The mathematical sciences have evolved significantly during the last two decades—particularly in regard to the role of mathematics in rapidly advancing areas of science, technology, and engineering, and to the research environment for U.S. mathematical scientists. These changes have contributed to an acceleration of developments in the mathematical sciences and to an enormous increase in the amount of mathematical research being pursued. Growth internationally in the number of mathematical conferences and workshops, sponsored by mathematical institutes, universities, and professional societies, has added greatly to the pace of mathematical activity worldwide.

Fundamental changes in many areas of science and technology—especially in biology and medicine, and in communications and computation—are presenting important new problems that require the expertise of mathematical scientists and also pose deep challenges for the field. Examples of such problems include:

- The protein-folding problem;
- Developments toward a unified field theory;
- Techniques for dealing with uncertainty in large-scale computer simulations;
- Compression, storage, enhancement, and reconstruction of visual images;
- The study of large, complex systems; and
- The mining and analysis of enormous data sets.

Some general areas of growth include financial mathematics, which is developing rapidly because of the need for improved decision making in an increasingly global marketplace; the modeling of very complex problems, such as the impact of human activities on the environment; and new and deep mathematical and statistical advances that address computational issues associated with the Human Genome Project.

In computer science, which traditionally has relied mainly on such branches of mathematics as logic, discrete mathematics, and number theory, areas such as topology, differential and conformal geometries, dynamical systems, and exploratory data analysis and mining have become increasingly relevant. The emergence of the Internet and the astonishing growth in its use have led to new applications of queuing theory, combinatorial optimization, and statistical modeling.

Reflecting an unprecedented resurgence of interplay between core mathematics and physics, a flow of mutually important ideas has been established between the areas of string theory and quantum field theory in physics and the mathematical areas of geometry, topology, and the theory of Lie groups and algebras. Also remarkably successful have been the development and application of wavelet analysis to data compression and image processing.

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3
New Challenges and Two New Types of Research Institutes in the Mathematical Sciences
Evolution in the Mathematical Sciences
The mathematical sciences have evolved significantly during the last two decades—particularly in regard to the role of mathematics in rapidly advancing areas of science, technology, and engineering, and to the research environment for U.S. mathematical scientists. These changes have contributed to an acceleration of developments in the mathematical sciences and to an enormous increase in the amount of mathematical research being pursued. Growth internationally in the number of mathematical conferences and workshops, sponsored by mathematical institutes, universities, and professional societies, has added greatly to the pace of mathematical activity worldwide.
Evolution of Science and Technology: Increasing Need for Mathematical Applications
Fundamental changes in many areas of science and technology—especially in biology and medicine, and in communications and computation—are presenting important new problems that require the expertise of mathematical scientists and also pose deep challenges for the field. Examples of such problems include:
The protein-folding problem;
Developments toward a unified field theory;
Techniques for dealing with uncertainty in large-scale computer simulations;
Compression, storage, enhancement, and reconstruction of visual images;
The study of large, complex systems; and
The mining and analysis of enormous data sets.
Some general areas of growth include financial mathematics, which is developing rapidly because of the need for improved decision making in an increasingly global marketplace; the modeling of very complex problems, such as the impact of human activities on the environment; and new and deep mathematical and statistical advances that address computational issues associated with the Human Genome Project.
In computer science, which traditionally has relied mainly on such branches of mathematics as logic, discrete mathematics, and number theory, areas such as topology, differential and conformal geometries, dynamical systems, and exploratory data analysis and mining have become increasingly relevant. The emergence of the Internet and the astonishing growth in its use have led to new applications of queuing theory, combinatorial optimization, and statistical modeling.
Reflecting an unprecedented resurgence of interplay between core mathematics and physics, a flow of mutually important ideas has been established between the areas of string theory and quantum field theory in physics and the mathematical areas of geometry, topology, and the theory of Lie groups and algebras. Also remarkably successful have been the development and application of wavelet analysis to data compression and image processing.

