4

Important Trends in the Mathematical Sciences

This chapter draws on the inputs gathered by the committee at its meetings and on the members’ collective experiences to identify trends that are affecting the mathematical sciences. These trends are likely to continue, and they call for adjustments in the way the enterprise—including individual professionals, academic departments, university administrators, professional societies, and funding agencies—supports the discipline and how the community adjusts for the needs between now and 2025. Recommendations on the necessary adjustments are included as appropriate.

INCREASING IMPORTANCE OF CONNECTIONS FOR MATHEMATICAL SCIENCES RESEARCH

Based on testimony received at its meetings, conference calls with leading researchers (see Appendix B), and the experiences of its members, the committee concludes that the importance of connections among areas of research has been growing over the past two decades or more. This trend has been accelerating over the past 10-15 years, and all indications are that connections will continue to be very important in the coming years. Connections are of two types:

•   The discipline itself—research that is internally motivated—is growing more strongly interconnected, with an increasing need for research to tap into two or more fields of the mathematical sciences;

•   Research that is motivated by, or applied to, another field of science, engineering, business, or medicine is expanding, in terms of



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4 Important Trends in the Mathematical Sciences This chapter draws on the inputs gathered by the committee at its meetings and on the members’ collective experiences to identify trends that are affecting the mathematical sciences. These trends are likely to con- tinue, and they call for adjustments in the way the enterprise—including individual professionals, academic departments, university administrators, professional societies, and funding agencies—supports the discipline and how the community adjusts for the needs between now and 2025. Recom- mendations on the necessary adjustments are included as appropriate. INCREASING IMPORTANCE OF CONNECTIONS FOR MATHEMATICAL SCIENCES RESEARCH Based on testimony received at its meetings, conference calls with lead- ing researchers (see Appendix B), and the experiences of its members, the committee concludes that the importance of connections among areas of research has been growing over the past two decades or more. This trend has been accelerating over the past 10-15 years, and all indications are that connections will continue to be very important in the coming years. Con- nections are of two types: • The discipline itself—research that is internally motivated—is grow- ing more strongly interconnected, with an increasing need for re- search to tap into two or more fields of the mathematical sciences; • Research that is motivated by, or applied to, another field of sci- ence, engineering, business, or medicine is expanding, in terms of 93

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94 THE MATHEMATICAL SCIENCES IN 2025 both the number of fields that now have overlaps with the math- ematical sciences and the number of opportunities in each area of overlap. The second of these trends was discussed in Chapter 3. This section elabo- rates on the trend toward greater connectivity on the part of the mathemati- cal sciences themselves. Internally driven research within the mathematical sciences is showing an increasing amount of collaboration and research that involves two or more fields within the discipline. Some of the most exciting advances have built on fields of study—for example, probability and combinatorics—that ­ had not often been brought together in the past. This change is non- trivial because large bodies of knowledge must be internalized by the investigator(s). The increased interconnectivity of the mathematical sciences commu- nity has led, as one would expect, to an increase in joint work. To cite just one statistic, the average number of authors per paper in the Annals of Mathematics has risen steadily, from 1.2 in the 1960s to 1.8 in the 2000s. While this increase is modest compared to the multiauthor traditions in many fields, it is significant because it shows that the core mathematics covered by this leading journal is trending away from the solitary researcher model that is embedded in the folklore of mathematics. It also suggests what has long been the experience of leading mathematicians: that the vari- ous subfields in mathematics depend on one another in ways that are unpre- dictable but almost inevitable, and so more individuals need to collaborate in order to bring all the necessary expertise to bear on today’s problems. While some collaborative work involves mathematical scientists of similar backgrounds joining forces to attack a problem of common inter- est, in other cases the collaborators bring complementary backgrounds. In such cases, the increased collaboration in recent years has led to greater cross-fertilization of fields—to ideas from one field being used in another to make significant advances. A few recent examples of this interplay among fields are given here. This list is certainly not exhaustive, but it indicates the vitality of cross-disciplinary work and its importance in modern math- ematical sciences. Example 1: Cross-Fertilization in Core Mathematics Recent years have seen major striking examples of ideas and results from one field of core mathematics imported to establish important results in other fields. An example is the resolution of the Poincaré conjecture, the most famous problem in topology, by using ideas from geometry and analy- sis along with results about a class of abstract metric spaces. As another

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IMPORTANT TRENDS IN THE MATHEMATICAL SCIENCES 95 example, there is mounting evidence that the same geometric flow techniques used in resolving the Poincaré conjecture can be applied to complex alge- braic manifolds to understand the existence and nonexistence of canonical m ­ etrics—for example, the Kähler-Einstein metric—on these manifolds. Re- cently, ideas from algebraic topology dating back to the 1960s, A∞-algebras and modules, have been used in the study of invariants of symplectic mani- folds and low-dimensional topological manifolds. These lead to the intro- duction of ever more sophisticated and powerful algebraic structures in the study of these topological and geometric problems. In a different direction, deep connections have recently been discov- ered between random matrix theory, combinatorics, and number theory. The zeros of the Riemann zeta function seem to follow—with astounding precision—a distribution connected with the eigenvalues of large random matrices. This distribution was originally studied as a way of understand- ing the spectral lines of heavy atoms. The same distribution occurs in many other areas—for example with respect to the standard tableaux of c ­ ombinatorics—and in the study of quantum chaos. There are also connections recently discovered between commutative algebra and statistics. Namely, to design a random walk on a lattice is equivalent to constructing a set of generators for the ideal of a variety given implicitly; that is, solving a problem in classical elimination theory. This is of importance in the statistics of medium-sized data sets—for example, con- tingency tables—where classical methods give wrong answers. The classical methods of elimination theory are hard; the modern technique of Groebner bases is now often used. Example 2: Interactions Between Mathematics and Theoretical High-Energy Physics Certainly one of the most important and surprising recent develop- ments in mathematics has been its interaction with theoretical high-energy physics. Large swathes of geometry, representation theory, and topology have been heavily influenced by the interaction of ideas from quantum field theory and string theory, and, in turn, these areas of physics have been informed by advances in the mathematical areas. Examples include the relationship of the Jones polynomial for knots with quantum field theory, and Donaldson invariants for 4-manifolds and the related Seiberg-Witten invariants. Then there is mirror symmetry, discovered by physicists, which, in each original formulation, led to the solution of one of the classical enumerative problems in algebraic geometry, the number of rational curves of a given degree in a quintic hypersurface in projective 4-space. This has been expanded conjecturally to a vast theory relating complex manifolds and symplectic manifolds. A more recent example of the cross-fertilization

