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Perspective on and Approach to Characterizing Institutes' Roles in the Mathematical Sciences

Throughout the 20th century, diverse strategies and circumstances have led to the development of a variety of mathematical research institutes, each designed to fit particular needs. Mathematical sciences research institutes of many kinds now play important and varied roles internationally. The committee considered U.S. mathematical research institutes in the context of this worldwide set, not only because of the complementary nature of various institutes, but also because a primary function of mathematical institutes is facilitating communication of mathematical developments worldwide, across a research enterprise that today is thoroughly international.

A Brief Historical View of Mathematical Sciences Institutes

Initial Mathematical Research Institutes

The concept of a mathematical research institute to host visiting mathematicians was realized early in the 20th century by the Swedish mathematician Gösta Mittag-Leffler and his wife, who founded the Mittag-Leffler Institute in 1916. Incorporated into the Swedish Royal Academy of Sciences in 1919, the Mittag-Leffler Institute fostered individual mathematical research. In the United States, the first mathematical research institute was created in 1930 as part of the Institute for Advanced Study (IAS) near Princeton University. Offering preeminent scholars (among them, Albert Einstein and John von Neumann) the opportunity to conduct long-term research in a quiet multidisciplinary atmosphere, the IAS today has several outstanding permanent members and a large number of long-term visitors.

Interactions at the IAS have fostered the development of deep mathematical ideas. An example is the Atiyah-Singer Index Theorem, which has repeatedly brought together analysis and geometry and provides an important link between mathematics and modern theoretical physics. As noted in the National Science Foundation's 1978 announcement calling for the establishment of new research institutes in the mathematical sciences, ''The IAS has been a major force in mathematical research since its very inception, and its effect on the development of U.S. mathematics has been decisive" (AMS, 1978; pp. 484–485).

Institutes Based on the IAS Model

Following the Second World War, several institutes were created along the IAS model. At the Institut des Hautes Études Scientifiques (IHÉS), founded in 1958 at the initiative of a private entrepreneur who wanted to create an IAS counterpart in Europe, activities reshaped algebraic geometry and led to the birth of the first models of turbulence and, thereby, recent theories of chaos. The Instituto de Mathematica Pura e Aplicada (IMPA), founded in Rio de Janeiro in 1957 with the IAS model in mind, became a focal point worldwide for the emerging field of dynamical systems and played a major role in establishing this new area of mathematical research by hosting research-oriented as well as instructional conferences. The Tata Institute, founded in 1945 in Bombay as an affiliated institution of the Indian Atomic Energy Commission,



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1 Perspective on and Approach to Characterizing Institutes' Roles in the Mathematical Sciences Throughout the 20th century, diverse strategies and circumstances have led to the development of a variety of mathematical research institutes, each designed to fit particular needs. Mathematical sciences research institutes of many kinds now play important and varied roles internationally. The committee considered U.S. mathematical research institutes in the context of this worldwide set, not only because of the complementary nature of various institutes, but also because a primary function of mathematical institutes is facilitating communication of mathematical developments worldwide, across a research enterprise that today is thoroughly international. A Brief Historical View of Mathematical Sciences Institutes Initial Mathematical Research Institutes The concept of a mathematical research institute to host visiting mathematicians was realized early in the 20th century by the Swedish mathematician Gösta Mittag-Leffler and his wife, who founded the Mittag-Leffler Institute in 1916. Incorporated into the Swedish Royal Academy of Sciences in 1919, the Mittag-Leffler Institute fostered individual mathematical research. In the United States, the first mathematical research institute was created in 1930 as part of the Institute for Advanced Study (IAS) near Princeton University. Offering preeminent scholars (among them, Albert Einstein and John von Neumann) the opportunity to conduct long-term research in a quiet multidisciplinary atmosphere, the IAS today has several outstanding permanent members and a large number of long-term visitors. Interactions at the IAS have fostered the development of deep mathematical ideas. An example is the Atiyah-Singer Index Theorem, which has repeatedly brought together analysis and geometry and provides an important link between mathematics and modern theoretical physics. As noted in the National Science Foundation's 1978 announcement calling for the establishment of new research institutes in the mathematical sciences, ''The IAS has been a major force in mathematical research since its very inception, and its effect on the development of U.S. mathematics has been decisive" (AMS, 1978; pp. 484–485). Institutes Based on the IAS Model Following the Second World War, several institutes were created along the IAS model. At the Institut des Hautes Études Scientifiques (IHÉS), founded in 1958 at the initiative of a private entrepreneur who wanted to create an IAS counterpart in Europe, activities reshaped algebraic geometry and led to the birth of the first models of turbulence and, thereby, recent theories of chaos. The Instituto de Mathematica Pura e Aplicada (IMPA), founded in Rio de Janeiro in 1957 with the IAS model in mind, became a focal point worldwide for the emerging field of dynamical systems and played a major role in establishing this new area of mathematical research by hosting research-oriented as well as instructional conferences. The Tata Institute, founded in 1945 in Bombay as an affiliated institution of the Indian Atomic Energy Commission,

