2
Examples of Linkages
This chapter summarizes the lessons the committee has learned from the 10 case studies of crossdisciplinary research compiled in Appendix A and the success factors that have been distilled from 11 examples of academic programs encouraging research and teaching at the mathscience interface. The case studies of mathscience research linkages provide compelling evidence of the synergism between science and mathematical sciences and the advances that can be made by such collaboration. They elucidate the factors that enable crossdisciplinary efforts as well as the barriers that inhibit them. The academic programs described in this chapter demonstrate the important role institutional structures play in the education of both undergraduate and graduate students and in the fostering of communication between mathematical scientists and other scientists.
CROSSDISCIPLINARY EFFORTS BY RESEARCHERS
Boxes 2.1 to 2.4 give four brief examples of what can be achieved through successful mathscience research linkages. Ten moredetailed case studies are set forth in Appendix A. Although they are not intended to be comprehensive, the four examples cut across the sciences and mathematical sciences and point out a variety of common features of mathscience linkages.
All the case studies in this report share some common ingredients for success. In every case, mathematical scientists and scientists saw problems that were attractive and important within their own disciplines. Mathematical scientists and scientists had the opportunity to interact over long periods of time, generally in a common location. There was often an institutional structure, in some cases provided by a funding agency, to maintain the collaboration during its initial phase. The scientists and mathematical scientists shared elements of a common language, by virtue of broad educational backgrounds.
The case studies expose significant barriers to crossdisciplinary work. A young scientist's job search was hampered by the crossdisciplinary nature of her work. A renowned mathematician working on a scientific problem was told by his peers that he was wasting his time.
BOX 2.1 Biostatistics The mathematical sciences have made significant contributions to many areas of science of special importance to mankind, and they, in turn, have been enriched by these contributions. One obvious example is the interactions between statistics and problems in medicine, epidemiology, and public health. These contributions have led to an entire subfield of the mathematical sciences, biostatistics, to address applications in these fields. One of the most important individuals in the development of statistics was John Graunt. In 1662, Graunt turned his attention to the Bills of Mortality, weekly reports by London parish clerks giving the number and the causes of death. These had been instituted to help authorities detect the onsets of epidemics. Graunt published an analysis of these data and developed what is now known as a life table, which allows for the calculation of life expectancy. His work was especially ingenious in that he did not have basic population data from censuses to work with. The ideas and methods Graunt established more than 300 years ago have been highly refined and now form the basis for the life insurance industry and modern survival analysis. The refinements of census methodology, especially in health and vital statistics, are one of the most important aspects of epidemiology, a body of methods designed to determine which group is likely to become ill, the reasons for illness, and what can be done to control an illness. An important method in medical science for determining the efficacy of treatments is the clinical trial. The initial formalization of the clinical trial method used the work of the famous statistician R.A. Fisher on randomization of treatments to comparable groups of experimental units. Fisher had developed these methods to obtain proper statistical tests of significance for agricultural variables such as soil type and fertilizer. With the inception of rigorous drug approval processes, the importance of clinical trials has intensified, as has the need for innovations to make the trials as informative and as efficient as possible. A variety of innovations have been introduced, for example sequential methods (see the case study on martingale theory in Appendix A). The methods developed to solve medical problems, such as determining the causes of certain diseases and evaluating the efficacy of various therapies, have become core elements of the discipline of statistics and have been applied to many other substantive areas. Another important aspect of biostatistics involves the mathematical and statistical modeling of biological and biomedical phenomena. Formal models have contributed greatly to an understanding of the time course of epidemics, the carcinogenesis process at the cellular level, the doseresponse behavior of animals or humans to drugs or toxic substances, the point processes of neuron firings, and the pharmacokinetics and pharmacodynamics of drugs in the bloodstream. Much of the early mathematical modeling involved relatively simple systems of differential equations or Markov process models. As the behavior of biological processes becomes known in much greater detail and more sophisticated scientific technologies are developed to measure biological processes, more complex mathematical models are required. For example, a new dimension of models is now being developed to gain an understanding of scientific phenomena such as protein folding, cognitive neuroscience, and genomics. These biological problems will ultimately create new branches of statistics and mathematics, and the mathematical and statistical sciences will ultimately help to improve our understanding of these areas. SOURCE: Based on Colton and Armitage (1998); Johnson and Kotz (1982); and Stigler (1986). 