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96 THE MATHEMATICAL SCIENCES IN 2025 between mathematics and physics is the reformulation of the geometric Langlands program in terms of quantum field theory. A very recent example involves the computation of scattering amplitudes in gauge theories, mo- tivated by the practical problem of computing backgrounds for the Large Hadron Collider. These computations use the tools of algebraic geometry and some methods from geometric number theory. Ideas of topology are very important in many areas of physics. Most notably, topological quan- tum field theories of the Chern-Simons form are crucial to understanding some phases of condensed matter systems. These are being actively explored because they offer a promising avenue for constructing quantum computers. Example 3: Kinetic Theory Kinetic theory is a good example of the interaction between areas of the mathematical sciences that have traditionally been seen as core and those that had been seen as applied. The theory was proposed by M ­ axwell and Boltzmann to describe the evolution of rarefied gases (not dense enough to be considered a “flow,” not dispersed enough to be just a system of particles, a dynamical system). Mathematically, the Boltzmann equation involves the spatial interaction (collisions) of probability densities of particles travelling at different velocities. The analytical properties of s ­olutions—their existence, regularity, and stability, and the phenomenon of shock ­ormation—were little understood until approximately 30 years f ago. Hilbert and Carleman worked on these problems for many years with little success, and attempts to understand the analytical aspects of the e ­quation—existence, regularity, stability of solutions, as well as possible shock formation—had not advanced very far. In the 1980s, the equation arose as part of the need for the modeling of the reentry dynamics of space flight through the upper atmosphere and it was taken up again by the mathematical community, particularly in France. That gave rise to 20 years of remarkable development, from the celebrated work of Di Perna-Lions (1988) showing the existence of solutions, to the recent contributions of Villani and his collaborators. In the meantime, the underlying idea of the modeling of particles interacting at a rarefied scale appeared in many other fields in a more complex way: sticky particles, intelligent particles, and so on, in the modeling of semi­onductors, traffic flow, flocking, and social c behavior, particularly in phenomena involving decision making. The sorts of connections exemplified here are powerful, because they establish alternative modes by which mathematical concepts may be ex- plored. They often inspire further work because surprising connections hint at deeper relationships. It is clear that the mathematical sciences have benefited in recent years from valuable, and perhaps surprising, connec- tions within the discipline itself. For example, the Langlands program in

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IMPORTANT TRENDS IN THE MATHEMATICAL SCIENCES 97 geometry is bringing together many different threads of mathematics, such as number theory, Lie theory, and representation groups, and as noted above it has more recently also been linked to physics. Another example is Demetrios Christodoulou’s work on the formation of black holes from a few years ago, which resolved a question that had been open for decades through a combination of insights from PDEs and differential geometry. Fields in the mathematical sciences are mature enough so that re­ searchers know the capabilities and limitations of the tools provided by their field, and they are seeking other tools from other areas. This trend seems to be flourishing, with the result that there is an increase in inter­ disciplinarity across the mathematical sciences. For example, there is greater interest in combinatorial methods which, 50 years ago, might not have been pursued because those methods may not have elegant structures and because computation may be required. The tendency decades ago was to make simplifying assumptions to eliminate the need for combinatorial calculations. But many problems have a real need for a combinatorial ap- proach, and many researchers today are willing to do those computations. Because of these interdisciplinary opportunities, more researchers are reach- ing out from areas that might in the past have been self-contained. Also, as discussed below, it is easier to collaborate these days because of the Internet and other communications technologies. In other research areas, opportunities are created when statistics and mathematics are brought together, in part because the two fields have complementary ways of describing phenomena. An example is found in envi­onmental sciences, where the synergies between deterministic math- r ematical models and statistics can lead to important insights. Such an ap- proach is helpful for, say, understanding the uncertainties in climate models, because of the value in combining insight about deterministic PDE-based models with statistical insights about the uncertainties. Because of these exciting opportunities that span multiple fields of the mathematical sciences, the amount of technical background needed by researchers is increasing. Education is never complete today, and in some areas older mathematicians may make more breakthroughs than in the past because so much additional knowledge is needed to work at the frontier. For this reason, postdoctoral research training may in the future become necessary for a greater fraction of students, at least in mathematics. The increase in postdoctoral study has been dramatic over the past 20 years, such that in the fall of 2010, 40 percent of recent Ph.D. recipients in the mathematical sciences were employed as postdoctoral researchers.1 Thus, the time from receipt of the Ph.D. to attaining a tenure-track academic 1  R. Cleary, J.W. Maxwell, and C. Rose, 2010, Report on the 2009-2010 new doctoral recipients. Notices of the AMS 58(7):944-954.

