Connections Between the Mathematical Sciences and Other Fields

In addition to ascertaining that the internal vitality of the mathematical sciences is excellent, as illustrated in Chapter 2, the current study found a striking expansion in the impact of the mathematical sciences on other fields, as well as an expansion in the number of mathematical sciences subfields that are being applied to challenges outside of the discipline. This expansion has been ongoing for decades, but it has accelerated greatly over the past 10-20 years. Some of these links develop naturally, because so much of science and engineering now builds on computation and simulation for which the mathematical sciences are the natural language. In addition, data-collection capabilities have expanded enormously and continue to do so, and the mathematical sciences are innately involved in distilling knowledge from all those data. However, mechanisms to facilitate linkages between mathematical scientists and researchers in other disciplines must be improved.

The impacts of mathematical science research can spread very in some cases, because a new insight can quickly be embodied in software without the extensive translation steps that exist between, say, basic research in chemistry and the use of an approved pharmaceutical. When mathematical sciences research produces a new way to compress or analyze data, value financial products, process a signal from a medical device or military system, or solve the equations behind an engineering simulation, the benefit can be realized quickly. For that reason, even government agencies or industrial sectors that seem disconnected from

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3
Connections Between the
Mathematical Sciences and Other Fields
INTRODUCTION
In addition to ascertaining that the internal vitality of the mathemati-
cal sciences is excellent, as illustrated in Chapter 2, the current study
found a striking expansion in the impact of the mathematical sciences
on other fields, as well as an expansion in the number of mathematical
sciences subfields that are being applied to challenges outside of the disci-
pline. This expansion has been ongoing for decades, but it has accelerated
greatly over the past 10-20 years. Some of these links develop naturally,
because so much of science and engineering now builds on computation and
simulation for which the mathematical sciences are the natural language.
In ddition, data-collection capabilities have expanded enormously and
a
continue to do so, and the mathematical sciences are innately involved in
distilling knowledge from all those data. However, mechanisms to facilitate
linkages between mathematical scientists and researchers in other disci-
plines must be improved.
The impacts of mathematical science research can spread very rapidly
in some cases, because a new insight can quickly be embodied in soft-
ware without the extensive translation steps that exist between, say,
b
asic research in chemistry and the use of an approved pharmaceutical.
When mathematical sciences research produces a new way to compress
or analyze data, value financial products, process a signal from a medi-
cal device or military system, or solve the equations behind an engineer-
ing simulation, the benefit can be realized quickly. For that reason, even
govern ent agencies or industrial sectors that seem disconnected from
m
58

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CONNECTION BETWEEN MATHEMATICAL SCIENCES AND OTHER FIELDS 59
the mathematical sciences have a vested interest in the maintance of a
strong mathe atical sciences enterprise for our nation. And because that
m
enterprise must be healthy in order to contribute to the supply of well-
trained individuals in science, technology, engineering, and mathematical
(STEM) fields, it is clear that everyone should care about the vitality of
the mathematical sciences.
This chapter discusses how increasing interaction with other fields has
broadened the definition of the mathematical sciences. It then documents
the importance of the mathematical sciences to a multiplicity of fields. In
many cases, it is possible to illustrate this importance by looking at major
studies by the disciplines themselves, which often list problems with a large
mathematical sciences component as being among their highest priorities.
Extensive examples of this are given in Appendix D.
BROADENING THE DEFINITION OF THE
MATHEMATICAL SCIENCES
Over the past decade or more, there has been a rapid increase in
the number of ways the mathematical sciences are used and the types of
mathematical ideas being applied. Because many of these growth areas
are fostered by the explosion in capabilities for simulation, computation,
and data analysis (itself driven by orders-of-magnitude increases in data
collection), the related research and its practitioners are often assumed to
fall within the umbrella of computer science. But in fact people with varied
backgrounds contribute to this work. The process of simulation-based sci-
ence and engineering is inherently very mathematical, demanding advances
in mathematical structures that enable modeling; in algorithm develop-
ment; in fundamental questions of computing; and in model validation,
uncertainty quantification, analysis, and optimization. Advances in these
areas are essential as computational scientists and engineers tackle greater
complexity and exploit advanced computing. These mathematical science
aspects demand considerable intellectual depth and are inherently interest-
ing for the mathematical sciences.
At present, much of the work in these growth areas—for example,
bioinformatics, Web-based companies, financial engineering, data analytics,
computational science, and engineering—is handled primarily by people
who would not necessarily be labeled “mathematical scientists.” But the
mathematical science content of such work, even if it is not research, is
considerable, and therefore it is critical for the mathematical sciences com-
munity to play a role, through education, research, and collaboration.
People with mathematical science backgrounds per se can bring different
perspectives that complement those of computer scientists and others, and
the combination of talents can be very powerful.