BOX 2.2 The Charneyvon Neumann Collaboration: Numerical Weather Prediction Historically, weather forecasting relied on a set of intuitive experiential skills. Partly as a result of the impetus given to meterology by the forecasting activities developed by the armed forces in World War II, forecasting was increasingly seen as a basic problem in fluid mechanics. Given the observed state of the atmosphere today, could one use the fundamental equations of fluid mechanics to calculate the state of the atmosphere tomorrow and the next day? A major step forward on this problem was taken at the Institute for Advanced Study in Princeton shortly after the war, when the mathematician John von Neumann was looking for a scientific problem to which his emerging interest in computer computation could be successfully applied. He focused on meteorology and weather prediction as a problem whose needs could be made to fit with the foreseen capabilities of computer calculations at that time. It was Vladimir Zworykin, a scientist at RCA and friend of von Neumann's, who was originally interested in the meterology problem. Von Neumann reached out to the meterology community for support and organized a conference that was attended by the leading figure in meteorology at that time, Carl Gustav Rossby. Rossby suggested that the core of the group around von Neumann consist of young meterologists interested in a mathematical approach to prediction, and the original funding proposal (to the Navy) called for a team with strong meteorological input. Jule Charney, a young postdoctoral fellow at the time, was suggested to von Neumann and became the leader of the meteorology group. Von Neumann had a very deep knowledge of physics as well as mathematics. Charney, although a meteorologist, had trained as an undergraduate in math and had started graduate school in mathematics before switching fields. Charney and von Neumann were therefore free of many of the communication barriers that might otherwise exist between meteorologists and mathematicians. Charney believed that von Neumann's willingness to work outside the traditional mathematics domain was related to his European training, which had a strong tradition of mathematicians being very interested in physics. Nonetheless, von Neumann was criticized by his mathematical colleagues, who felt he was wasting his time in mundane concerns rather than contributing strictly to pure mathematics. The key problem was how to approach the calculation of viscous fluid flow in response to pressure and pressure changes. The formal mathematics are contained in the NavierStokes equations, but these equations were clearly beyond the reach of the computers of the day. Even the socalled primitive equations in which the hydrostatic approximation is made are an enormous challenge since they contain fasttimescale phenomena like gravity/acoustic waves largely irrelevant for the weather problem. Von Neumann's interest centered on the computational problem, and he was pushing a direct attack on the primitive equations. But Charney, the meteorologist, outlined a way of dealing with the original NavierStokes equations by a method now called singular perturbation theory to abstract a simplified, consistent, and powerful approximation to the NavierStokes equations that is today called the quasigeostrophic approximation. And this was the path the Princeton group followed with success. One can identify three principal ingredients for the success of the endeavor: (1) a sympathetic funding source, in this case the Navy Office of Research and Inventions, (2) a committed mathematician with a good grasp of physical principles and a leadership role in the mathematical side of the problem, and (3) scientists, in this case meteorologists, with a strong mathematical background prepared to profit from the interest of the mathematical community in their problem. SOURCE: Based on Lindzen et al. (1980) and Platzman (1979). 
BOX 2.3 Biomedical Imaging and Mathematics The history of mathematics in biomedical imaging illustrates how mathematicians and speciality scientists can make rapid progress when they work in teams. Most of the early work on modern medical image reconstruction was developed very slowly by individuals working independently. At first, success was stifled by the lack of mathematical input, but later on, partnerships between mathematicians and medical scientists resulted in immediate successes. The mathematical formulations underpinning the threedimensional image reconstruction techniques now known as Xray computerassisted tomography (Xray CT, also known as CAT scan), positron emission tomography (PET), single photon emission tomography (SPECT), and magnetic resonance imaging (MRI) were laid by Johann Radon in 1917, but the Radon transform was not discovered until 60 years later. The first success in reconstruction tomography involving elegant mathematical applications was that of physicist and radioastronomer Ronald Bracewell, who in 1956 used the Fourier projection (the central slice theorem) as the basis for reconstructing the regions of microwave radiation emitted from the Sun disk. The connection between Radon's mathematics and Bracewell's early work was not made until 20 years later, in the mid1970s. The development of medical reconstruction tomography proceeded independently of Bracewell's contributions. Medical computed tomography began to be developed in the early 1960s and proceeded slowly because there was little mathematical input. The earliest Xray CT demonstration was by a neurologist, William Oldendorf, who in 1961 singlehandedly engineered an Xray reconstruction of the transverse section of an object consisting of iron and aluminum nails. Although an inventive experimental study, it utilized a crude method of simple back projection. The patented invention that resulted was deemed impractical because it required lengthy analysis. Oldendorf worked without the input of a mathematician and without any knowledge of the work of Radon or Bracewell. In 1963, David Kuhl, a physician, and Roy Edwards, an engineer, invented a method of imaging radionuclide distributions. They even performed clinical studies in patients 9 years before the first patient Xray tomogram. Since the mathematics needed for an accurate mapping had not been incorporated into their method and computer operating systems in 1963 were unable to quickly perform even simple back projection, the resolution of Kuhl's scanner was only as good as that obtained with existing methods of radionuclide imaging. A crucial mathematical contribution to the reconstruction problem was made in 1963 and 1964 by the physicist/mathematician Allan Cormack. His contributions were directly motivated by two problems. First, medical radiotherapy treatment required the ability to determine the body's attenuation coefficient distributions so that externally applied radiation could be targeted at the tumor. Second, a mathematical algorithm was needed for reconstructing the threedimensional distribution of radionuclide concentrations from data collected by a PET instrument developed in 1962. Independent of the above developments, Godfrey Hounsfield, a computer engineer and industrial researcher, invented the first practical device for performing Xray computerassisted tomography on human beings. In 1967, oblivious of the earlier mathematics of Radon, Bracewell, or Cormack and of the instruments developed by Oldendorf and Kuhl, Hounsfield used an Xray source and Xray detector with a test bed for obtaining projections through a cadaver brain. Nine hours of data acquisition and two hours of computation were required to obtain a single twodimensional plane from multiple onedimensional profiles or projections. Hounsfield used a simple arithmetic reconstruction technique that was merely an iterative estimation method of solving a series of simultaneous equations (i.e., each equation represented a line integral of attenuation coefficients through the head). This method was entirely independent of and different from the method published 8 years earlier by Cormack. 