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98 THE MATHEMATICAL SCIENCES IN 2025 position has lengthened. While these trends clearly create researchers with stronger backgrounds, the length of time for becoming established as a researcher could lessen the attractiveness of this career path. At the same time, more mathematical scientists are now addressing ap- plications, such as those in computer science. This work builds successfully on the deep foundations that have been constructed within mathematics. For example, results of importance to computer science have been achieved by individuals who are grounded in discrete mathematics and combinatorics and who may not have had previous exposure to the particular application. These increasing opportunities for interdisciplinary research pose some challenges for individuals and the community. Interdisciplinary work is facilitated by proximity, and even a walk from one corridor to another can be a hindrance. So attention must be paid to fostering collaboration, even within a single department. When the connections are to be established across disciplines, this need is even more obvious. Ideally, mathematical scientists working in biology, for example, will spend some of their time visiting experimental labs, as will mathematical scientists working with other disciplines. But to make this happen, improved mechanisms for con- necting mathematical scientists with potential collaborators are needed, such as research programs that bring mathematical scientists and collabo- rators together in joint groups. Such collaborations work best when the entire team shares one primary goal—such as addressing a question from biology—even for the team members who are not biologists per se. But to make this work we need adjustments to reward systems, especially for the mathematical scientists on such teams. One leading researcher who spoke with the committee observed how mathematicians at Microsoft Research are often approached by people from applied groups, which is a fortunate result of the internal culture. One value that mathematicians can provide in such situations that is often under­ stimated is that they can prove negative results: that is, the impos- e sibility of a particular approach. That knowledge can redirect a group’s efforts by, say, helping them realize they should attack only a limited ver- sion of their problem or stopping them from expending more resources on a hopeless task. This ultimately increases productivity because it helps the organization to focus resources better. This is a contribution of mathematics beyond product-focused work or algorithm development. Another set of challenges to interdisciplinary students and re­ earchers s stems from their lack of an obvious academic home. Who is in their com- munity of peers? Who judges their contributions? How are research pro- posals and journal submissions evaluated? At the National Institutes of Health, for example, the inclusion of mathematical scientists in the study sections that review proposals with mathematical or statistical content is an important step, though it is not a perfect process. Tenure review in large

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IMPORTANT TRENDS IN THE MATHEMATICAL SCIENCES 99 universities can also be problematic for interdisciplinary junior faculty. While it takes longer to build a base of interdisciplinary knowledge, once that base is built it can open doors to very productive research directions that are not feasible for someone with a more conventional background. Universities are changing slowly to recognize that interdisciplinary faculty members can produce both better research and better education. There is a career niche for such people, but it could be improved. For example, it might be necessary to relax the tenure clock for researchers pursuing interdisciplinary topics, and a proper structure must be in place in order to conduct appropriate tenure reviews for them. This is one way to break down academic silos. INNOVATION IN MODES FOR SCHOLARLY INTERACTIONS AND PROFESSIONAL GROWTH Mathematical Science Institutes A major change in the mathematical sciences over the past decade and more has been the increasing number of mathematical science institutes and their increased influence on the discipline and community. In 1981 there was only one such institute in the United States—the Institute for Advanced Study in Princeton, which has a very different character from the institutes created after it. The National Science Foundation/Division of Mathematical Sciences (NSF/DMS) now supports eight mathematical sciences institutes in the United States;2 other entities heavily involved in the mathematical sciences include the Clay Institute for Mathematics, the Simons Center for Geometry and Physics, and the Kavli Institute for Theo- retical Physics. In the last 20 years we have seen new institutes appear in Japan, England, Ireland, Canada, and Mexico, to name just a few countries, joining older institutes in France, Germany, and Brazil. Overall, there are now some 50 mathematical science institutes in 24 different countries.3 These institutes have made it easier for mathematical scientists to form and work in collaborative teams that bridge two or more fields or that connect the mathematical sciences to another discipline. Typical goals of most U.S. mathematical science institutes include the following: • Stimulate research, collaboration, and communication; • Seed and sustain important research directions; • Promote interdisciplinary research; 2  Representing 12.5 percent ($32.5 million) of DMS’s 2012 budget request. 3  Notices of the AMS, August 2011, pp. 973-976.

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100 THE MATHEMATICAL SCIENCES IN 2025 • Build research teams, including collaborations with industry, gov- ernment; laboratories, and international colleagues; • Enrich and invigorate mathematics education at all levels; • Provide postdoctoral training; and • Expand mathematics opportunities for underrepresented groups.4 The institutes promote research and collaboration in emerging areas, encourage continued work on important problems, tackle large research agendas that are outside the scope of individual researchers, and help to maintain the pipeline of qualified researchers for the future. Many of the institute programs help researchers broaden their expertise, address- ing the need for linking multiple fields that was emphasized above and in Chapter 3. For example, every year the Institute for Mathematics and its Applications (IMA) offers intensive two-week courses aimed at helping to introduce established researchers to new areas; recent courses have focused on mathematical neuroscience, economics and finance, applied algebraic topology, and so on. All of these institutes have visiting programs, often around a specific theme that varies from year to year, and they invite math- ematical scientists from around the world to visit and participate in the programs. This has led to an enormous increase in the cross-fertilization of ideas as people from different places and different disciplines meet and exchange ideas. In addition, it is quite common for these institutes to record the lectures and make them freely available for downloading. Real-time streaming of lectures is just starting to emerge. All of these steps help to strengthen the cohesiveness of the community. Beyond this, the institutes frequently allow researchers to meet people they would not otherwise meet. This is especially crucial in connecting researchers from other disciplines with the right mathematical scientists. Often, scientists, engineers and medical researchers do not know what mathematics and statistics are available that might be relevant to their problem, and they do not know whom to turn to. Likewise, mathematical scientists are often sitting on expertise that would be just what is needed to solve an outside problem, but they are unaware of the existence of these problems or of who might possess the relevant data. Arguably, the institutes have collectively been one of the most impor- tant vehicles for culture change in the mathematical sciences. Some illustra- tions of the impact of mathematical science institutes follow.5 To help the mathematical sciences build connections, the IMA reaches out so that some 40 percent of the participants in its programs come from 4  Thesegoals are explicit for the NSF-supported institutes. 5  Thecommittee thanks IMA director Fadil Santosa, IPAM director Russel Caflisch, SAMSI director Richard Smith, and MSRI director Robert Bryant for helpful inputs to this section.