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60 THE MATHEMATICAL SCIENCES IN 2025
There is no precise definition of “the mathematical sciences.” The
followng definition was used in the 1990 report commonly known as the
i
David II report after the authoring committee’s chair, Edward E. David:
The discipline known as the mathematical sciences encompasses core (or
pure) and applied mathematics, plus statistics and operations research,
and extends to highly mathematical areas of other fields such as theoreti-
cal computer science. The theoretical branches of many other fields—for
instance, biology, ecology, engineering, economics—merge seamlessly with
the mathematical sciences.1
The 1998 Odom report implicitly used a similar definition, as embodied
in Figure 3-1, adapted from that report.
Figure 3-1 captures an important characteristic of the mathematical
sciences—namely, that they overlap with many other disciplines of science,
engineering, and medicine, and, increasingly, with areas of business such
as finance and marketing. Where the small ellipses overlap with the main
ellipse (representing the mathematical sciences), one should envision a mu-
tual entwining and meshing, where fields overlap and where research and
people might straddle two or more disciplines. Some people who are clearly
affiliated with the mathematical sciences may have extensive interactions
and deep familiarity with one or more of these overlapping disciplines.
And some people in those other disciplines may be completely comfortable
in mathematical or statistical settings, as will be discussed further. These
interfaces are not clean lines but instead are regions where the disciplines
blend. A large and growing fraction of modern science and engineering is
“mathematical” to a significant degree, and any dividing line separating
the more central and the interfacial realms of the mathematical sciences
is sure to be arbitrary. It is easy to point to work in theoretical physics or
theoretical computer science that is indistinguishable from research done
by mathematicians, and similar overlap occurs with theoretical ecology,
mathematical biology, bioinformatics, and an increasing number of fields.
This is not a new phenomenon—for example, people with doctorates in
mathematics, such as Herbert Hauptman, John Pople, John Nash, and
Walter Gilbert, have won Nobel prizes in chemistry or economics—but it is
becoming more widespread as more fields become amenable to mathemati-
cal representations. This explosion of opportunities means that much of
twenty-first century research is going to be built on a mathematical science
foundation, and that foundation must continue to evolve and expand.
1 NRC, 1990, Renewing U.S. Mathematics: A Plan for the 1990s. National Academy Press,
Washington, D.C.

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CONNECTION BETWEEN MATHEMATICAL SCIENCES AND OTHER FIELDS 61
…
Mathematical
Sciences
…
FIGURE 3-1 The mathematical sciences and their interfaces. SOURCE: Adapted
from National Science Foundation, 1998, Report of the Senior Assessment Panel for
the International Assessment of the U.S. Mathematical Sciences, NSF, Arlington, Va.
Note that the central ellipse in Figure3-1 is not subdivided. The com-
Figure 3-1
mittee members—like many others who have examined the mathematical
sciences—believe that it is important to consider the mathematical sciences
as a unified whole. Distinctions between “core” and “applied” mathe atics
m
increasingly appear artificial; in particular, it is difficult today to find an
area of mathematics that does not have relevance to applications. It is true
that some mathematical scientists primarily prove theorems, while others
primarily create and solve models, and professional reward systems need to
take that into account. But any given individual might move between these
modes of research, and many areas of specialization can and do include
both kinds of work. Overall, the array of mathematical sciences share a
commonality of experience and thought processes, and there is a long his-
tory of insights from one area becoming useful in another.
Thus, the committee concurs with the following statement made in the
2010 International Review of Mathematical Sciences (Section 3.1):

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62 THE MATHEMATICAL SCIENCES IN 2025
A long-standing practice has been to divide the mathematical sciences into
categories that are, by implication, close to disjoint. Two of the most com-
mon distinctions are drawn between “pure” and “applied” mathematics,
and between “mathematics” and “statistics.” These and other categories
can be useful to convey real differences in style, culture and methodology,
but in the Panel’s view, they have produced an increasingly negative effect
when the mathematical sciences are considered in the overall context of
science and engineering, by stressing divisions rather than unifying prin-
ciples. Furthermore, such distinctions can create unnecessary barriers and
tensions within the mathematical sciences community by absorbing energy
that might be expended more productively. In fact, there are increasing
overlaps and beneficial interactions between different areas of the math-
ematical sciences. . . . [T]he features that unite the mathematical sciences
dominate those that divide them.2
What is this commonality of experience that is shared across the math-
ematical sciences? The mathematical sciences aim to understand the world
by performing formal symbolic reasoning and computation on abstract
structures. One aspect of the mathematical sciences involves unearthing and
understanding deep relationships among these abstract structures. Another
aspect involves capturing certain features of the world by abstract structures
through the process of modeling, performing formal reasoning on these ab-
stract structures or using them as a framework for computation, and then
reconnecting back to make predictions about the world—often, this is an
iterative process. A related aspect is to use abstract reasoning and structures
to make inferences about the world from data. This is linked to the quest to
find ways to turn empirical observations into a means to lassify, order, and
c
understand reality—the basic promise of science. Through the mathematical
sciences, researchers can construct a body of knowledge whose interrelations
are understood and where whatever understanding one needs can be found
and used. The mathematical sciences also serve as a natural conduit through
which concepts, tools, and best practices can migrate from field to field.
A further aspect of the mathematical sciences is to investigate how
to make the process of reasoning and computation as efficient as possible
and to also characterize their limits. It is crucial to understand that these
different aspects of the mathematical sciences do not proceed in isolation
from one another. On the contrary, each aspect of the effort enriches the
others with new problems, new tools, new insights, and—ultimately—new
paradigms.
Put this way, there is no obvious reason that this approach to knowledge
should have allowed us to understand the physical world. Yet the entire
2 Engineering
and Physical Sciences Research Council (EPSRC), 2010, International Review
of Mathematical Science. EPSRC, Swindon, U.K., p. 10.