Eventually, both Cormack and Hounsfield were recognized with a Nobel Prize for their contributions to the development of medical imaging. But it is clear that the ideas and mathematics were independently discovered by these and other scientists mentioned in this example. The key to the application of computed tomography in hospitals was the computer. In 1970, when practical disc operating systems first became available, it was immediately recognized that by using the fast Fourier transform and algorithms based on the work of Radon, Bracewell, and Cormack, a practical medical Xray CAT scanner could be manufactured. In the early 1970s, a number of mathematicianscientist pairs commenced a series of discoveries that led to modern medical threedimensional imaging. Noteworthy contributions circa 1974 came from three teams. First there were the contributions of Lawrence Shepp to the practical implementation of reconstruction. That work was clearly the result of his partnership with physicist Jerome Stein and physician Sadek Hilal in the filtered back projection method of reconstruction now used. Then, the team of Robert Marr, Paul Lauterbur, and Lawrence Shepp showed the power of arithmetic and Fourier techniques in threedimensional reconstruction from projections in MRI. Grant Gullberg, Ronald Huesman, and Thomas Budinger, another team of mathematician, physicist, and physician, showed solutions to the attenuated Radon transform problem in SPECT. Currently, MRI and SPECT are being applied by teams of mathematicians, computational scientists, and statisticians to problems ranging from earthquake prediction to understanding and treating mental disorders, heart disease, and cancer. The message from this historical synopsis is that mathematicians and scientists working in separate locations and on seemingly unrelated scientific objectives related to the mathematical inverse problem made slow and generally unrecognized progress. But when both mathematicians and scientists worked together, as did the three teams cited above, progress was rapid and almost immediately significant. SOURCE: Based on Herman (1979); Natterer (1986); and Deans (1983). 
BOX 2.4 Economics and Game Theory Modern game theory has provided economists with mathematical tools for investigating resource allocation conflicts between groups of adversarial agents (“players”). These agents can be firms competing for market share, governments vying for advantages in trade, or firms and workers bargaining over labor contracts. Mathematician John von Neumann and economist Oskar Morgenstern were founders of modern game theory, and their collaboration offers an intriguing illustration of the linkages between mathematics and sciences. Mathematician John Nash extended their work in ways that made game theory applicable to a rich collection of conflicts of vital interest to the field of economics. Morgenstern and von Neumann's book, Theory of Games and Economic Behavior, was published in 1944. At the time, Morgenstern was in the Economics Department at Princeton and von Neumann was at the Institute for Advanced Study. Morgenstern was skeptical of the then preeminent role of the Keynesian paradigm because of its naive treatment of incentives and individual decision making. This skepticism was clearly evident in his earlier writings. It was his collaboration with von Neumann, however, that allowed Morgenstern to translate his skepticism into an alternative approach to economic modeling. While the formal results in the book were due to von Neumann, Morgenstern's perspective was vital for attracting the attention of economists. Indeed some economists were quick to recognize the potential importance of game theory to their discipline, although it would take decades before this view was widely held. Von Neumann was a 
polymath with a brilliant grasp of many categories of knowledge as well as mathematics. His previous interest in and exposure to economics undoubtedly smoothed communication between him and Morgenstern. In fact, von Neumann had already made contributions to economic dynamics before his collaboration with Morgenstern. While the work of the two men provided the impetus for the use of game theory in economics, its formal results were limited in scope. Morgenstern and von Neumann's work set the stage for John Nash, who made seminal contributions to game theory while he was a graduate student in mathematics at Princeton University in the late 1940s and early 1950s. In his PhD dissertation in mathematics, Nash developed formally the concept of an nperson game and an equilibrium point of that game. He established the existence of equilibria in this setting, extending von Neumann's existence result for twoplayer, zerosum games. Thus, Nash's formulation was much richer and opened the way to the use of game theory in understanding many problems that are now central to the field of economics. In related work, Nash formally posed the notion of bargaining between two players and provided a set of axioms that rationalized a solution to the “bargaining problem.” His papers showed how game theory could be used to study economic conflicts that formerly had been viewed as beyond the reach of standard formal economic analysis. What is intriguing about Nash is that his contributions, which eventually were recognized with a Nobel Prize in economics, were made while he was a graduate student in mathematics at Princeton. Part of his inspiration came from the von NeumannMorgenstern book, which had been published just a few years before Nash's arrival at Princeton and had attracted considerable interest among mathematicians at Princeton and elsewhere. More important input came from a seminar in game theory that mathematicians Tucker (who later became Nash's advisor), Kuhn, and Gale were running at Princeton when Nash was a graduate student. Although von Neumann was initially dismissive of some of Nash's ideas, both Tucker and Gale were supportive. Neither, of course, could anticipate the eventual influence of Nash's work on economics and other social sciences. While it was Nash's cleverness and creativity that allowed him to make a major advance over von Neumann's existence result zerosum games, the broad view of mathematics at Princeton and the rather indirect influence of the economist Morgenstern helped to foster this work. In 1994 Nash shared the Nobel Prize in economics with John Harsanyi and Reinhard Selten. All three researchers did important work on the theory of noncooperative games, but it was Nash's work that laid the foundations for this line of research. His work also preceded Harsanyi's and Selten's work by almost two decades. The twodecade time lag between Nash's work on game theory and the important extensions by Harsanyi, Selten, and others leads to speculation about what might have been gained by restructuring research environments in economics and mathematics. Would the integration of game theory into economic analysis have proceeded at a faster pace had there been a more interactive research environment for economists and mathematicians? The likely answer is yes. The critical roles of von Neumann and Nash in the initial development of game theory provide powerful support for this conjecture. SOURCE: Based on Gul (1997); Leonard (1995); and Nasar (1998). 