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IMPORTANT TRENDS IN THE MATHEMATICAL SCIENCES 101 outside the mathematical sciences. In this way, it has helped to nucleate new communities and networks in topics such as mathematical materials science, applied algebraic geometry, algebraic statistics, and topological methods in proteomics. The Institute for Pure and Applied Mathematics (IPAM) hosts a similar percentage of researchers from other disciplines. The institutes have had success in initiating new areas of research. For example, IPAM worked for 9 years to nucleate and then nurture a new focused area of privacy research, starting with a workshop on contempo- rary methods in cryptography in 2002. That led to a 2010 workshop on statistical and learning-theoretic challenges in data privacy, which brought together data privacy and cryptography researchers to develop an ap- proach to data privacy that is motivated and informed by developments in cryptography, one of them being mathematically rigorous concepts of data security. A second follow-on activity was a 2011 workshop on mathemat- ics of information-theoretic cryptography, which saw algebraic geometers and computer scientists working on new approaches to cryptography based on the difficulty of compromising a large number of nodes on a network. Another IPAM example illustrates that the same process can be important and effective in building connections within the discipline. A topic called “expander graphs” builds on connections that have emerged among dis- crete subgroups of Lie groups, automorphic forms, and arithmetic on the one hand, and questions in discrete mathematics, combinatorics, and graph theory on the other. In 2004 IPAM held the workshop Automorphic Forms, Group Theory, and Graph Expansion, which was followed by a program on the subject at the Institute for Advanced Study in 2005 and a second IPAM workshop, Expanders in Pure and Applied Mathematics, in 2008. Similarly, the Statistical and Applied Mathematical Sciences Institute (SAMSI) has worked to develop the general topic of low-dimensional structure in a high- dimensional system. Many problems of modern statistics arise when the number of available variables is comparable with or larger than the number of independent data points (often referred to as the p > n problem). Tradi- tional methods for dealing with such problems involve techniques such as variable selection, ridge regression, and principal components regression. Beginning in the 1990s, more modern methods such as lasso regression and wavelet thresholding were developed. These ideas have now been extended in numerous directions and have attracted the attention of researchers in computer science, applied mathematics, and statistics, in areas such as manifold learning, sparse modeling, and the detection of geometric struc- ture. This is an area with great potential for interaction among statisticians, applied mathematicians, and computer scientists. The Mathematical Sciences Research Institute (MSRI) is focused pri- marily on the development of fundamental mathematics, specifically in a ­ reas in which mathematical thinking can be applied in new ways. Pro-

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102 THE MATHEMATICAL SCIENCES IN 2025 grams have spanned topics such as mathematical biology, theoretical and applied topology, mathematics of visual analysis, analytic and compu- tational elliptic and parabolic equations, dynamical systems, geometric evolution equations, parallel computing for mathematics, computational finance, statistical computing, multiscale methods, climate change model- ing, algebraic geometry, and advances in algebra and geometry. In spite of its primary focus on the mathematical sciences per se, MSRI has long included a robust set of outreach activities. For example, its 2006 program Computational Aspects of Algebraic Topology explored ways in which the techniques of algebraic topology are being applied in various contexts related to data analysis, object recognition, discrete and compu- tational geometry, combinatorics, algorithms, and distributed computing. That program included a workshop focused on application of topology in science and engineering, which brought together people working in problems ranging from protein docking, robotics, high-dimensional data sets, and sensor networks. In 2007, MSRI organized and sponsored the World Congress on Computational Finance, in London, which brought together both theoreticians and practitioners in the field to discuss its cur- rent problems. MSRI has also sponsored a series of colloquia to acquaint mathematicians with fundamental problems in biology. An example is the 2009 workshop jointly sponsored by MSRI and the Jackson Laboratories on the topic of mathematical genomics. Both MSRI and SAMSI have helped build bridges between statisticians and climate scientists through at least six programs focused on topics such as new methods of spatial statistics for climate change applications, data assimilation, analysis of climate models as computer experiments, chaotic dynamics, and statistical methods for combining ensembles of climate models. IPAM and SAMSI provide two additional illustrations of how the institutes build new connections to other disciplines. After a professor of Scandinavian languages at UCLA participated in a 2007 IPAM program on knowledge and search engines, which introduced him to researchers and methods from modern information theory, he went on to organize two workshops in 2010 and 2011, on networks and network analysis for the humanities; the workshops were funded by the National Endow- ment for the Humanities and cosponsored by IPAM. They led to the exploration of new data analysis tools by many of the humanists who participated. ­ AMSI’s example comes from a more established area, the S interaction between statistics and social sciences, which SAMSI has sup- ported through several activities, such as a workshop to explore compu- tational methods for causal modeling and for the analysis of transactions ­ and social relationships. The IMA has a long history of outreach to industry, for instance through its Industrial Postdoctoral Fellowship program and other activities

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IMPORTANT TRENDS IN THE MATHEMATICAL SCIENCES 103 that bring cutting-edge mathematical sciences to bear on significant indus- trial problems. Examples include uncertainty quantification in the automo- tive industry and numerical simulation of ablation surgery. Overall, IMA has trained over 300 postdoctoral fellows since 1982, and about 80 percent of them are now in academic positions. The IMA also offers programs for graduate students, most notably its regular workshops on mathematical modeling in industry, in which students work in teams under the guidance of industry mentors on real-world problems from their workplace. Through this program, many mathematical scientists have been exposed early in their careers to industrial problems and settings. As an example of value provided beyond academe, IPAM has been instrumental in introducing modern imaging methods to the National G ­ eospatial-Intelligence Agency (NGA). Several individuals from NGA at- tended IPAM’s 2005 summer school, Intelligent Extraction of Information from Graphs and High-­ imensional Data, which convinced them and their D agency to explore further. Subsequently, NGA organized a series of three workshops at IPAM on advancing the automation of image analysis. This led to hiring by NGA of several new mathematics Ph.D.s with expertise in image analysis, as well as a major funding initiative. IPAM has similarly held workshops for the Office of Naval Research (ONR) on aspects of machine reasoning, which may lead to an ONR funding initiative. IPAM’s program Multiscale Geometry and Analysis in High Dimensions led to the ­ explosive growth of applications of compressed sensing, followed by a large funding program at DARPA. In addition to the institutes, NSF/DMS and other financial supporters of mathematics have in recent years created other funding programs to en- courage and nurture research groups, which help investigators to address broad and cross-cutting topics. Changing Models for Scholarly Communication The first thing that comes to mind when one thinks of interconnectivity these days is the Internet and the World Wide Web. These affect practically all human activity, including the way that mathematical scientists work. The maturation of the Internet has led in the past 15-20 years to the avail- ability of convenient software tools that painlessly lead to the quick dis- semination of research results (consider for instance the widely used arXiv preprint server, http://arxiv.org/), the sharing of informal ideas through blogs and other venues, and the retrieval of information through efficient search engines. These new tools have profoundly changed both the modes of collaboration and the ease with which mathematical scientists can work across fields. The existence of arXiv has had a major influence on scholarly communication in the mathematical sciences, and it will probably become