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CONNECTION BETWEEN MATHEMATICAL SCIENCES AND OTHER FIELDS 63
mathematical sciences enterprise has proven not only extraordinarily effec-
tive, but indeed essential for understanding our world. This conundrum is
often referred to as “the unreasonable effectiveness of mathematics,” men-
tioned in Chapter 2.
In light of that “unreasonable effectiveness,” it is even more striking
to see, in Figure 3-2, which is analogous to Figure 3-1, how far the math-
ematical sciences have spread since the Odom report was released in 1998.
Reflecting the reality that underlies Figure 3-2, this report takes a very
inclusive definition of “the mathematical sciences.” The discipline encom-
passes the broad range of diverse activities related to the creation and analy-
sis of mathematical and statistical representations of concepts, systems, and
processes, whether or not the person carrying out the activity identifies as a
M Fin ic s
ar an m er
Ma ke ce ono put ce al n g
nu tin
g
... Ec om ien ctriceeri
fac C Sc Ele gin
tur ng
i ng En vil eeri
i gin
C n
E al
Def
ens hanic
Mecineering
e
Eng
Geosciences
Communications
Mathematical
Astronomy
Sciences
ation Phy
Informessing sics
Proc Ma
ter
nt ials
n me Ch
em
er ta i l
cia ks istr
Ent Sotwor e ... Eco Biolo y
Ne icin log gy
M ed y
FIGURE 3-2 The mathematical sciences and their interfaces in 2013. The number
of interfaces has increased since the time of Figure 3-1, and the mathematical sci-
ences themselves have broadened in response. The academic science and engineering
enterprise is suggested by the right half of the figure, while broader areas of human
endeavor are indicated on the left. Within the academy, the mathematical sciences
are playing a more integrative and foundational role, while within society more
broadly their impacts affect all of us—although that is often unappreciated because
it is behind the scenes. This schematic is notional, based on the committee’s varied
and subjective experience rather than on specific data. It does not attempt to repre-
sent the many other linkages that exist between academic disciplines and between
those disciplines and the broad endeavors on the left, only because the full interplay
is too complex for a two-dimensional schematic.

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64 THE MATHEMATICAL SCIENCES IN 2025
mathematical scientist. The traditional areas of the mathematical sciences
are certainly included. But many other areas of science and engineering
are deeply concerned with building and evaluating mathematical models,
exploring them computationally, and analyzing enormous amounts of ob-
served and computed data. These activities are all inherently mathe atical
m
in nature, and there is no clear line to separate research efforts into those
that are part of the mathematical sciences and those that are part of com-
puter science or the discipline for which the modeling and analysis are
performed.3 The committee believes the health and vitality of the discipline
are maximized if knowledge and people are able to flow easily throughout
that large set of endeavors.
So what is the “mathematical sciences community”? It is the collection
of people who are advancing the mathematical sciences discipline. Some
members of this community may be aligned professionally with two or
more disciplines, one of which is the mathematical sciences. (This alignment
is reflected, for example, in which conferences they attend, which journals
they publish in, which academic degrees they hold, and which academic
departments they belong to.) There is great value in the mathematical sci-
ences welcoming these “dual citizens”; their involvement is good for the
mathematical sciences, and it enriches the ways in which other fields can
approach their work.
The collection of people in the areas of overlap is large. It includes
s
tatisticians who work in the geosciences, social sciences, bioinformatics,
and other areas that, for historical reasons, became specialized offshoots of
statistics. It includes some fraction of researchers in scientific computing and
computational science and engineering. It includes number theorists who
contribute to cryptography, and real analysts and statisticians who contrib-
ute to machine learning. It includes operations researchers, some computer
scientists, and physicists, chemists, ecologists, biologists, and economists
who rely on sophisticated mathematical science approaches. Some of the
engineers who advance mathematical models and computational simulation
are also included. It is clear that the mathematical sciences now extend far
beyond the definitions implied by the institutions— cademic departments,
a
funding sources, professional societies, and principal journals—that support
the heart of the field.
As just one illustration of the role that researchers in other fields play
in the mathematical sciences, the committee examined public data4 on
N
ational Science Foundation (NSF) grants to get a sense of how much of
the research supported by units other than the NSF Division of Mathe
3 Most of the other disciplines shown in Figure 3-2 also have extensive interactions with
other fields, but the full interconnectedness of these endeavors is omitted for clarity.
4 Available at http://www.nsf.gov/awardsearch/.