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Engineering in all its sub-areas requires sophisticated mathematics and statistics to formulate, analyze, and solve today's problems. Successful mathematical models have saved billions of dollars by replacing expensive, and sometimes very risky, scaled-down or full-scale experimentation.
Effects of Technology Development on Research in the Mathematical Sciences
Dramatic technological changes, especially in computation and communications, have revolutionized mathematical research itself, even in its core and more traditional areas. The growing power of computers now often enables large-scale experimentation to verify hypotheses and provides tools for the visualization of sophisticated geometric objects and for the manipulation or analysis of algebraic expressions. The result is the expanding field of experimental mathematics. The revolution in communications has also dramatically changed the way in which mathematical research is carried out. Electronic mail and facsimile machines have drastically reduced communication turn-around time, thus facilitating collaborations between mathematical scientists who are physically located at different sites around the world. Indeed, from 1981 to 1993, the percentage of papers written by U.S. mathematicians involved in an international co-authorship almost doubled, from 13% to nearly 25% (NSF, 1998, p. 11). The development of the Internet has also made mathematical information much more accessible to the research community in the form of electronic archives of preprints, electronic journals, and so forth.
However, good communications technology and videoconferencing cannot replace the extended person-to-person experience that is of crucial importance in mathematical research. Nor can it substitute for a stimulating significant immersion in a new and different, supportive, and idea-charged research environment.
Growth and Change in the U.S. Mathematical Sciences Research Community
With the end of the Cold War, many highly qualified mathematicians, including leaders in their fields, relocated from the former Soviet Union and countries in Eastern Europe, as well as from China, to the United States. The influx of talented foreign-born graduate students to U.S. university mathematical sciences departments has increased as well. The best among these foreign students often find jobs and stay in the United States upon completing their degree programs. (For average stay rates, see Figure 7, p. 40, in COSEPUP, 1997). In 1996, non-U.S. citizens earned nearly 57% of the total doctoral degrees awarded in mathematical and computer sciences (AMS, 1996).
The size of the mathematical research community has increased substantially since the competition that resulted in the establishment of the Mathematical Sciences Research Institute and the Institute for Mathematics and Its Applications. In 1993, 22,820 PhD mathematicians were employed in the United States (NSF, 1998, p. 3), compared with about 13,000 in 1979 (COSEPUP, 1997, Figure B-7, p. 62). Of those mathematicians employed in 1993, 14,670 were employed by universities and 4-year colleges, and more than 9,500 were active in research. The number of PhDs awarded annually by U.S. mathematics departments has grown from 800 in 1986 to 1,240 in 1995 (COSEPUP, 1997, Figure 3, p. 35). A more limited job market in academia has tended to cause more PhD mathematicians to seek jobs in industry or finance, which argues for the need for more varied research opportunities for graduate students and younger mathematicians. (For data on employment trends for PhD mathematicians, see Table B-1, p. 63, in COSEPUP, 1997.)
The growth of the international mathematical community (according to the World Directory of Mathematicians, 1998, some 50,000 mathematicians in 69 countries are now involved in research activities worldwide, a figure that does not include the global community of

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statisticians), and the need for the research enterprise to move ahead more quickly to contribute needed expertise, have added pressure to develop centrally collected resources and tools for mathematical scientists. This need has been felt in all mathematically developed countries.
Another change that affects the design of institutes' programs is that more and more researchers now have a working spouse, which can limit researchers' availability for long-term visits at the crucial early stages of their career, before they have secured a permanent position. This factor can also make it more difficult for senior mathematical scientists to relocate during their sabbatical years. In response, most broadly based mathematical research institutes now offer shorter-term workshops or visits that, to some degree, are intended to accommodate this change.
How to Address New Challenges
New Requirements
The technological and logistical problems now facing society are increasingly complex, requiring joint and sustained efforts by researchers from multiple disciplines. The mathematical sciences are almost always basic to, and often pivotal in, the search for ways to address these problems. For example, to maintain their position in a highly competitive international economy, U.S. industry and business must be continuously innovative in product development while minimizing the time from concept to product. Equally pressing needs for technical and scientific research in the environmental, social, medical, and biological sciences can be met only if the United States maintains a world-class science and technology research base. That in turn requires the vigorous participation of the mathematical sciences.
Need for Focused Exploration of Topics That Are Becoming Mathematical
New mathematical and statistical results must be developed constantly to address ever more complicated practical problems and applications and their associated increasingly difficult theoretical questions. Researchers and engineers often find that existing mathematical theories and techniques are not sufficient to completely analyze their mathematical models, to fully substantiate proposed methods of approximation, solution, or simulation, or to deal with very large and complex bodies of data. In practice, non-classical conditions or constraints are often encountered that have not been covered by existing mathematical theories.
Increased interaction in addition to improved communication between core mathematicians and other mathematical sciences researchers, and between mathematical scientists and researchers in quantitative areas of science, engineering, medicine, and technology, is needed to identify and answer important theoretical questions that are basic to realizing the full promise of emerging new knowledge.
Realizing this promise requires concentrated and dedicated support for and nurturing of highly promising theoretical and mathematical foundations once their potential contribution is recognized. Researchers in an area where mathematical promise is emerging must have the opportunity for intense, prolonged communication and cooperative work with mathematical sciences researchers. A concentration of such cooperative efforts has great potential to yield a bountiful harvest of far-reaching developments if stimulated at the right point of scientific development.