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IMPORTANT TRENDS IN THE MATHEMATICAL SCIENCES 105 Widespread dissemination of research results has made it easier for anyone to borrow ideas from other fields, thereby creating new bridges between subdisciplines of the mathematical sciences or between the math- ematical sciences and other fields of science, engineering, and medicine. For example, new research directions can be seen where ideas from abstract probability theory prove to have very deep consequences in signal process- ing and tantalizing applications in signal acquisition, and where tools from high-dimensional geometry can change the way we perform fundamental calculations, such as solving systems of linear equations. Effortless access to information has spurred the development of communities with aston- ishingly broad collective expertise, and this access has lowered barriers between fields. In this way, theoretical tools find new applications, while applications renew theoretical research by offering new problems and sug- gesting new directions. This cycle is extremely healthy. A recent paper9 evaluated an apparent shift in collaborative behavior within the mathematical sciences in the mid-1990s. At that time, the net- works of researchers in core and applied mathematics moved from being centered primarily around a small number of highly prolific authors toward networks displaying more localized connectivity. More and stronger col- laboration was in evidence. Brunson and his collaborators speculated that a cause of this trend was the rise of e-communications and the Web—for example, arXiv went online in 1993 and MathSciNet in 1996—because ap- plied subdisciplines, which historically had made greater use of computing resources, showed the trend most strongly. The Internet provides a ready mechanism for innovation in communica- tion and partnering, and novel mechanisms are likely to continue to appear. As just one more example, consider the crowdsourcing, problem-solving venture called InnoCentive. It is an example of another new Web-enabled technology that may have real impacts on the mathematical sciences by pro- viding opportunities to learn directly of applied challenges from other dis- ciplines and to work on them. InnoCentive is backed by venture capitalists with the goal of using “crowdsourcing”—Web-based methods for parceling out tasks to anyone who wishes to invest time in hopes of achieving results and then receiving payment—to solve problems for corporate, government, and nonprofit clients. When checked on March 16, 2012, the company’s Web site listed 128 challenges that were either open or for which submis- sions were being evaluated. Of these, 13 were flagged as having mathemati- cal or statistical content. Examples of the latter included challenges such as the development of an algorithm to identify underlying geometric features 9  Brunson, J.C., S. Fassino, A. McInnes, M. Narayan, B. Richardson, C. Franck, P. Ion, and R. Laubenbacker, 2012, Evolutionary Events in a Mathematical Sciences Research Collabo- ration Network. Manuscript submitted for publication. arXiv:1203.5158 [physics.soc-ph].

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106 THE MATHEMATICAL SCIENCES IN 2025 in noisy two-dimensional data and the creation of a model to predict par- ticle size distributions after milling. The fee for supplying a best solution to these problems typically ranged from $15,000 to $30,000, and most challenges attracted hundreds of potential solvers. There are debates about whether crowdsourcing is a healthy trend for a research community. And while InnoCentive does offer opportunities for mathematical scientists to engage in a broad range of problems, the engage- ment is at arm’s length, at least initially. However, crowdsourcing is one more Web-based innovation that may affect mathematical scientists, and the community should be aware of it. THE MATHEMATICAL SCIENCES SHOULD MORE THOROUGHLY EMBRACE COMPUTING Computing is often the means by which the mathematical sciences are applied to other fields. For example, mathematical scientists collaborate with astrophysicists, neuroscientists, or materials scientists to develop new models and their computationally feasible instantiations in software in order to simulate complex phenomenology. The mathematical sciences can obviously contribute to creating mathematical and statistical models. But they can also contribute to the steps that translate those models into computational simulations: discretizations, middleware (such as gridding algorithms), numerical methods, visualization methods, and computational underpinnings. Whenever a computer simulation is used, the question of validation is critical. Validation is an essential part of simulation, and the mathematical sciences provide an underlying framework for validation. Because of that, validation is also an exciting growth area for the mathematical sciences. Owing to the multidisciplinary nature of validation studies, mathematical scientists may increasingly engage with behavioral scientists, domain scien- tists, and risk and decision analysts along this growing frontier. This spectrum of challenges is often labeled “scientific computing,” an area of study in its own right and an essential underpinning for simulation- based engineering and science. It is, however, somewhat of an academic orphan: that is, a discipline that does not fit readily into any one academic department and one for which academic reward systems do not align well with the work that needs to be carried out. Scientific computing experts must know enough about the domain of the work to ensure that the soft- ware is properly approximating reality, while also understanding relevant applied mathematics and the subtleties of a computer’s architecture and compiler. But those disciplines do not normally assign a high priority to the combination of skills possessed by the scientific computing expert, nor is the development of critical and unique software rewarded as an academic