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CONNECTION BETWEEN MATHEMATICAL SCIENCES AND OTHER FIELDS 65
matical Sciences (DMS) has resulted in publications that appeared in jour-
nals readily recognized as mathematical science ones or that have a title
strongly suggesting mathematical or statistical content. While this exercise
was necessarily subjective and far from exhaustive, it gave an indication
that NSF’s support for the mathematical sciences is de facto broader than
what is supported by DMS. It also lent credence to the argument that the
mathematical sciences research enterprise extends beyond the set of indi-
viduals who would traditionally be called mathematical scientists. This
exercise revealed the following information:
• Grants awarded over the period 2008-2011 by NSF’s Division of
Computing and Communication Foundations (part of the Direc-
torate for Computer and Information Science and Engineering) led
to 262 publications in the areas of graphs and, to a lesser extent,
foundations of algorithms.
• Grants awarded over 2004-2011 by the Division of Physics led to
148 publications in the general area of theoretical physics.
• Grants awarded over 2007-2011 by the Division of Civil, Mechani-
cal, and Manufacturing Innovation in NSF’s Engineering Directorate
led to 107 publications in operations research.
This cursory examination also counted 15 mathematical science publica-
tions resulting from 2009-2010 grants from NSF’s Directorate for Biological
Sciences. (These publication counts span different ranges of years because
the number of publications with apparent mathematical sciences content
varies over time, probably due to limited-duration funding initiatives.) For
comparison, DMS grants that were active in 2010 led to 1,739 publications.
Therefore, while DMS is clearly the dominant NSF supporter of mathe
matical science research, other divisions contribute in a nontrivial way.
Analogously, membership figures from the Society for Industrial and
Applied Mathematics (SIAM) demonstrate that a large number of indi
viduals who are affiliated with academic or industrial departments other
than mathematics or statistics nevertheless associate themselves with this
mathematical science professional society. Figure 3-3 shows the departmen-
tal affiliation of SIAM’s nonstudent members.
A recent analysis tried to quantify the size of this community on the
interaces of the mathematical sciences.5 It found that faculty members in
f
50 of the top U.S. mathematics departments—who would therefore be in the
central disk in Figure 3-2—have published in aggregate some 64,000 research
papers since 1971 that have been indexed by Zentralblatt MATH (and thus
5 Joseph Grcar, 2011, Mathematics turned inside out: The intensive faculty versus the
extensive faculty. Higher Education 61(6): 693-720.

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66 THE MATHEMATICAL SCIENCES IN 2025
Physical
Sciences
Engineering 4%
14% Computer
Mathematics,
Science &
Applied
Information
Mathematics,
Sciences
Statistics, & Biological 9%
Other Sciences
Mathematical 1% Operations
Sciences Research,
59% Other Management &
10% Industrial
Engineering
3%
FIGURE 3-3 SIAM members identify the primary department with which they are
affiliated. This figure shows the fraction of 6,269 nonstudent members identifying
with a particular category.
Figure 3-3
can be inferred to have mathematical content). Over the same period, some
75,000 research papers indexed by Zentralblatt MATH were published by
faculty members in other departments of those same 50 universities. The
implication is that a good deal of mathematical sciences esearch—as much
r
as half of the enterprise—takes place outside departments of mathematics.6
This also suggests that the scope of most mathematics departments may not
mirror the true breadth of the mathematical sciences.
That analysis also created a Venn diagram, reproduced here as Fig-
ure 3-4, that is helpful for envisioning how the range of mathematical sci-
ence research areas map onto an intellectual space that is broader than that
covered by most academic mathematics departments. (The diagram also
shows how the teaching foci of mathematics and nonmathematics depart-
ments differ from their research foci.)
IMPLICATIONS OF THE BROADENING OF
THE MATHEMATICAL SCIENCES
The tremendous growth in the ways in which the mathematical sciences
are being used stretches the mathematical science enterprise—its people,
teaching, and research breadth. If our overall research enterprise is operat-
6 Some of those 75,000 papers are attributable to researchers in departments of statistics or
operations research, which we would clearly count as being in the central disk of Figure 3-2.
But the cited paper notes that only about 17 percent of the research indexed by Zentralblatt
MATH is classified as dealing with statistics, probability, or operations research.

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CONNECTION BETWEEN MATHEMATICAL SCIENCES AND OTHER FIELDS 67
FIGURE 3-4 Representation of the research and teaching span of top mathematics
departments and of nonmathematics departments in the same academic institutions.
Subjects most published are shown in italics; subjects most taught are underscored.
Figure 3-4
SOURCE: Joseph Grcar, 2011, Mathematics turned inside out: The intensive faculty
versus the extensive faculty. Higher Education 61(6):693-720, Figure 8. The num-
Bitmapped
bers correspond to the following Zentralblatt MATH classifications:
05 Combinatorics 60 Probability theory
11 Number theory 62 Statistics
14 Algebraic geometry 65 Numerical analysis
15 Linear, multilinear algebra 68 Computer science
20 Group theory 74 Mechanics of deformable solids
26 Real functions 76 Fluid mechanics
32 Several complex variables 80 Classical thermodynamics
34 Ordinary differential equations 81 Quantum theory
35 Partial differential equations 86 Geophysics
37 Dynamical systems 90 Operations research
42 Fourier analysis 91 Game theory, economics
46 Functional analysis 92 Biology
53 Differential geometry 93 Systems theory, control
57 Manifolds, cell complexes 94 Information and communications
58 Global analysis