LESSONS LEARNED
From the 10 case studies in Appendix A, the committee compiled a list of two kinds of factors: those that enable mathscience collaborations and other crossdisciplinary pursuits and those that militate against them (Box 2.5). Next, the committee categorized impediments to crossdisciplinary research and education and elaborated on them: careerrelated obstacles, obstacles related to the research culture, and resourcerelated obstacles.
BOX 2.5 Lessons Learned from the 10 Case Studies Enabling Factors
Impeding Factors

CareerRelated Obstacles
A career path involving crossdisciplinary research and teaching has risks, ranging from diminished recognition of teaching efforts to delayed or denied promotions. The criteria for hiring and promotion often do not credit or reward those considering crossdisciplinary work as highly as those who keep working within the discipline. Although some researchers might welcome this challenge, it could be a significant impediment to others in the early stages of their academic careers.
Crossdisciplinary research is timeconsuming. The time invested learning another discipline or establishing a viable research collaboration means that the time from startup to first publishable result can be significantly longer than for singlediscipline research. This is particularly difficult for junior faculty on the tenure clock. Developing and teaching crossdisciplinary courses is also timeconsuming. Even where tenure is not a factor, few departments have a good mechanism for assigning teaching credit for such courses or for advising students from outside the researcher's home department.
Crossdisciplinary research is not generally published in prestigious, disciplineoriented journals. This is particularly problematic for junior researchers, who generally must demonstrate contributions to the discipline via publication in respected journals. The multidisciplinary journals Science, Nature, and Proceedings of the National Academy of Sciences welcome crossdisciplinary papers, but they do not offer a large enough forum for the variety of crossdisciplinary research reports expected in the future. Even for these journals, it is harder to find qualified reviewers for crossdisciplinary research than for disciplinespecific research, as few reviewers span fields.
The potential for diminished recognition can also deter collaborations. The academic structure rewards those who evince independent thinking and creativity, so the need to be a sole or the lead author to advance one's career is a fact of life. It is, as well, difficult to have balance in collaborative relationships. Research problems are usually perceived as either primarily science or primarily math problems, leading one researcher to be perceived as a lesser partner in the collaboration.
Cultural Obstacles
The culture of a discipline is such that it encourages and reinforces relationships between researchers within the discipline. Departmental meetings and seminars, professional society gatherings, and conferences bring together colleagues in a single discipline. This is an effective way to maintain the health of a discipline, but it also means that the typical researcher does not have much opportunity to meet and network with colleagues from other disciplines.
A lack of shared knowledge and a common professional language also inhibits collaborations. The case studies demonstrate that researchers may need to persist over a number of years in their attempts to communicate a problem to their colleagues in another discipline before there is a common appreciation of the essence of the problem. This may happen because the researchers do not yet understand one another's discipline well enough to fully grasp the depth of the problem, or it may happen simply because the potential collaborators do not understand each other's jargon. Potential collaborators need not be expert in each other's fields, but they must at least understand enough of the other discipline to recognize the contribution it can make to their research problem.
Resource Obstacles
At the level of the individual investigator, crossdisciplinary efforts by their nature generally require more time to start up and bear fruit than singlediscipline efforts. This additional time generally translates into a need for more money to initiate and succeed at a crossdisciplinary research project than would be needed for a similar disciplinary project. This need for greater resources can be a barrier.
At the level of individual disciplines, a scarcity of human resources can limit the possibilities for crossdisciplinary research. For example, graduate student enrollment in mathematics departments has not kept pace with enrollment in the sciences or in other mathematical sciences. This dearth of human resources makes some mathematics departments,
focused as they are on maintaining a critical mass of researchers to keep their basic discipline healthy, reticent to share their students by allowing them to participate in crossdisciplinary endeavors.
Across disciplines, the committee recognizes that some efforts are under way to move funding from the traditional disciplines to crossdisciplinary efforts. However, such efforts must be undertaken with caution. Some disciplines can benefit from shifting some of their resources from strictly disciplinary to crossdisciplinary activities. Others, such as mathematics, barely have the critical mass to support their core discipline and cannot benefit from such shifts. Some disciplines will, moreover, need additional monies to develop effective interdisciplinary ties.
PROGRAMS THAT FOSTER CROSSDISCIPLINARY RESEARCH
The committee then examined some programs and institutes designed to foster linkages between the sciences and mathematical sciences. Here, too, it took a case study approach: the programs can be usefully examined to identify elements that contribute to their success and that can be emulated. The 11 examples cited below are not intended to be in any way exhaustive and were chosen largely because committee members had personal experience with their effectiveness.
The Mathematical Sciences Research Institute^{1}
The Mathematical Sciences Research Institute (MSRI) of Berkeley, California, exists to further mathematical research through broadly based programs in the mathematical sciences and closely related activities. From its beginning, in 1982, MSRI has been primarily funded by the NSF, with additional support from other government agencies, private foundations, and academic and commercial sponsors. It is administered by a board of trustees drawn from academia, government laboratories, and the business world, with input from its committee of sponsoring institutions. These include about 35 of the leading mathematics research departments in the country, corporate sponsors, and several cooperating institutes and private foundations. A scientific advisory council of leading mathematical scientists oversees its scientific programs. MSRI's basic elements are mathematical programs and workshops and postdoctoral training.
Each year MSRI hosts between two and four major research programs covering a wide variety of topics in pure and applied mathematics. Many of these fullyear and halfyear programs involve researchers from the sciences, engineering, and commercial applied mathematics. Researchers interested in participating submit applications; they can apply to participate in the full program or for 2 or 3 months. The MSRI program provides time, space, services, and at least partial salary to the researchers accepted. It also hosts 20 to 30 postdoctoral students each year for a semester or a year, bringing these students into contact with leaders in their fields of study. MSRI is successful at least in part because it brings together researchers from different disciplines for an extended period of time, in an environment that encourages interaction.