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has played an important role in augmenting the international visibility of Indian mathematics and establishing in India an outstanding strength in algebra and algebraic geometry. Mathematical Sciences in Other Institutes and Research Centers The early results of mathematical sciences research conducted at Bell Laboratories (now part of Lucent Technologies), founded in 1925, included Walter Shewhart's statistical process control charts of the 1920s, Claude Shannon's information theory and Richard Hamming's error-correcting codes in the 1940s, and the pioneering work of David Slepian on algebraic coding theory in the 1950s. From its creation in 1931 to today, the Indian Statistical Institute has had an important influence on the development of statistics as a field. The world's first statistical institute when founded by P.C. Mahalanobis, then working part-time in a single room, it currently has more than 250 faculty members and over 1,000 support staff in four major Indian cities. It promotes and pursues all aspects of statistical research, applications, and education. An example of a different model of mathematical institute, one with a large permanent staff, is the Steklov Mathematical Institute of the Russian Academy of Sciences in Moscow, established as a distinct entity in 1937. It is also noteworthy that many of the former Soviet Union's mathematicians have worked in institutes devoted to other scientific disciplines or applied technologies. With its launching of the Centre National de la Recherche Scientifique (CNRS) in the late 1940s, France put in place another research structure whereby approximately 10 percent of that nation's mathematical researchers are funded by the CNRS. These research faculty work in university departments rather than in independent institutes. Conference Centers Also inaugurated in the 1940s was a completely new type of institute, the Oberwolfach Mathematisches Forschungsinstitut in Germany. This conference center holds 50 week-long meetings annually, each on a specific theme. In the Second World War's aftermath, Oberwolfach was remarkably effective in reestablishing Germany as a major force in mathematics worldwide. Its success has been attributed to attracting top specialists to serve as session organizers, using a simple scheme for workshops (strictly limited in size), providing superb facilities (including an excellent library and good housing), and encouraging interactions among German researchers and between German mathematicians and the many visiting foreign researchers. In 1981, France created a similar conference center, the Centre International de Rencontres Mathématiques (CIRM), near Marseilles. Supported by the Ministry of Research and the CNRS, CIRM hosts week-long conferences at a well-designed facility. Worldwide Growth After Mid-Century The worldwide technology and information revolution in the last half-century has been stimulated and sustained by a global strengthening of and growth in scientific research capability. As part of their efforts to strengthen and nurture their strategic scientific and technological resources, a number of nations created mathematical research institutes. Located at Kyoto University, the Research Institute for Mathematical Sciences was established as a cooperative research institute in 1964 with research in the traditional major areas of pure and applied mathematics, as well as in mathematical physics, fluid mechanics, mathematical engineering, and theoretical computer science.