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IMPORTANT TRENDS IN THE MATHEMATICAL SCIENCES 107 achievement. At different academic institutions, scientific computing might at times be found within departments of computer science or applied math- ematics or at other times be spread across a range of science and engineer- ing departments. In the latter case, scientific computing experts often are faced with the misalignment of rewards and incentives. Recognizing that many aspects of scientific computing are at heart very mathematical, departments in the mathematical sciences should play a role in seeing that there is a central home for scientific computing research and education at their institutions, whether or not it is within their own depart- ments. Computation is central to the future of the mathematical sciences, and to future training in the mathematical sciences. Another aspect of computing vis-á-vis the mathematical sciences is the availability of more and better data, including computer-generated data. There is a long and strong tradition in mathematics to discount empirical evidence; while occasionally someone will refer to empirical observations as having suggested a research direction, even that is rare. But today, with advanced computing, we have the opportunity to generate a great deal of empirical evidence, and this trend is growing. However, some mathemati- cal scientists will be deterred from seizing this opportunity unless we over- come the tradition and embrace phenomenology-driven research in addition to the theorem/proof paradigm. Computational resources open the door to more data-driven mathematical exploration. More generally, because computation is often the means by which methods from the mathematical sciences are applied in other disciplines and is also the driver of many new applications of the mathematical sciences, it is important that most mathematical scientists have a basic understanding of scientific computing. Academic departments may consider seminars or other ways to make it easy for mathematical scientists to learn about and keep up with the rapidly evolving frontiers of computation. Some mathematical sciences research would benefit from the most advanced computing resources, and not many mathematical scientists are currently exploiting those capabilities. Because the nature and scope of computation is continually changing, there is a need for a mechanism to ensure that mathematical sciences researchers have access to computing power at an appropriate scale. NSF/DMS should consider instituting pro- grams to ensure that state-of-the art computing power is widely available to mathematical science researchers. FUNDING IMPLICATIONS OF INCREASING CONNECTIVITY OF THE MATHEMATICAL SCIENCES The broadening of the mathematical sciences discussed in Chapter 3 raises the question of whether the mathematical science enterprise is ex-

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108 THE MATHEMATICAL SCIENCES IN 2025 panding along with its opportunities. This is very difficult to answer, but the committee is concerned that the accelerating reach of the mathemati- cal sciences is not being matched by a commensurate increase in financial support. A key message of the 1984 David report (see Appendix A) was that, in order to maintain a pipeline of students to replenish the research capabilities in place at that time, federal funding for mathematical sciences research would need to double. That doubling, in inflation-adjusted dollars, ­ finally occurred late in the 1990s, as shown in Table 4-1. However, that doubling goal was based on the David report’s estimate of what it would take to adequately fund the 2,600 mathematical scientists who were ac- tively performing research in 1984. In light of the dramatic expansion of research opportunities described in Chapter 3 and the commensurately larger pipeline of students requiring mathematical science education, the David report’s goals are clearly not high enough to meet today’s needs. While the funding for NSF/DMS did reach the goals of the David report, and later exceeded them, the overall growth in federal research funding for the mathematical sciences has not been on the same scale as the growth in intellectual scope documented in this report. The recent addition of a large private source, through the Simons Foundation, while very welcome, can only be stretched so far. As shown in Table 4-1, Department of Defense (DOD) funding for the mathematical sciences has increased only about 50 percent in con- stant dollars since 1989. It is difficult to discern trends at the Department ­ of Energy (DOE) or the National Institutes of Health (NIH), and the “other agency” category remains oddly underfunded given the pervasive- ness of the mathematical sciences. Apropos that last point, some science a ­ gencies—such as the National Oceanic and Atmospheric Administration, the National Aeronautics and Space Administration, the Environmental ­ Protection Agency, and the U.S. Geological Survey at present have few direct interfaces with the mathematical sciences community and provide very little research support for that community. And although many agen- cies that deal with national security, intelligence, and financial regulation rely on sophisticated computer simulations and complex data analyses, only a fraction of them are closely connected with mathematical scientists. As a result, the mathematical sciences are unable to contribute optimally to the full range of needs within these agencies, and the discipline—especially its core areas—is as a result overly dependent on NSF. Conclusion 4-1: The dramatic expansion in the role of the mathe­ matical sciences over the past 15 years has not been matched by a comparable expansion in federal funding, either in the total amount or in the diversity of sources. The discipline—especially the core areas—is still heavily dependent on the NSF.

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IMPORTANT TRENDS IN THE MATHEMATICAL SCIENCES 109 TABLE 4-1  Federal Funding for the Mathematical Sciences (in millions) 2010 1984 1989 1998 Estimate 2012 (Constant (Constant (Constant (Constant Estimate 2011 2011 2011 2011 (Actual Dollars) Dollars) Dollars) Dollars) Dollars) NSF  DMS 113 143 147 251 238  Other 11 17  —  —  —    Subtotal 124 160 147 251 238 DOD  AFOSR 28 36 48 53 47  ARO 19 25 19 12 16  DARPA  — 19 31 12 28  NSA  — 6  — 7 6  ONR 33 28 12 20 24    Subtotal 80 113 111 105 121 DOE   Applied mathematics  —  —  — 45 46  SciDAC  —  —  — 51 44  MICS  —  — 194  —  —    Subtotal 8 14 194 96 90 NIH  NIGMS  —  —  — 51  —  NIBIB  —  —  — 40  —    Subtotal  —  —  — 91  — Other agencies 6 3  —  —  — Total 217 289 451 543 449 NOTE: Inflation calculated using the higher education price index (HEPI) outlined in Com- mon Fund Institute, “2011 Update: Higher Education Price Index,” Table A. Available at http://www.commonfund.org/CommonFundInstitute/HEPI/HEPI%20Documents/2011/ CF_HEPI_2011_FINAL.pdf. Acronyms: AFOSR, Air Force Office of Scientific Research; ARO, Army Research Office; DARPA, Defense Advanced Research Projects Agency; MICS, Mathematical, Information, and Computational Sciences Division; NIBIB, National Institute of Biomedical Imaging and Bioengineering; NIGMS, National Institute of General Medical Sciences; NSA, National Security Agency; ONR, Office of Naval Research; SciDAC, Scientific Discovery Through Advanced Computing. SOURCES: 1984 and 1989 data from David II; 1998 data from Daniel Goroff, 1999, Math- ematical sciences in the FY 2000 budget, Notices of the AMS 46(6):680-682; 2010 and 2012 data from Samuel Rankin, III, 2011, Mathematical sciences in the FY 2012 budget, AAAS Report XXXVI: Research and Development FY 2012:225-230.