^{1} 
This description is drawn primarily from material on the MSRI Web site, <http://www.msri.org>. 
Each of the main programs also organizes an introductory workshop and one or more specialized workshops. These serve a broader community than the community of those who can come to MSRI for a longer period. In addition, MSRI hosts workshops on a variety of topics, from commercial applications of mathematics to the latest success in pure research. It also sponsors activities designed to develop human resources in the mathematical sciences, to communicate mathematics within and outside of the mathematics community, and to increase public awareness and appreciation of research in the mathematical sciences.
Largescale programs at MSRI that directly involved other sciences include Mathematical Biology (1992), Strings in Mathematics and Physics (1991), Symplectic Geometry and Mechanics (1988–1989), and Mathematical Economics (1985–1986). More recently, there have been shorter events on financial mathematics, cryptography, physical oceanography, genomics, scientific imaging, and parallel computing. Programs planned for the near future include the mesoscopic structure of materials, computeraided design, and quantum computing. MSRI recently established a subcommittee of its scientific advisory committee explicitly charged with identifying areas where collaborations between math and other sciences could be fruitfully encouraged and with developing program possibilities in these areas.
National Computational Science Alliance^{2}
The National Computational Science Alliance (the Alliance) is a partnership between a number of institutions. Headquartered at the National Center for Supercomputing Applications (NCSA) at the University of Illinois at UrbanaChampaign (UIUC), it has as its primary goal fostering work in computational science and computational mathematics. The Alliance's efforts divide into four areas: highperformance computing resources, consulting, and training; enabling technologies development; applications technologies development; and educational applications.
Of particular interest are the Alliance's efforts in applications technology development. These efforts bring scientists, engineers, and computational scientists together to customize computational infrastructures for use by specific scientific and engineering communities. Efforts focus on six areas: chemical engineering, cosmology, environmental hydrology, molecular biology, nanomaterials, and scientific instruments. Work in each of these areas is carried out by teams whose members come from all parts of the United States. The teams are made up of scientists or engineers in the relevant scientific discipline and computational scientists and mathematicians. They are headed by a representative from the relevant scientific discipline and anchored by staff at NCSA who are assigned to the team. The goal in each area is to create computing infrastructure that can be used by the scientific community to attack important problems in various fields.
One example of the results of the Alliance's efforts is the Biology Workbench. This technology provides a pointandclick Webbased interface to more than 100 public domain databases of interest to molecular biologists and to software for sequencing DNA and identifying protein structures. The Biology Workbench frees biologists from needing to understand the various formats and syntaxes of individual databases in order to access the necessary information. It should give molecular biologists the ability to find, sort, and use the growing
^{2} 
Further information on the NCSA and the Alliance can be found at <http://www.ncsa.uiuc.edu>. 
quantities of data generated by the community more efficiently and effectively than had been previously possible, allowing them to focus on their biology.
The interdisciplinary approach is a key theme of NCSA's adaptation of computational methods and is built into NCSA policy. Staff are alert to new possibilities for research linkages, and NCSA encourages facility users, Alliance partners, and staff to meet and talk across disciplinary lines and to consider novel research approaches. The interdisciplinary environment facilitates the transfer of technology and approaches between fields—for example, the molecular biology team adapted software developed by the nanomaterials team and applied it to the analysis of ion channels in cells.
The Institute for Mathematics and Its Applications, University of Minnesota^{3}
The Institute for Mathematics and Its Applications (IMA) was established in 1982 with the mission of closing the gap between theories and their applications. This entails two tasks: (1) identifying problems and areas of mathematical research needed in other sciences and (2) encouraging the participation of mathematicians in these areas of application by providing settings conducive to the solution of such problems and by demonstrating that firstrate mathematics can make a real impact in the sciences.
The IMA scientific programs allow mathematicians and other scientists to share a stimulating research environment. Researchers who have spent time at IMA cite its success in building contacts between researchers of varied backgrounds and experiences; they also remark on the sense of community engendered there. Yearly programs are chosen for the purpose of encouraging interaction between mathematicians and scientists from academia, industry, and government laboratories and opening up new opportunities for the mathematical sciences. The topic of the year is usually divided into two or three subtopics, each of which the program concentrates on for 1 to 3 months and for which it holds a number of workshops. A typical yearly program is designed around a group of senior scientists who agree to be in residence for 3 to 10 months. This allows program continuity as well as scientific guidance for postdoctoral members. The IMA also runs a series of shorter programs during the summer. The annual programs have topics such as mathematics in biology, reactive flow and transport phenomena, mathematics in multimedia applications, and mathematics in the geosciences. Summer programs cover narrower topics, such as energy and environmental models for decision making under uncertainty, and codes, systems, and graphical models.
Postdoctoral fellows are a key component of IMA programs. Selected by open competition, they bring flexibility and enthusiasm to the program and challenge the senior members. Their participation is critical to the mission of the IMA, since it is expected that they will use their experience to become leaders in the mathematicalscientific community. Some of them work half of the time with industry and are supported by it.
Other IMA programs include the Industrial Problems Seminar, in which industrial scientists are invited to present industrial problems to IMA researchers; the IMA Participating Corporations Program, a formalized relationship between the IMA and industrial scientists; and IMA Participating Institutions, a consortium of universities that provides valuable support and guidance.