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The Centre de Recherches Mathématiques, founded in 1968 in affiliation with the Université de Montréal, represented an institute of a new type in that positions there are temporary and are filled in close association with the host university. The International Stephan Banach Center, established in 1972 in close physical proximity to the Mathematical Institute of the Polish Academy of Sciences, organizes workshops and featured periods devoted to exploration of special themes. During the Cold War, it played an important role in bringing together mathematicians from the East and the West. As a result of NSF's 1980 call for proposals for new mathematical research institutes, the Mathematical Sciences Research Institute (MSRI) in Berkeley, California, and the Institute for Mathematics and Its Applications (IMA) in Minneapolis, Minnesota, were created in 1982. Neither has a permanent faculty, and both have a number of universities as sponsors. Each has developed independently and serves different segments of the mathematical community. Both institutes have succeeded in attracting world-class researchers to lead their thematic programs and thus have become focal points of world mathematical research life. Their impacts on the various segments of the mathematical community are discussed in Chapter 2. Inspired by the Institute for Advanced Studies and the Institut des Hautes Études Scientifiques, Bonn's Max-Planck-Institut für Mathematik was founded in 1982, further enhancing the city's prominent role in mathematical research. More recently, additional mathematical sciences institutes have sprung up around the world, each one differing slightly from its predecessors. The International Center for Theoretical Physics (ICTP) in Trieste, Italy, opened a mathematical section in 1986; the Euler International Mathematical Institute was founded in 1988 in Saint Petersburg; the Centre Émile Borel commenced in 1990 in Paris; the International Centre for Mathematical Sciences was founded in 1990 by Edinburgh University, Heriot-Watt University, the ICTP, and government agencies; the Fields Institute, now headquartered in Toronto, started in 1992, as did the Isaac Newton Institute for the Mathematical Sciences, in Cambridge, England; and the Erwin Schrödinger Institut opened in 1993 in Vienna. Recent Trends Most of the institutes created in the past 10 years for mathematical sciences research have more innovative features than did earlier institutes. For example, the NSF-funded Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), founded in 1989 as a center for the advancement of science and technology with a national scope, is a collaborative effort involving academic participants from Rutgers and Princeton universities and corporate researchers from AT&T Labs-Research, Bell Labs, SAIC/Bellcore, and the NEC Research Institute. As another example, the Park City (Utah) Mathematical Institute, now affiliated with the IAS, was established in 1991 to stimulate vertical integration through interactions among researchers, graduate students, undergraduate students, high school teachers, and undergraduate faculty. The NSF Science and Technology Center (STC) for Computation and Visualization of Geometric Structures at the University of Minnesota, generally known as the Geometry Center, was founded to use visualization as a tool in mathematical research and to communicate mathematical research developments. It was established in 1989 as part of the first round of NSF's STC awards. Although the STCs were envisioned as having 11-year lifetimes (subject to triennial reviews), NSF announced in mid-1996 that the Geometry Center would be phased out; its NSF funding ceased on August 31, 1998. Certain aspects of that center's experience could be worth considering in efforts to plan future mathematical sciences institutes, including: What attention does an institute pay to reaching out and including the larger mathematical research

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community in the institute's efforts? How does an institute maintain awareness of potentially changing expectations when various changes in its operating environment occur, such as changes within funding agencies? How can an institute ensure that what it does is considered to be valuable by the mathematical research community? The Max-Planck-Institut für Mathematik in den Naturwissenschaften, which opened in 1996 in Leipzig (Germany), develops interactions with other disciplines via longer-term visits (typically 5 years). A collaborative initiative, the Pacific Institute in the Mathematical Sciences, emerged in 1997 in Alberta and British Columbia (Canada) to establish new types of contacts with industry and with other disciplines, as had been pioneered at the Institute for Mathematics and Its Applications, through special training sessions and seminars. Supported by U.S. statistical societies as well as universities near its location in Research Triangle Park, North Carolina, the National Institute of Statistical Sciences (NISS) was created in 1991 to focus on project-oriented research while emphasizing feedback to improve statistical methodology. A similar entity devoted to stochastic problems connected to industrial and societal applications, Eurandom, is developing today in the Netherlands. In addition to collaborative institute efforts targeted to industries or other science, engineering, and technology users, there seems to be growing interest in the establishment of mathematical research units within high-technology companies. Hewlett-Packard opened a mathematical sciences basic research institute in 1993, and Microsoft recently established a fundamental research group dedicated to fairly unfettered pursuit of mathematical questions at the foundations of computer science. As recognition expands of the mathematical nature of problems that arise in engineering and scientific fields, more mathematical institutes and industry research organizations will likely be established around the world to link the industrial, engineering, technological, and academic arenas. A Rough Classification of Existing Mathematical Sciences Institutes While realizing the variety and plurality of types of mathematical institutes, the committee offers five basic categories that provide a rough taxonomy of most existing institutes: 1.   IAS-like institutes that principally host large numbers of rigorously selected visitors for significant periods (for instance, a term or a year), and that have a small group of permanent members of exceptional mathematical stature; 2.   Broadly based institutes, such as the Mathematical Sciences Research Institute and the Institute for Mathematics and Its Applications, which have charters to promote diverse areas of mathematical research and mathematical applications and which annually offer scientific programs and workshops of periods from a few months to a year, selected by an advisory committee to emphasize specific topics; 3.   Conference centers, which host short meetings of intermediate size aimed at maximizing interactions among participants; 4.   Institutes that serve as gateways to other entities, such as an industry, a country as a whole, or an entire subdiscipline of the mathematical sciences; and 5.   Institutes designed to promote vertical integration and interaction among the mathematical enterprise's different cohorts (such as researchers, postdoctoral students, undergraduate faculty, high school teachers, graduate students, and undergraduate students).

