safety factors
to protect against unusual events such as power outages. Because
of concerns about excessive energy usage, Alice’s utility companies
are investing in new mathematical and statistical techniques for
planning, monitoring, and controlling future energy systems.
• When Bob and Alice need medical or dental attention, they explic-
itly benefit from sophisticated applications of the mathematical sci-
ences. Everyone has heard of X-rays, CT scans, and MRIs, but few
people realize that modern medical and dental image analysis and
interpretation depend on complicated mathematical concepts, such

OCR for page 15

24 THE MATHEMATICAL SCIENCES IN 2025
as the Radon and Fourier transforms, whose theory was initially
developed during the nineteenth century. This example illustrates
the crucial observation that research in the mathematical sciences
has a very long shelf life in the sense that, because of their abstract
nature, discoveries in the mathematical sciences do not become
obsolete. Hence, a fresh insight about their application may arise
several decades (or more) after their publication.
• When Alice’s doctor prescribes a new medication, she depends on
decisions by pharmaceutical companies and the government about
the effectiveness and safety of new drugs and chemical treatments—
and those decisions in turn depend on ever-mproving statistical and
i
mathematical methods. Companies use mathematical models to pre-
dict how possible new drug molecules are expected to interact with
the body or its invaders and combinatorial and statistical methods
to explore the range of promising permutations.
• When Alice and Bob order products online, the processes used
for inventory management and control, delivery scheduling, and
pricing involve ingredients from the mathematical sciences such as
random matrices, scheduling and optimization algorithms, decision
theory, statistical regression, and machine learning.
• If Alice or Bob borrows money for a house, car, education, or to
pay off a credit card or invest savings in stocks, bonds, real estate,
mutual funds, or their 401(k)s, the mathematical sciences are hard
at work in the financial markets and the related micro- and macro
economics. Mathematical methods, statistical projections, and com-
puter modeling based on data are all necessary to function, prosper,
and plan for daily life and retirement in today’s array of global
markets. Today, many tools are brought directly to indi iduals in
v
customized applications for virtually every type of personal com-
puter and personal communication device.
• When Alice enters an airport or bus station, surveillance cameras are
likely to record her movements as well as those of everybody in the
area. The enormous task of processing and assessing the images from
multiple closed-circuit recordings is sometimes done by mathemati-
cal tools that automatically analyze movement patterns to determine
which people are likely to be carrying hidden weapons or explosives.
Similar techniques are being applied in stores and shopping centers
to assess which people are likely to be shoplifters or thieves.
• Even when Bob stops at the supermarket on the way home, he can-
not escape the mathematical sciences, which are used by retailers
to place products in the most appealing locations, to give him a
set of discount coupons chosen based on his past shopping history,
and to price items so that total sales revenue will be maximized.

OCR for page 15

INTRODUCTION 25
BOX 1-1
Four Facts Most People Don’t Know
About the Mathematical Sciences
Mathematical scientists have varied careers and styles of work. They do not spend
all their time calculating—though some do quite a bit—nor do most of them toil
in isolation on abstract theories. Most engage in collaborations of some sort.
While the majority are professors, there are also many mathematical scientists in
pharmaceutical and manufacturing industries, in government and national defense
laboratories, in computing and Internet-based businesses, and on Wall Street.
Some mathematicians prove theorems, but many others engage in other aspects
of quantitative modeling and problem solving. Mathematical scientists contribute
to every field of science, engineering, and medicine.
The mathematical sciences are always innovating. They do not consist of a fixed
collection of facts that are learned once and thereafter simply applied. While
theorems, once proved, may continue to be useful on a time-scale of centuries,
new theorems are constantly being discovered, and adapting existing knowledge
to new contexts is a never-ending process.
The United States is very good at the mathematical sciences. In spite of concerns
about the average skill of precollege students, the United States has an admirable
record of attracting the best mathematical and statistical talent to its universities,
and many of those people make their homes here after graduation. Assessments
of capabilities in mathematical sciences research find the United States to be at
or near the top in all areas of the discipline.
Mathematical scientists can change course during their careers. Because the
mathematical sciences deal with methods and general principles, researchers
need not maintain the same focus for their entire careers. For example, a stat-
istician might work on medical topics, climate models, and financial engineer-
ing in the course of one career. A mathematician might find that insights from
research in geometry are also helpful in a materials science problem, or in
a research challenge from brain imaging. And new types of mathematical science
jobs are constantly being created.
This chapter concludes with Box 1-1, “Four Facts Most People Don’t
Know About the Mathematical Sciences,” which illustrates some attributes
of today’s mathematical sciences.
STRUCTURE OF THE REPORT
Chapter 2 discusses recent accomplishments of the mathematical sci-
ences and the general health of the discipline. While the situation is very

OCR for page 15

26 THE MATHEMATICAL SCIENCES IN 2025
good at present, stresses and challenges are on the horizon. Chapter 3 sum-
marizes the current state of the mathematical sciences. Chapter 4 draws
from the inputs to the study and from committee members’ own experi-
ences to identify trends that are affecting the mathematical sciences. It also
identifies emerging stresses and challenges. Chapter 5 discusses the pipeline
that prepares people for mathematical science careers, while Chapter 6 dis-
cusses the ramifications of emerging changes in the academic environment.