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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

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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.

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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.

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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.

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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

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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.

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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.

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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

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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

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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.

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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

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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.