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110 THE MATHEMATICAL SCIENCES IN 2025 In 2010, NSF/DMS was externally reviewed by a Committee of ­ isitors V (COV) to address its balance, priorities, and future directions, among other things. This COV review culminated in a report that found that DMS is underfunded and that, in spite of an overall budget increase, most of the in- creased funds went to interdisciplinary programs while funding of the core DMS programs stayed constant. The COV recommended that more money be directed to core areas.10 In its response to the COV report, DMS pointed out that funding for core areas increased significantly from 2006 to 2007, with additional (but small) increases in the following 2 years.11 DMS is faced with an innate conflict: As the primary funding unit charged with maintaining the health of the mathematical sciences, it is naturally driven to aid the expansions discussed in Chapter 3; yet it is also the largest of a very few sources whose mission includes supporting the foundations of the discipline, and thus it plays an essential role with respect to those foundations. As noted in Chapter 3, some mathematical scientists receive research support from other parts of NSF and from nonmath units in other federal funding agencies, but there are only anecdotal accounts of this. With limited data it is difficult to get a full picture of the totality of funding for the broader mathematical sciences community—the community that is an intellectually coherent superset of those researchers who sit in departments of mathematics or statistics—and to determine whether the funding is adequate and appropriately balanced. Nor can we say whether it is keep- ing pace with the expanding needs of this broader community. There are challenges inherent in supporting a broad, loosely knit community while maintaining its coherence, and the adequacy and balance of funding is a foremost concern. As noted in Chapter 3, funding of excellence wherever it is found should still be the top priority. A VISION FOR 2025 Finding: Mathematical sciences work is becoming an increasingly in- tegral and essential component of a growing array of areas of investi­ gation in biology, medicine, social sciences, business, advanced design, climate, finance, advanced materials, and much more. This work in- volves the integration of mathematics, statistics, and computation in the broadest sense, and the interplay of these areas with areas of potential application; the mathematical sciences are best conceived of as including 10  NSF/DMS Committee of Visitors, NSF/DMS, 2010, Report of the 2010 Committee of Visitors. Available at http://www.nsf.gov/attachments/117068/public/DMS_COV_2010_­ final_report.pdf. 11  NSF, 2010, Response to the Division of Mathematical Sciences Committee of Visitors Report. Available at http://www.nsf.gov/mps/advisory/covdocs/DMSResponse_2010.pdf.

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IMPORTANT TRENDS IN THE MATHEMATICAL SCIENCES 111 all these components. These activities are crucial to economic growth, national competitiveness, and national security. This finding has rami- fications for both the nature and scale of funding of the mathematical sciences and for education in the mathematical sciences. Conclusion 4-2: The mathematical sciences have an exciting opportu- nity to solidify their role as a linchpin of twenty-first century research and technology while maintaining the strength of the core, which is a vital element of the mathematical sciences ecosystem and essential to its future. The enterprise is qualitatively different from the one that prevailed during the latter half of the twentieth century, and a differ- ent model is emerging—one of a discipline with a much broader reach and greater potential impact. The community is achieving great success within this emerging model, as recounted in this report. But the value of the mathematical sciences to the overall science and engineering enter­ prise and to the nation would be heightened by increasing the number of mathematical scientists who share the following characteristics: • T  hey are knowledgeable across a broad range of the discipline, beyond their own area(s) of expertise; • T  hey communicate well with researchers in other disciplines; • T  hey understand the role of the mathematical sciences in the wider world of science, engineering, medicine, defense, and business; and • T  hey have some experience with computation. It is by no means necessary or even desirable for all mathematical sci- entists to exhibit these characteristics, but the community should work toward increasing the fraction that does. To move in these directions, the following will need attention: • The culture within the mathematical sciences should evolve to en- courage development of the characteristics listed in Conclusion 4-2. • The education of future generations of mathematical scientists, and of all who take mathematical sciences coursework as part of their preparation for science, engineering, and teaching careers, should be reassessed in light of the emerging interplay between the math- ematical sciences and many other disciplines. • Institutions, for example, the funding mechanisms and reward systems—should be adjusted to enable cross-disciplinary careers when they are appropriate. • Expectations and reward systems in academic mathematics and statis­ics departments should be adjusted so as to encourage a t

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112 THE MATHEMATICAL SCIENCES IN 2025 broad view of the mathematical sciences and to reward high-­ quality work in any of its areas. • Mechanisms should be created that help connect researchers out- side the mathematical sciences with mathematical scientists who could be appropriate collaborators. Funding agencies and academic departments in the mathematical sciences could play a role in lowering the barriers between researchers and brokering such con- nections. For academic departments, joint seminars, cross-listing of courses, cross-disciplinary postdoctoral positions, collaboration with other departments in planning courses, and courtesy appoint- ments are useful in moving this process forward. • Mathematical scientists should be included more often on the p ­ anels that design and award interdisciplinary grant programs. Because so much of today’s science and engineering builds on advances in the mathematical sciences, the success and even the validity of many projects depends on the early involvement of mathematical scientists. • Funding for research in the mathematical sciences must keep pace with the opportunities. While there are limits to the influence that it can have on the direction and character of research in the mathematical sciences and on the culture of the mathematical sciences community, the NSF can exercise leadership and serve as an enabler of positive developments. Successful examples include the flourishing Research Experiences for Undergraduates program and NSF’s portfolio of mathematical science institutes. The NSF can, through funding opportunities, enhance the pace of change and facilitate bottom-up developments that capitalize on the energy of members of the ­ ommunity— c examples include open calls for workforce proposals, grants to enable the development of new courses and curricula; grants that support interdisci- plinary research and research between disciplines within the mathematical sciences, grants that enable individuals to acquire new expertise; and pro- grams that make it easier for young people to acquire experience in industry and to acquire international experience. The trends discussed in this chapter may appear quite disruptive to many core mathematicians, or even irrelevant. To address that possibility, the committee closes with a personal reflection by the study vice-chair, Mark Green, in Box 4-1, “Core Mathematicians and the Emerging Math- ematical Landscape.”