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Need for an Infrastructure for Mathematical Sciences Experimentation and for Sharing of Tools
The mathematical sciences have always had experimental components. Historically, experimentation has often been (and is increasingly today) an important approach to mathematical discovery. Experimental mathematics does not replace rigorous mathematics; rather, it is a tool for mathematical scientists to test new phenomena or conjectures, so that they can later be proved rigorously.
Thanks to the increasing accessibility of powerful computers, mathematical experimentation has developed rapidly in recent years and will likely increase further. However, even powerful existing commercial software tools for computation frequently cannot handle many of the necessary calculations when the mathematical examples involve massive amounts of data, and a mathematical scientist must then create pseudo-code for the requisite algorithms or even do the programming. This generally reduces the time that can be devoted to the main thrust of the research. Inefficient use of talent is exacerbated when two or more groups write code for the same or very similar calculations.1 In addition, many of the admirable specialized software tools developed all over the world are at best available for the short period during which the creator is actively engaged in the specific research that motivated the development of the software. A central place is needed where mathematical scientists can learn about specialized and general software that has already been developed in various research areas and explore how well it fits a particular research need. Such a center should also support the use of this software.
Also needed are better and more readily available opportunities for computer-based mathematical experimentation. Further, development of computational software and other computer-based resources needs to be systematically encouraged, and accessibility to them also markedly improved. A systematic effort to encourage development would contribute not only to fundamental mathematical research but also to applications in medicine, science, and technology. A collateral benefit would be the increased visibility of the role of the mathematical sciences in the development of computer-based resources.
A Proposal for Two New Types of Institutes
The committee proposes two new types of mathematical institutes to help address the needs identified above. The current U.S. research institutes in the mathematical sciences are chartered to support and stimulate a broad cross section of mathematical research and applications. The two proposed new types of institutes are viewed as differing in nature from the class of broadly based institutes, and would play quite a different role.
The first new type of institute, described in more detail below, would facilitate the engagement of the mathematical sciences, and particularly core mathematics, with emerging scientific, engineering, medical, and technological fields of national importance. These "emerging-fields" research institutes (there could be more than one based on this model) would each be devoted to long-term, intensive research in a single field whose interface with the mathematical sciences is emerging and shows great promise, but for which the interface has not yet reached the critical mass needed to have a profound, far-reaching impact.
The second type of institute would be a research institute for experimental mathematics and the use of electronic tools in the mathematical sciences, abbreviated in the discussion below as "e-MSI." This institute would provide the beginnings of an infrastructure for the experimental component of mathematical sciences research, including in particular core mathematics research. As such, it would offer access to and support for mathematical software tools used in such
1
For example, at least four groups now have coded fast algorithms for Gröbner bases so as to be able to handle calculations beyond the scope of standard computer algebra packages.

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research, as well as access to mathematical resources needed to meet the ever increasing demands from the other sciences, engineering, medicine, and technology for mathematical sciences research.
The committee recognizes that some in the community will interpret its recommendations as suggesting new uses of scarce resources that might otherwise be dedicated to mathematics pursued in the traditional individual investigator manner. The committee believes, however, that its recommendations offer the promise not only of strengthening the mathematical sciences research enterprise, but also of attracting new partners to the mathematical research community and possibly expanding the funding base.
Research Institute for Mathematical Sciences in Emerging Fields
The mission of a research institute for mathematical sciences in an emerging field would be to accelerate the pace of already-begun highly promising interactions and developments between the mathematical sciences and scientific, technological, medical, or engineering fields of national importance.
Envisioning an Emerging-Field Mathematical Research Institute
The committee envisions an emerging-field mathematical research institute as having integrated teams consisting of mathematical sciences researchers and researchers in what might be referred to as a "companion field"—a scientific, technological, medical, or engineering area in which promising mathematical developments are emerging. The teams would devote concentrated long-term, dedicated effort to the challenging fundamental mathematical and statistical questions arising in the companion field. The institute would be co-directed by a mathematical scientist and a scientific researcher in the companion field, both highly regarded in their respective fields. The mission of the institute would include, at a minimum:
The development of mathematical theories, statistical methods, and computational algorithms for the solution of fundamental problems in the companion field; and
The education and involvement of mathematical scientists and companion-field scientists in the solution of these problems.
The effectiveness of such institutes would depend critically on creating communication links at the broadest level between mathematical and companion-field scientists, both within the institute and also involving the mathematical sciences community at large. The most effective operational framework for a particular emerging-field mathematical institute should be decided by a competition soliciting proposals to establish such an institute addressing a specific emerging focus area. Some examples of possible focus areas would be the mathematical sciences and the biomedical sciences, the mathematical sciences and the environmental sciences, and the mathematical sciences and the information sciences.
The committee envisions that each such institute would have a flexible but limited life span (depending on the companion field) of perhaps 5 to 10 years, and that several such institutes would be established over a period of time, each addressing one focus area.
The committee also anticipates that concentrated interactions, and the subsequent increased integration of the mathematical sciences with other fields, would increase the visibility of the mathematical sciences, expand their scope, and help make mathematical sciences a more attractive career choice. Furthermore, since mathematical sciences PhDs are now seeking jobs in business and industry in greater numbers than in the past, providing opportunities for future