^{3} 
This description is drawn primarily from material on the IMA Web site, <http://www.ima.umn.edu>. 
Research accomplishments have included mathematical advances in nonlinear waves, dynamical systems, and probability. The Industrial Problems Seminar alone generates about 10 published mathematical articles yearly.
Factors that contribute to IMA's success include (1) its ability to bring researchers together to interact for a period of weeks or months, (2) the presence of young researchers who have a less conservative view of research, and (3) the mentoring of postdoctoral fellows by senior researchers.
The Courant Institute of Mathematical Sciences^{4}
New York University's Courant Institute of Mathematical Sciences is a leading center for research and graduate education in mathematics and computer science. Over the past 50 years, it has contributed to U.S. science by promoting an integrated view of the mathematical and computational sciences as a single, unified field. The Courant Institute has played a central role in the development of analysis, differential equations, applied mathematics, and computer science. Its research activity ranges from the theoretical to the applied. It covers a broad frontier that includes pure mathematics and computer science, as well as applications of mathematics and computation to the biological, physical, and economic sciences.
Richard Courant came to New York University in 1934 as a visiting professor, having left his position as director of the Mathematics Institute at the University of Göttingen, in Germany. In 1935, he was invited to build up the Department of Mathematics at the Graduate School of Arts and Science. In 1937, he was joined by Kurt O. Friedrichs and James J. Stoker. Together with a few of the faculty members already in the department, they formed a closely knit research group. During World War II, under the sponsorship of the Office of Scientific Research and Development, the team became the nucleus of an expanded group that undertook mathematically challenging problems arising from various war projects. However, in contrast to most other ad hoc teams, it did not abandon basic research and advanced instruction. After the war, support from the Office of Naval Research and other government agencies maintained the group and encouraged its growth. The name Institute for Mathematics and Mechanics was adopted in 1946. The Atomic Energy Commission installed a stateoftheart electronic computer at New York University in 1952. This led to the creation of the Courant Mathematics and Computing Laboratory, which has functioned for many years under the auspices of the U.S. Department of Energy.
Central to the scientific life of the Courant Institute is its lively program of research seminars. Their purpose is to stimulate education and research at the level where the two are synonymous. Seminars promote the formation of working groups by drawing students and visitors into contact with ongoing research activities. They also keep the Courant community abreast of new developments around the world. In recent years, there have been regular seminars in applied mathematics, analysis, computational geometry, computer science, magnetofluid dynamics, probability and statistical physics, numerical analysis, programming languages, and robotics. Additional seminars are organized each year depending on the interests
^{4} 
This program description is derived primarily from the Courant Institute brochure, at <http://www.cims.nyu.edu/information/brochure>. 
of the faculty and postdoctoral visitors; recent examples include mathematical biology, materials science, nonlinear waves, and turbulence.
Each year the Courant Institute is host to a large number of visiting scientists. Some are senior distinguished scientists on leave from their home institutions. Others are postdoctoral visitors with external support, typically from the NSF or a comparable foreign agency such as Canada's Natural Sciences and Engineering Research Council (NSERC), Italy's Consiglio Nazionale delle Ricerche (CNR), or France's Centre National de la Recherche Scientifique (CNRS). Still others are appointed to assist with one of the institute's many research projects. In addition, there are the Courant Institute instructorships and the Visiting Membership Program. Both serve to bring recent PhDs to the institute. These programs provide postdoctoral training and research support by involving young scientists in the institute's varied research activities. The programs have existed in one form or another since 1956, financed by various government agencies, industrial organizations, and private foundations. The NSF is currently the principal source of funding. Courant Institute instructorships are ordinarily for a 2year term; they carry a teaching load of one course per semester. Visiting memberships are ordinarily for a 1year term, but they carry no teaching duties; extension or renewal may be possible.
Current research efforts at Courant span a broad range of mathematics and applications. For example, a multidisciplinary effort has recently been launched to study the interactions of ice shelves with the atmosphere and ocean—an important unresolved scientific issue in current climate models. Simulation of complex biofluid dynamics phenomena, including blood flow through the heart, wave propagation along the basilar membrane of the inner ear, and the flight of insects, is being advanced using the immersed boundary method. The institute's recent focus on physiological neuroscience is complemented by research in computer vision. Work in computational physics includes projects in materials science, computational biochemistry, quantum mechanics, and electromagnetic scattering. This dynamic, crossdisciplinary institute is ranked among the top 10 mathematics departments nationwide (NRC, 1994).
The Courant Institute also follows a model of bringing a mix of senior and junior researchers together for an extended period of time, in a setting that provides many opportunities for interaction and mentoring. In addition to the longterm researchers, the steady flow of seminar speakers who come to the institute further stimulates the mix of ideas.
The University of Michigan Department of Mathematics^{5}
The University of Michigan has a strong tradition of core mathematics research and training. However, in the early 1990s, department faculty determined that there was insufficient research and educational interaction with other departments to ensure the longterm health of the department. The department believed that both mathematics and the physical, biological, social, and management sciences flourish if there is genuine communication between them.
Furthermore, it felt that improved educational cooperation would result from individual faculty contacts with colleagues from other departments. Department leadership believed that the best way to create such connections was by recruiting faculty whose research interests match those of
^{5} 
Information on the University of Michigan Department of Mathematics can be found at <http://www.math.lsa.umich.edu>. A detailed discussion of the success of this department can be found in AMS (1999). 
colleagues from other science and engineering departments. At the urging of the department chair and with the support of the university administration, the department began its Interdisciplinary Initiative in 1993. Eight tenured faculty positions were committed to new faculty whose research interests cut across disciplinary boundaries. Four of these faculty positions have been filled.