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This rough classification merely isolates certain common aspects of various institutes and should not be interpreted as implying what might be fundamental parameters. In characterizing institutes, it is of course also necessary to take into consideration the methods by which programs are selected, the time scales for decision making on programs and applications, and the nature and importance of the mathematical issues and questions being investigated. The committee did not attempt to define a priori what constitutes a mathematical sciences research institute. For the purposes of this study, however, the committee limited its consideration to research institutes that have public service as part of their mission in the sense that they are constituted to serve a significantly broader community than just that within their host institution(s). The institutes considered in this report are those that are qualitatively different from a collection of individually funded investigators and which purposely create conditions that encourage a mixing of mathematical scientists with different backgrounds. Brief Summary of Input to the Committee In the course of its study, the committee requested input from a large number of leaders in the mathematical sciences. This section summarizes some of the thoughts received that are relevant to current and potential roles for mathematical sciences research institutes in the United States. More detail on the call for input and the results received is contained in the appendix. The input received by the committee suggests that the general mathematical sciences community is extremely well disposed toward existing research institutes in the mathematical sciences. However, there was also overwhelming common sentiment expressed by respondents that neither continued funding of existing institutes nor the creation of new institutes should occur at the expense of individual investigator research grants. It was the near-unanimous opinion of respondents that there was no optimal number of institutes. The prevailing opinion of respondents was that a collection of mathematical institutes, properly constituted, would be well positioned to help the mathematical sciences meet new challenges and take better advantage of the opportunities now facing the community. Among the new challenges identified by some respondents was the need for interdisciplinary programs that link cutting-edge mathematical sciences research to the broader physical, social, and engineering sciences communities; among the opportunities, many respondents noted that institutes, in addition to their role in research, can additionally serve as training grounds that broaden the horizons of young researchers. There was no enthusiasm among respondents for a U.S. counterpart to an Oberwolfach conference-oriented institute. One concern expressed was the cost, including the cost of bringing geographically dispersed U.S. mathematical sciences researchers to one conference center. Others observed that there are already a great many conferences, and some respondents saw Oberwolfach as a model better suited to intradisciplinary, rather than interdisciplinary, research. The prevailing sentiment of respondents was that there has been a continual and dramatic decrease in funding for basic research in the mathematical sciences, with one consequence being the virtual disappearance of fully funded sabbatical periods. Both applied and core mathematicians noted a need for increased financial support for senior mathematical scientists to spend sabbatical periods at an institute. The replies reflected a general feeling that new funding for new institutes might help alleviate the decrease in basic research funding in the mathematical sciences at all levels (from postdoctoral programs to research opportunities for senior researchers). Concerning how institutes might address such needs or evolve in changing times, some individuals expressed the opinion that an institute would be a good mechanism for fostering new and significant advances in mathematical modeling for experimental sciences, while others believed that now is an appropriate time to consider a new type of institute devoted in large part

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to the dramatically increased role of computers in computation, large-scale modeling and simulation, industrial engineering and manufacturing, and so on. Also, new institutes could help address what now have become significant barriers to professional growth and achievement resulting from the greater than ever generation of research mathematicians by PhD-granting institutions.