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mathematical sciences PhD students to become well versed in both the culture and the emerging mathematical research challenges of one or more companion fields would likely provide them with more career options upon completion of their doctoral degrees.
While it is true that for more than a decade various efforts have been made to enhance the connections of the mathematical sciences with other areas, they have met with varying degrees of success. Perhaps the most common shortcoming has been that a connection was either too brief or relatively superficial. Those potential weaknesses could be obviated by an emerging-field institute emphasizing substantive, long-term interactions.
U.S. research institutes for mathematical sciences in emerging fields would be similar in some respects to the United Kingdom's Interdisciplinary Research Centres (IRCs), which were initiated under the aegis of the U.K. Engineering and Physical Sciences Research Council and also through the Biotechnology and Biological Science Research Council. IRCs were conceived as new forms of research organizations that would enable a concentration of expensive equipment and facilities, and that would draw together a range of disciplines to tackle cutting-edge science and technology problems. However, it was intended from the start that IRCs would seek industrial support and become self-supporting (via industrial funding) within 10 years. Expressions of industry interest are the basis for establishing IRCs, and future IRCs must seek substantial initial industry investment, with success in securing that support taken as a measure of an IRC's value.
The proposed emerging-fields institutes would share with the IRCs the characteristic of being true partnerships between collaborating fields, aiming to maximize research potential and to have a substantial effect on both fields. However, because an emerging-field institute would emphasize long-term, intensive work on fundamental, core mathematical issues that arise in connection with a single area of focus that has a promising interface with the mathematical sciences, the impetus for the research performed there would be thoroughly different from that at an IRC. Nevertheless, given the few similarities, it would be prudent to take note of the United Kingdom's experience with IRCs when establishing U.S. institutes for mathematical sciences in emerging fields, while keeping very clearly in mind their crucial and substantial differences.
Illustration of a Potential Emerging-Field Mathematical Institute
The scope of an emerging-field mathematical institute might be illustrated by, for example, a hypothetical mathematical sciences and biomedical sciences institute. Areas of activities might include human physiology, where mathematical modeling is beginning to make meaningful contributions. For instance, such tools as stochastic processes, dynamical systems, and partial differential equations are used in models of cell biology to better understand how cells perform tasks in response to chemicals that the immune system produces (or to drugs), how blood vessels grow within tumors, or how cell membranes in neurons receive or transmit electrical signals. Fluid dynamics is used in analyzing blood flow or the flow of hormones in the human body. Control theory is a mathematical tool that provides insight into how brain subsystems receive messages and convert them into action through the nervous system and muscle contractions, and it also has useful applications in the development of drug therapies. In genomics, mathematics and statistics play an increasingly important role in addressing issues that involve massive data sets, pattern recognition, large-scale optimization, and topological questions about the conformation of long molecules.
A substantial community of mathematical biologists and biomedical scientists, in mathematical sciences departments, medical schools, national laboratories, and industry, now employ mathematical theory. Although scattered, this community appears well positioned to soon reach a critical mass and to have a significant impact in the biomedical sciences. A research institute for mathematical sciences in emerging fields with this focus would noticeably enlarge the community of mathematical researchers, particularly young researchers, working in this area.