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68 THE MATHEMATICAL SCIENCES IN 2025
ing well, the researchers who traditionally call themselves mathematical
s
cientists—the central ellipse in Figure 3-2—are in turn stimulated by the
challenges from the frontiers, where new types of phenomena or data stimu-
late fresh thinking about mathematical and statistical modeling and new
technical challenges stimulate deeper questions for the mathematical sciences.
Many people with mathematical sciences training who now work at
those frontiers—operations research, computer science, engineering, biology,
economics, and so on—have told the committee that they appreciate the
grounding provided by their mathematical science backgrounds and that,
to them, it is natural and healthy to consider the entire family tree as being
a unified whole. Many mathematical scientists and academic math depart-
ments have justifiably focused on core areas, and this is natural in the sense
that no other community has a mandate to ensure that the core areas remain
strong and robust. But it is essential that there be an easy flow of concepts,
results, methods, and people across the entirety of the mathematical sciences.
For that reason, it is essential that the mathematical sciences community
actively embraces the broad community of researchers who contribute intel-
lectually to the mathematical sciences, including people who are profession-
ally associated with another discipline.
Anecdotal information suggests that the number of graduate students
receiving training in both mathematics and another field—from biology to
engineering—has increased dramatically in recent years. This trend is rec-
ognized and encouraged, for example, by the Simons Foundation’s Math+X
program, which provides cross-disciplinary professorships and support for
graduate students and postdoctoral researchers who straddle two fields. If
this phenomenon is as general as the committee believes it to be, it shows
how mathematic sciences graduate education is contributing to science and
engineering generally and also how the interest in interfaces is growing. In
order for the community to rationally govern itself, and for funding agen-
cies to properly target their resources, it is necessary to begin gathering
data on this trend.
Recommendation 3-1: The National Science Foundation should sys-
tematically gather data on such interactions—for example, by surveying
departments in the mathematical sciences for the number of enroll-
ments in graduate courses by students from other disciplines, as well as
the number of enrollments of graduate students in the mathematical sci-
ences in courses outside the mathematical sciences. The most effective
way to gather these data might be to ask the American Mathematical
Society to extend its annual questionnaires to include such queries.
Program officers in NSF/DMS and in other funding agencies are aware
of many overlaps between the mathematical sciences and other disciplines,

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82 THE MATHEMATICAL SCIENCES IN 2025
diffeomorphisms. Shape analysis also comes into play in virtual surgery,
where surgical outcomes are simulated on the computer before being tried
on a patient, and in remote surgery for the battlefield. Here one needs to
combine mathematical modeling techniques based on the differential equa-
tions describing tissue mechanics with shape description and visualization
methods.
As our society is learning somewhat painfully, data must be protected.
The need for privacy and security has given rise to the areas of privacy-
preserving data mining and encrypted computation, where one wishes to
be able to analyze a data set without compromising privacy, and to be able
to do computations on an encrypted data set while it remains encrypted.
CONTRIBUTIONS OF MATHEMATICAL SCIENCES
TO SCIENCE AND ENGINEERING
The mathematical sciences have a long history of interaction with
other fields of science and engineering. This interaction provides tools and
insights to help those other fields advance; at the same time, the efforts of
those fields to push research frontiers routinely raise new challenges for
the mathematical sciences themselves. One way of evaluating the state
of the mathematical sciences is to examine the richness of this interplay.
Some of the interactions between mathematics and physics are described in
Chapter 2, but the range extends well beyond physics. A compelling illus
tration of how much other fields rely on the mathematical sciences arises
from examining those fields’ own assessments of promising directions and
identifying the directions that are dependent on parallel progress in the
mathematical sciences. A number of such illustrations have been collected
in Appendix D.
CONTRIBUTIONS OF MATHEMATICAL SCIENCES TO INDUSTRY
The role of the mathematical sciences in industry has a long history,
going back to the days when the Egyptians used the 3-4-5 right triangle to
restore boundaries of farms after the annual flooding of the Nile. That said,
the recent period is one of remarkable growth and diversification. Even in
old-line industries, the role of the mathematical sciences has expanded. For
example, whereas the aviation industry has long used mathematics in the
design of airplane wings and statistics in ensuring quality control in pro-
duction, now the mathematical sciences are also crucial to GPS and navi-
gation systems, to simulating the structural soundness of a design, and to
optimizing the flow of production. Instead of being used just to streamline
cars and model traffic flows, the mathematical sciences are also involved in
the latest developments, such as design of automated vehicle detection and

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CONNECTION BETWEEN MATHEMATICAL SCIENCES AND OTHER FIELDS 83
avoidance systems that may one day lead to automated driving. Whereas
statistics has long been a key element of medical trials, now the mathemati-
cal sciences are involved in drug design and in modeling new ways for drugs
to be delivered to tumors, and they will be essential in making inferences
in circumstances that do not allow double-blind, randomized clinical trials.
The financial sector, which once relied on statistics to design portfolios
that minimized risk for a given level of return, now makes use of statistics,
m
achine learning, stochastic modeling, optimization, and the new science
of networks in pricing and designing securities and in assessing risk.
What is most striking, however, is the number of new industries that the
mathematical sciences are a part of, often as a key enabler. The encryption
industry makes use of number theory to make Internet commerce possible.
The “search” industry relies on ideas from the mathematical sciences to make
the Internet’s vast resources of information searchable. The social networking
industry makes use of graph theory and machine learning. The animation and
computer game industry makes use of techniques as diverse as differential
geometry and partial differential equations. The biotech industry heavily
uses the mathematical sciences in modeling the action of drugs, searching
genomes for genes relevant to human disease or relevant to bioengineered
organisms, and discovering new drugs and understanding how they might
act. The imaging industry uses ideas from differential geometry and signal
processing to procure minimally invasive medical and industrial images and,
within medicine, adds methods from inverse problems to design targeted
radi tion therapies and is moving to incorporate the new field of computa-
a
tional anatomy to enable remote surgery. The online advertising industry uses
ideas from game theory and discrete mathematics to price and bid on online
ads and methods from statistics and machine learning to decide how to target
those ads. The marketing industry now employs sophisticated statistical and
machine learning techniques to target customers and to choose locations for
new stores. The credit card industry uses a variety of methods to detect fraud
and denial-of-service attacks. Political campaigns now make use of complex
models of the electorate, and election-night predictions rely on integrating
these models with exit polls. The semiconductor industry uses optimization
to design computer chips and in simulating the manufacture and behavior
of designer materials. The mathematical sciences are now present in almost
every industry, and the range of mathematical sciences being used would have
been unimaginable a generation ago.
This point is driven home by the following list of case studies assembled
for the SIAM report Mathematics in Industry.13 This list is just illustrative,
but its breadth is striking:
13 Societyfor Industrial and Applied Mathematics, 2012, Mathematics in Industry. SIAM,
Philadelphia, Pa.