The initiative has had a significant impact on the department and its crossdisciplinary interactions. A new and wellattended seminar in applied and crossdisciplinary mathematics with participants from a broad range of departments has been created. There are more joint research proposals being submitted and to a broader range of funding agencies. In addition, it has had a large impact on the curriculum. Several new courses at the graduate and advanced undergraduate level have been created, in addition to a new applied calculus honors sequence at the freshmansophomore level and an undergraduate specialization in mathematical biology. Further, mathematics faculty recruited for the interdisciplinary initiative are playing a key role in the collaborative efforts of the department and the College of Engineering to review and possibly restructure the sophomore calculus sequence, the one taken by most engineering students. Also, a new doctoral program that encourages crossdisciplinary work is being set up.
Department of Statistics, Carnegie Mellon University^{6}
The Carnegie Mellon University PhD program in statistics has a strong emphasis on crossdisciplinary training. The program is unusual in that training in crossdisciplinary applications of statistics is required of all students, along with training in probability and statistical theory and statistical computing.
Approximately half of the firstyear curriculum is theoretical and half focuses on the practice of statistics. Some of the applied subject matter courses involve problems that arise from the faculty's collaborative research involvement. In the required course on statistical practice, students gain experience in working on real applications. The applications cover a wide range of disciplines, and even though a student addresses only one or two applications, he or she learns about other applications from other students' experiences.
In the second year of the program, the advanced data analysis course exposes students to a substantial crossdisciplinary experience. The course has two main components that run throughout the year. One is a discussion of different types of data analysis problems, along with current tools and techniques. The second requires each student to engage in a major scientific research project and to analyze the data generated by that project. The student is supervised by a committee, one member of which is the scientist whose data is being analyzed. This culminates in a major report (which often becomes a research publication) and a presentation on the project to the entire department.
The third and fourth years of the PhD program focus on dissertation research; however, students also select a seminar course. A variety of such courses are offered. Some of them focus on landmark papers in statistics, some explore the latest developments in a subfield (such as
Bayesian statistics, spatial statistics, or dynamic graphics), while others deal with application areas of faculty interest, such as statistics applied to cognitive neuroscience and neuroimaging. Because among them the faculty have many crossdisciplinary research interests, students can also continue to gain a variety of additional crossdisciplinary research experiences by working as research assistants.
The primary intent of the graduate program is to train mathematical scientists for a variety of careers. PhD graduates conduct research in industrial and government laboratories, pursue research and teaching careers in traditional university settings, or pursue careers in the financial sector. The success of the program is due in large part to a strong faculty consensus on the importance of crossdisciplinary training. With knowledge increasing so rapidly, students may be frustrated by how little they can learn of it as graduate students, particularly if they are pursuing crossdisciplinary research. It is important, therefore, for them to receive consistent messages and good examples from the faculty concerning the importance of true crossdisciplinary experiences to their future success as statisticians.
University of Chicago Math Concentration of Specialization in Economics^{7}
Undergraduate training of candidates for top PhD programs in economics by U.S. universities falls short of the training in European systems. Many foreign undergraduate students have substantially more mathematics training than U.S. students: they begin a specialized technical curriculum earlier and so have more opportunity to take courses beyond calculus, including real, complex, and functional analysis and ordinary differential equations.
In response to this situation, the University of Chicago has created an option that encourages college students who aspire to either a PhD in economics or a quantitative research position in private industry to take more mathematics. A math concentration with a specialization in economics incorporates seven quarterlong courses outside the Mathematics Department (intermediate economics courses, mathematical probability, statistics, and econometrics) with 10 courses from the undergraduate curriculum for a BS in mathematics. Courses from the Statistics Department are also included in the concentration. The program requires analysis and differential equations courses as well as two quarters of basic algebra.
This initiative came about because the directors of the undergraduate mathematics studies were willing to work with the undergraduate economics program directors, who recognized that quantitatively oriented students are likely to receive better training in mathematics from mathematicians than from economists. It brings technically strong mathematics students into a substantive science, in this case economics. Students with this mix of courses have gone on to graduate studies in economics and statistics. Others use this training to gain employment in research departments of private industries.
^{7} 
A description of the requirements of this specialization can be found at <http://www2.college.uchicago.edu/catalog9900/htm/Math99.html>. 
The University of Arizona Program in Applied Mathematics^{8}
The Applied Mathematics Program at the University of Arizona was established 20 years ago by faculty from the Mathematics Department and other departments in the Colleges of Science and Engineering. Its mission is to provide graduate training leading to MS and PhD degrees and intellectual leadership in applied mathematics research and education at the university. The program recruits students from diverse undergraduate and master's backgrounds in the mathematical, physical, and engineering sciences. The students are supported through research assistantships, special fellowships, and teaching assistantships in the Mathematics Department.
The first year consists of basic mathematical coursework, and the student designs a program of study for subsequent years. Good faculty mentoring in an environment supportive of crossdisciplinary research is an important part of this process. Courses are offered by the Mathematics Department and other departments, and the program has also developed a number of uniquely crossdisciplinary training activities, such as an experimental teaching laboratory and a biomathematics initiative.
Students are ultimately able to apply a combination of mathematical sophistication and disciplinary knowledge to their chosen research area. They have produced dissertation research in such diverse areas as medical imaging, geoscience, atmospheric science, chemical engineering, aeronautical engineering, planetary science, neuroscience, molecular biology, and ecology and have gone on to research positions in academia, government laboratories, and industry.