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IMPORTANT TRENDS IN THE MATHEMATICAL SCIENCES 113 BOX 4-1 Core Mathematicians and the Emerging Mathematical Landscape by Mark Green Al Thaler, a long-time program officer at the NSF, once told me, “The twenty- first century is going to be a playground for mathematicians.” Events have more than justified his prediction. Core mathematics is flourishing, new ways of using ­ the mathematical sciences continue to develop, more and more disciplines view the mathematical sciences as essential, and new areas of mathematics and statis­ics are emerging. Where do core mathematicians fit into all this? t Of course, their main role in research is to continue to produce excellent core mathematics. That said, for those who are interested, the intellectual challenges coming from other fields, and the opportunities they present for core mathemati- cians are extraordinary. Core mathematicians often have a knowledge base and a set of insights and instincts that would be of immense value in other fields. As someone who was trained in the late 1960s and early 1970s at two out- standing U.S. mathematics departments, the things I was exposed to reflected the emphasis of that time. I had no classes—even in the lower division—that dealt with probability or statistics. Although it was offered, I took no discrete mathematics. The only algorithm I saw at either place was Gaussian elimination, and the last algorithm before that was long division. Throughout much of my career I didn’t experience these as gaping lacunae, but it is definitely not how I would train a student these days. When I became the director of an NSF mathematical science institute, one of the first things I started doing was reading Science and Nature, two places where scientists publish work they feel will be of interest to the broader scientific community. What you see immediately is that almost none of the articles are specifically about mathematics, but that the majority make use of sophisticated mathematical and statistical techniques. This is not surprising—when a new idea in core mathematics first sees the light of day, researchers in other fields do not know how to make use of it, and by the time the new idea has made its way into wider use, it is no longer new. The overwhelming impression one comes away with from reading these journals is what an explosively creative golden age of science we are living through and just how central a role the mathematical sciences are playing in making this possible. Examples abound. An extensive and very beautiful literature developed on Erdős-Rényi random graphs—these are graphs where a fraction p of the n(n – 1)/2 possible edges of a graph with n vertices is filled in at random, each edge being equally likely. However, as very large naturally occurring graphs—citations and col- laborations, social networks, protein interaction networks, the World Wide Web— became widely available for study, it became apparent that they did not look at all like Erdős-Rényi random graphs. It is still hotly contested what class or classes of probabilistically generated graphs best describe those that actually occur. This is the embryonic stage of an emerging area of study, with questions about how best continued

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114 THE MATHEMATICAL SCIENCES IN 2025 BOX 4-1 Continued to design a network so that it has certain properties, how best to ascertain the structure of a network by the things one can actually measure, and how to search the network most effectively. There are purely mathematical questions relating to how to characterize asymptotically different classes of large random graphs and, for such classes, potential analogs of the theorems about Erdős-Rényi graphs and the cutoff value of p for which there is an infinite connected component of the graph. With the advent of digital images, the question of how to analyze them—to get rid of noise and blurring, to segment them into meaningful pieces, to figure out what objects they contain, to recognize both specific classes of objects such as faces and to identify individual people or places—poses remarkably interesting mathematical and statistical problems. Core mathematicians are aware of the extraordinary work of Fields medalist David Mumford in algebraic geometry, but many may be unaware of his seminal work in image segmentation (the Mumford- Shah algorithm, for example). Approaches using a moving contour often involve geometrically driven motion—for example, motion by curvature—and techniques such as Osher-Vese based in analysis involve decompositions of the image inten- sity function into two components, one minimizing total variation (this piece should provide the “cartoon”) and one minimizing the norm in the dual of the space of functions of bounded variation (this piece should provide the “texture”). In machine learning, the starting point for many algorithms is finding a mean- ingful notion of distance between data points. In some cases, a natural distance suggests itself—for example, the edit distance for comparing two sequences of nucleotides in DNA that appear in different species where the expected relation- ship is by random mutation. In other cases, considerable insight is called for—to compare two brain scans, one needs to “warp” one into the other, requiring a distance on the space of diffeomorphisms, and here there are many interesting candidates. For large data sets, the distance is sometimes found using the data set itself—this underlies the method of diffusion geometry, which relates the distance between two data points to Brownian motion on the data set, where only a very local notion of distance is needed to get started. There are interesting theoretical problems about how various distances can be bounded in terms of one another, and to what extent projections from a high-dimensional Euclidean space to a lower-dimensional one preserves distances up to a bounded constant. This is one facet of dimensionality reduction, where one looks for lower-dimensional structures on which the dataset might lie. Many of these problems are part of large and very general issues—­ ealing d with “big data,” understanding complex adaptive systems, and search and knowl- edge extraction, to name a few. In some cases, these represent new areas of mathematics and statistics that are in the process of being created and where the outlines of an emerging field can only be glimpsed “through a glass, darkly.” Research in core mathematics has a long track record of bringing the key issues in an applied problem into focus, finding the general core ideas needed, and thereby enabling significant forward leaps in applications. We take this for granted when continued

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IMPORTANT TRENDS IN THE MATHEMATICAL SCIENCES 115 BOX 4-1 Continued we look at previous centuries, but the same phenomenon is providing opportuni- ties and challenges today. There is a long list of ways mathematics is now being used, and the types of fundamental mathematics that are needed spans almost every field of core m ­ athematics—algebra, geometry, analysis, combinatorics, logic. A sampling of these uses, described in nontechnical language, can be found in the companion volume to this report, Fueling Innovation and Discovery: The Mathematical Sci- ences in the 21st Century. Whether or not one gets directly involved in these developments, it would be very useful to the profession if core mathematicians were to increase their level of awareness of what is going on out there. As educators, professors want to continue to instill in their students the clarity and rigor that characterizes core mathematics. But they must do this cognizant of the fact that what students need to learn has vastly expanded, and multiple educational paths must be available to them. There is, of course, some intellectual investment for core mathematicians involved in teaching these courses, but in the end they will have a wider and more varied range of choices of what to teach, and they will be enlarging the number of ecological niches available to mathematicians in both the academic world and the outside world.