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84 THE MATHEMATICAL SCIENCES IN 2025
• Predictive analytics,
• Image analysis and data mining,
• Scheduling and routing of deliveries,
• Mathematical finance,
• Algorithmic trading,
• Systems biology,
• Molecular dynamics,
• Whole-patient models,
• Oil basin modeling,
• Virtual prototyping,
• Molecular dynamics for product engineering,
• Multidisciplinary design optimization and computer-aided design,
• Robotics,
• Supply chain management,
• Logistics,
• Cloud computing,
• Modeling complex systems,
• Viscous fluid flow for computer and television screen design,
• Infrastructure management for smart cities, and
• Computer systems, software, and information technology.
The reader is directed to the SIAM report to see the details of these case
studies,14 which provide many examples of the significant and cost-effective
impact of mathematical science expertise and research on innovation, eco-
nomic competitiveness, and national security.
Another recent report on the mathematical sciences in industry came
to the following conclusions:
It is evident that, in view of the ever-increasing complexity of real life
applications, the ability to effectively use mathematical modelling, simula-
tion, control and optimisation will be the foundation for the technological
and economic development of Europe and the world.15
Only [the mathematical sciences] can help industry to optimise more and
more complex systems with more and more constraints.16
However, that report also points out the following truism:
[Engineering] designers use virtual design environments that rely heavily
on mathematics, and produce new products that are well recognised by
14 Ibid., pp. 9-24.
15 European Science Foundation, 2010, Mathematics and Industry. Strasbourg, France, p. 8.
16 Ibid., p. 12.

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CONNECTION BETWEEN MATHEMATICAL SCIENCES AND OTHER FIELDS 85
management. The major effort concerned with the construction of reli-
able, robust and efficient virtual design environments is, however, not
recognised. As a result, mathematics is not usually considered a key tech-
nology in industry. The workaround for this problem usually consists of
leaving the mathematics to specialised small companies that often build
on mathematical and software solutions developed in academia. Unfortu-
nately, the level of communication between these commercial vendors and
their academic partners with industry is often at a very low level. This, in
turn, leads to the observation that yesterday’s problems in industry can be
solved, but not the problems of today and tomorrow. The latter can only
be addressed adequately if an effort is made to drastically improve the
communication between industrial designers and mathematicians.17
One way to address this communication challenge, of course, is to include
high-caliber mathematical scientists within corporate R&D units.
The mathematical functions of greatest value in these and other success-
ful applications were characterized by R&D managers in 1996 as follows:18
• Modeling and simulation,
• Mathematical formulation of problems,
• Algorithm and software development,
• Problem-solving,
• Statistical analysis,
• Verifying correctness, and
• Analysis of accuracy and reliability.
To this list, the European Science Foundation report adds optimization,
n
oting (p. 12) that “due to the increased computational power and the
achievements obtained in speed-up of algorithms . . . optimization of prod-
ucts has [come] into reach. This is of vital importance to industry.” This
is a very important development, and it opens up new challenges for the
mathematical sciences, such as how to efficiently explore design options
and how to characterize the uncertainties of this computational sampling
of the space.
These sorts of opportunities were implicitly recognized in a report
from the Chinese Academy of Sciences (CAS), Science and Technology in
China: A Roadmap to 2050.19 That report identified eight systems of im-
portance for socioeconomic development: sustainable energy and resources,
17 Ibid.,
p. 9.
18 Societyfor Industrial and Applied Mathematics, 1996, Mathematics in Industry. Available
at http://www.siam.org/reports/mii/1996/listtables.php#lt4.
19 CAS, 2010, Science & Technology in China: A Roadmap to 2050. Springer. Available
online at http://www.bps.cas.cn/ztzl/cx2050/nrfb/201008/t20100804_2917262.html.