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It would also greatly influence and contribute to research progress in areas of the biomedical sciences such as human physiology, cell biology, and genomics.
Recommendation: Start a Process to Establish Emerging-Field Institutes
The committee recommends that the National Science Foundation's Division of Mathematical Sciences embark on a process of cooperation with other funding units or agencies to establish a well-chosen and critically focused research institute for mathematical sciences devoted to intensive research in a field whose emerging interface with the mathematical sciences shows great potential for the incorporation of mathematical ideas to achieve important societal advances.
Research Institute for Experimental Mathematics and Electronic Tools in the Mathematical Sciences
The mission of a research institute for experimental mathematics and electronic tools in the mathematical sciences, or e-MSI, would be to collect, maintain, and utilize advances in computer technology and software to stimulate and support the experimental and computational aspects of the mathematical sciences, and to reduce infrastructural barriers to breakthroughs and progress in mathematical sciences research. Examples of such barriers are the lack of access to some appropriate software tools and resources for mathematical experimentation, the physical isolation of many mathematical researchers dispersed throughout the United States, and inadequate access to activities and recent developments in the mathematical sciences research community.
An institute is the proper venue for an e-MSI because the effort and resources needed to effectively identify, assess, acquire, and support the rapidly growing number of tools for mathematical experimentation are too great for individual investigators or groups. A dedicated, substantial investment—which an institute would provide, but which individual or group investigator programs would not—will be required to effectively address the need for a proper infrastructure for mathematical sciences experimentation, tools, and research. An institute also offers the prospect of continuance and survivability that, by their nature, individual or group investigator programs cannot.
Envisioning an Institute for Experimental Mathematics and Electronic Tools in the Mathematical Sciences
An e-MSI would:
Be a national laboratory for mathematical sciences experimentation, playing a role for the mathematical sciences similar to that played by the major facilities for other experimental sciences. It would promote the use of experimentation in the development of new mathematical theories, statistical methods, and computational algorithms for the solution of difficult problems in the mathematical sciences; support the development and maintenance of novel software tools for mathematical experimentation; and make such tools available to the extended mathematical sciences research community.
Provide additional infrastructure to the mathematical sciences community by serving as the focal point for access to experimental and computational resources produced by the worldwide community for research in the mathematical sciences. These resources are dispersed throughout the world but are needed to meet the ever increasing demands for

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mathematical sciences research from the other sciences, engineering, medicine, and technology.
An e-MSI would emphasize providing tools for individual mathematical sciences researchers through the most effective use of electronic resources. Its structure could take any of several possible forms. For example, it might have one director, a technical staff with several members, and visiting research mathematicians. Or it might be a virtual entity, cooperatively overseen by a consortium of institutions with its directorship periodically rotating among designated representatives from the institutions. A competition for proposals would undoubtedly elicit a wide variety of innovative alternatives.
Research at an e-MSI would focus on the role of experimentation in the development of new mathematical sciences results and on the use of mathematical experiments to construct and test conjectures and to generate counter-examples. Such an institute should serve as a forum where experiments can be described, conjectures posed, techniques debated, and standards set. Many of the researchers involved with an e-MSI would be individuals who already pursue some of these goals but seek opportunities to share others' tools and pool their experience. The e-MSI is envisioned as being an entity whose establishment would support and facilitate curiosity-driven mathematical sciences research.
An outgrowth of these activities would be software tools, numerical and non-numerical algorithms, and visualization software that would both support this type of mathematical sciences research and be made available to the mathematical sciences research community at large. A primary mission of an e-MSI would be to collect and maintain a mathematical software library of tools for mathematical experimentation and to produce new software in support of research in the mathematical sciences.
Currently, there exist various electronic research resources that are not easily accessible. It is increasingly difficult for researchers to keep track of this information or to even know where to look for it. No concerted effort has been made to coordinate it and make all of it readily available to all members of the mathematical research community. Anticipated advances in network communications would enable an e-MSI to provide additional access to such resources to aid in the development of mathematical researchers' capabilities and stimulate new mathematical sciences research. The institute would solicit and accumulate cutting-edge resources developed in various fields in the mathematical sciences.
The committee envisions that an e-MSI would be initiated by the National Science Foundation for a minimum operating period of 5 years. Whatever its ultimate form, an e-MSI's design should reflect careful attention to the experiences (including successes, difficulties, and any failures) of previous or continuing institutes or centers that could be viewed as being similar in some way. Examples might include the Geometry Center and the center for Discrete -Mathematics and Theoretical Computer Science, among others.
Recommendation: Establish a Research Institute for Experimental Mathematics and Electronic Tools in the Mathematical Sciences
The committee recommends that the National Science Foundation's Division of Mathematical Sciences establish a research institute for experimental mathematics and electronic tools in the mathematical sciences. Its mission would be to promote the experimental component of mathematical sciences research, to facilitate the development of new computer-based tools, and to provide visibility and accessibility to existing tools and to existing research resources that are scattered throughout the world.