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86 THE MATHEMATICAL SCIENCES IN 2025
advanced materials and intelligent manufacturing, ubiquitous information
networking, ecological and high-value agriculture and biological industry,
health assurance, ecological and environmental conservation, space and
ocean exploration, and national and public security. To support the matura-
tion of these systems, the report goes on to identify 22 science and technol-
ogy initiatives. Of these, three will build on the mathematical sciences: an
initiative to create “ubiquitous informationized manufacturing system[s],”
another to develop exascale computing technology, and a third in basic
cross-disciplinary research in mathematics and complex systems.
The last-mentioned initiative is intended to research the “basic princi-
ples behind various kinds of complex systems,” and the report recommends
that major efforts be made in the following research directions:
• Mathematical physics equations;
• Multiscale modeling and computation of complex systems;
• Machine intelligence and mathematics mechanization;
• Theories and methods for stochastic structures and data;
• Collective behaviors of multiagent complex systems, their control
and intervention; and
• Complex stochastic networks, complex adaptive systems, and re-
lated areas.
In particular, the report recommends that due to the fundamental impor-
tance of complex systems, the Chinese government should provide sus-
tained and steady support for research into such systems so as to achieve
major accomplishments in this important field.
One more example of the role of the mathematical sciences in indus-
try comes from the NRC report Visionary Manufacturing Challenges for
2020,20 which identified R&D that would be necessary to advance national
capabilities in manufacturing. A number of these capabilities rely on re-
search in modeling and simulation, control theory, and informatics:
• ltimately, simulations of manufacturing systems would be based on a
U
unified taxonomy for process characteristics that include human char-
acteristics in process models. Other areas for research include a general
theory for adaptive systems that could be translated into manufacturing
processes, systems, and the manufacturing enterprise; tools to opti-
mize design choices to incorporate the most affordable manufacturing
a
pproaches; and systems research on the interaction between workers
and manufacturing processes for the development of adaptive, flexible
controls. (p. 39)
20 NRC, 1998, Visionary Manufacturing Challenges for 2020. National Academy Press,
Washington, D.C.

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CONNECTION BETWEEN MATHEMATICAL SCIENCES AND OTHER FIELDS 87
• odeling and simulation capabilities for evaluating process and enter-
M
prise scenarios will be important in the development of reconfigurable
enterprises. . . . Virtual prototyping of manufacturing processes and
systems will enable manufacturers to evaluate a range of choices for
optimizing their enterprises. Promising areas for the application of
modeling and simulation technology for reconfigurable systems include
neural networks for optimizing reconfiguration approaches and artifi-
cial intelligence for decision making. . . . Processes that can be adapted
or readily reconfigured will require flexible sensors and control algo-
rithms that provide precision process control of a range of processes
and environments. (pp. 39-40)
• esearch in enterprise modeling tools will include ‘soft’ modeling (e.g.,
R
models that consider human behavior as an element of the system
and models of information flow and communications), the optimiza-
tion and integration of mixed models, the optimization of hardware
systems, models of organizational structures and cross-organizational
behavior, and models of complex or nonlinear systems and processes.
(p. 44)
• uture information systems will have to be able to collect and sift
F
through vast amounts of information. (p. 44)
Today’s renewed emphasis on advanced manufacturing, as exempli-
fied by the recent report Capturing Domestic Competitive Advantage in
Advanced Manufacturing,21 also relies implicitly on advances in the math-
ematical sciences. The 11 cross-cutting technology areas identified in that
report (p. 18) as top candidates for R&D investments rely in multiple ways
on modeling, simulation, and analysis of complex systems, analysis of large
amounts of data, control, and optimization.
CONTRIBUTIONS OF MATHEMATICAL SCIENCES
TO NATIONAL SECURITY
National security is another area that relies heavily on the mathemati-
cal sciences. The National Security Agency (NSA), for example, employs
roughly 1,000 mathematical scientists, although the number might be half
that or twice that depending on how one defines such scientists.22 They
include people with backgrounds in core and applied mathematics, prob-
ability, and statistics, but people with computer science backgrounds are
not included in that count. NSA hires some 40-50 mathematicians per year,
and it tries to keep that rate steady so that the mathematical sciences com-
21 White House, Office of Science and Technology Policy, 2012.
22 AlfredHales, former head of the Institute for Defense Analyses, Center for Communi-
cations Research–La Jolla, presentation to the committee in December 2010. IDA’s La Jolla
center conducts research for the NSA.

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88 THE MATHEMATICAL SCIENCES IN 2025
munity knows it can depend on that level of hiring. The NSA is interested
in maintaining a healthy mathematical sciences community in the United
States, including a sufficient supply of well-trained U.S. citizens. Most of
NSA’s mathematical sciences hiring is at the graduate level, about evenly
split between the M.S. and Ph.D. levels. The agency hires across almost all
fields of the mathematical sciences rather than targeting specific subfields,
because no one can predict the mix of skills that will be important over an
employee’s decades-long career. For example, few mathematicians would
have guessed decades ago that elliptic curves would be of vital interest to
NSA, and now they are an important specialty underlying cryptology.
While cryptology is explicitly dependent on mathematics, many other
links exist between the mathematical sciences and national security. One
example is analysis of networks (discussed in the next section), which is
very important for national defense. Another is scientific computing. One of
the original reasons for John von Neumann’s interest in creating one of the
first computers was to be able to do the computations necessary to simulate
what would happen inside a hydrogen bomb. Years later, with atmospheric
and underground nuclear testing banned, the country relies once again
on simulations, this time to maintain the readiness of its nuclear arsenal.
Because national defense relies in part on design and manufacturing of
cutting-edge equipment, it also relies on the mathematical sciences through
their contributions to advanced engineering and manufacturing. The level
of sophistication of these tools has ratcheted steadily upward. The math-
ematical sciences are also essential to logistics, simulations used for training
and testing, war-gaming, image and signal analysis, control of satellites and
aircraft, and test and evaluation of new equipment. Figure 3-5, reproduced
from Fueling Innovation and Discovery: The Mathematical Sciences in the
21st Century,23 captures the broad range of ways in which the mathemati-
cal sciences contribute to national defense.
New devices, on and off the battlefield, have come on stream and
furnish dizzying quantities of data, more than can currently be analyzed.
Devising ways to automate the analysis of these data is a highly mathemati-
cal and statistical challenge. Can a computer be taught to make sense of a
satellite image, detecting buildings and roads and noticing when there has
been a major change in the image of a site that is not due to seasonal varia-
tion? How can one make use of hyperspectral data, which measure light
reflected in all frequencies of the spectrum, in order to detect the smoke
plume from a chemical weapons factory? Can one identify enemy vehicles
and ships in a cluttered environment? These questions and many others are
inherently dependent on advances in the mathematical sciences.
23 National Research Council, 2012. The National Academies Press, Washington, D.C.

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CONNECTION BETWEEN MATHEMATICAL SCIENCES AND OTHER FIELDS 89
A very serious threat that did not exist in earlier days is that crucial net-
works are constantly subject to sophisticated attacks by thieves, mischief-
makers, and hackers of unknown origin. Adaptive techniques based on the
mathematical sciences are essential for reliable detection and prevention of
such attacks, which grow in sophistication to elude every new strategy for
preventing them.
The Department of Defense has adopted seven current priority areas for
science and technology investment to benefit national security.
1. Data to decisions. Science and applications to reduce the cycle time
and manpower requirements for analysis and use of large data sets.
2. Engineered resilient systems. Engineering concepts, science, and
design tools to protect against malicious compromise of weapon
systems and to develop agile manufacturing for trusted and assured
defense systems.
3. Cyber science and technology. Science and technology for efficient,
effective cyber capabilities across the spectrum of joint operations.
4. Electronic warfare/electronic protection. New concepts and technol-
ogy to protect systems and extend capabilities across the electro-
magnetic spectrum.
5. Countering weapons of mass destruction (WMD): Advances in
DOD’s ability to locate, secure, monitor, tag, track, interdict, elimi-
nate, and attribute WMD weapons and materials.
6. Autonomy. Science and technology to achieve autonomous sys-
tems that reliably and safely accomplish complex tasks in all
environments.
7. Human systems. Science and technology to enhance human-
machine interfaces, increasing productivity and effectiveness across
a broad range of missions.24
While the mathematical sciences are clearly of importance to the first
and third of these priority areas, they also have key roles to play in sup-
port of all of the others. Advances in the mathematical sciences that allow
simulation-based design, testing, and control of complex systems are essen-
tial for creating resilient systems. Improved methods of signal analysis and
processing, such as faster algorithms and more sensitive schemes for pat-
tern recognition, are needed to advance electronic warfare and protection.
Rapidly developing tools for analyzing social networks, which are based on
novel methods of statistical analysis of networks, are being applied in order
24 Department of Defense, 2012, Memorandum on Science and Technology (S&T) Priori
ties for Fiscal Years 2013·17 Planning. Available at http://acq.osd.mil/chieftechnologist/
publications/docs/OSD%2002073-11.pdf. Accessed May 3, 2012.

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90 THE MATHEMATICAL SCIENCES IN 2025
Mathematical Sciences Inside... The Bat
The mathematical sciences underpin many of the technologies on which national defense
depends. Cutting-edge mathematics and statistics lie behind smart sensors and ad-
vanced control and communications. They are used
throughout the research, development, engineering,
and test and evaluation process. They are embed-
ded in simulation systems for planning and for war-
fighter training. Since World War II, the mathemati-
cal sciences have been key contributors to national
defense, and their utility continues to expand. This
graphic illustrates some of those impacts.
The mathematical sciences are used in planning
logistics, deployments, and scenario evaluations
for complex operations.
Mathematical simulations allow
predictions of the spread of smoke and
chemical and biological agents in urban
terrain.
Mathematics is used to design
Mathematics and statistics advanced armor.
underpin tools for control
and communications in
tactical operations.
34
FIGURE 3-5 Mathematical sciences inside the battlefield.

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CONNECTION BETWEEN MATHEMATICAL SCIENCES AND OTHER FIELDS 91
Inside... The Battleﬁeld
Signal analysis and control
theory are essential for drones.
Large-scale computational codes are
used to design aircraft, simulate ﬂight
paths, and train personnel.
Signal processing facilitates
communication capabilities.
Mobile translation systems employ voice
recognition software to reduce language Satellite-guided weapons utilize GPS
barriers when human linguists are not for highly-precise targeting, while
available. More generally, math-based mathematical methods improve
simulations are used in mission and ballistics.
specialty training.
Modeling and simulation facilitates
trade-off analysis during vehicle
design, while statistics underpins
test and evaluation.
35

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92 THE MATHEMATICAL SCIENCES IN 2025
to improve our capabilities in counter-WMD. And mathematical science
methods of machine learning are necessary for improving our capabilities
in autonomy and human-machine interfaces. Computational neuroscience,
which relies heavily on the mathematical sciences, is also a promising area
for future developments in human-machine interfaces.
The realm of threat detection in general requires a multiplicity of tech-
niques from the mathematical sciences. How can one most quickly detect
patterns that might indicate the spread of a pathogen spread by bioterror-
ism? How does one understand the structure of a terrorist network? Can
one design a power grid or a transportation network in such a way that it is
maximally resistant to attack? A new threat, cyberwarfare, grows in mag-
nitude. Fighting back will involve a multipronged response: better encryp-
tion, optimized design of networks, and the burgeoning mathematics- and
statistics-intensive field of anomaly detection.