There are approximately 50 active faculty members in the program from 15 departments in the colleges of Science, Engineering, and Medicine. The program falls under the jurisdiction of the Vice President for Research and the Graduate College, an administrative structure that helps it remain independent of any particular college. Although the Applied Mathematics Program is administratively and financially separate from the Mathematics Department, the two work together closely. This longterm, positive relationship has strengthened both units and is a key factor in the program's success.
^{8} 
Further information on this program is available at <http://w3.arizona.edu/~applmath/home.shtml>. A detailed discussion of this program and its successes can be found in AMS (1999). 
Spelman College Center for Scientific Applications of Mathematics^{9}
The Center for Scientific Applications of Mathematics (CSAM) at Spelman College was launched 3 years ago to increase the number of AfricanAmericans pursuing scientific careers by enhancing the scientific development of students at the undergraduate level. The primary objectives are to support research training and the development of stronger research programs at Spelman, as well as the professional development of faculty and students; to support curricular developments that incorporate interdisciplinary areas of study; and to develop partnerships with other educational, governmental, and industrial institutions that advance Spelman's scientific program.
CSAM attempts to break down the traditional barriers between academic departments. Its associated faculty sponsor a biweekly seminar to promote the sharing of research and curricular ideas across departmental lines. New seminars, courses, and course modules have emerged. The most significant results of crossing disciplinary barriers are the collegial connections that have been established and that lead to new collaborations.
Teams of students and faculty engage in research projects over a 12month period, fulltime during the summers. Approximately 60 percent of the fulltime science and mathematics faculty have been involved in this aspect of the program. A visitor's program brings outstanding scientists to campus who can enlighten students on the interdisciplinary nature of today's scientific enterprise and share information on graduate programs and career paths. Through weekly seminars and presentations of their research at local, regional, and national conferences, students enhance their professional development.
Funding for this effort began when the W.K. Kellogg Foundation established CSAM as a Kellogg Center of Excellence in Science. Continuation funding was made available by Eastman Kodak. Factors that have been credited for the successful start of the program include (1) the vision and vitality of a senior mathematics faculty member who also has administrative responsibilities at the college, (2) a cooperative and dedicated faculty, and (3) a supportive senior administrative staff. CSAM provides a venue for faculty from different departments to interact repeatedly over time as they pursue the center's mission. Through CSAM, the college has begun publication of the Spelman Science and Mathematics Journal, an interdisciplinary undergraduate journal designed to enhance the technical writing skills of students and to increase communication on innovation in science education.
The Department of Energy Computational Science Graduate Fellowship^{10}
The Department of Energy (DOE) supports a broad spectrum of basic and applied research in science and engineering at its national laboratories and through an extensive grants and contract program with universities and the private sector. Highperformance computing is an integral part of DOE missions in climate change, biological systems, materials science, and national defense, as well as in many other areas. Accordingly, DOE has developed a program to encourage and sponsor graduate education in the application of mathematics and computational
^{9} 
Based on a program description found in Spelman Science and Mathematics Journal 1(1). This regularly published journal can be found online at <http://www.spelman.edu/ssmj>. 
^{10} 
This program description is based on information found at <http://www.krellinst.org/CSGF>. 
science to broad areas of scientific and technological development. The goal of the Computational Science Graduate Fellowship (CSGF) is to improve the quality and quantity of young scientists engaged in pursuits that depend on skills and understanding in mathematics and computing.
Predoctoral students at U.S. universities who are in their first or second year of graduate study in the physical, engineering, computer, mathematical, or life sciences may apply for the CSGF. The program provides tuition, an $1,800 per month stipend, an allowance for miscellaneous services, including travel, and computer purchase matching funds. Support can be for up to 4 years and must be renewed each year. The fellowship program requires a program of study that will provide a solid background in three areas: (1) a scientific or engineering discipline, (2) computer science, and (3) applied mathematics. A fellow's major field must fall in one of these categories, and the program of study must demonstrate breadth through substantial academic achievement in the other two. Submission of a program of study signifies a commitment to complete the courses listed if the fellowship is awarded. Changes in the program of study may be made only with the advance consent of the program's advisory committee.
A practicum (research assignment) at a DOE research laboratory is required of every fellow for at least one 3month period during the term of the fellowship, with additional funding available to cover any extra expenses. The practicum is normally undertaken during the summer, as early as possible during the fellowship term.
Computational science graduate fellows become part of a group of researchers learning to solve problems outside disciplinary boundaries. A biennial conference brings participants together to share ideas and support one another. The most recent conference featured presentations on the use of computation in such areas as ecology, biomechanics, hydrology, signal processing, plasma physics, and enzymology, illustrating the diverse areas in which the program is able to develop talent.
The value to the graduate student is a disciplined educational program with quality mentorship from mathematicians and computer scientists, as well as career opportunities at DOE national laboratories and the industrial or educational institutions where their skills are appreciated and sought. Currently 37 fellows are enrolled at 22 major universities. There are 11 practicum sites, including large, multipurpose laboratories with active DOE projects.
Woods Hole Geophysical Fluid Dynamics Summer Program^{11}
For nearly four decades, the Geophysical Fluid Dynamics Summer Program, located at the Woods Hole Oceanographic Institution, has brought together mathematicians and fluid dynamicists from diverse areas of frontier research and from many institutions. Each 10week summer program focuses on a particular research theme. The course begins with an introductory formal lecture series, followed by a summerlength sequence of research seminars. Graduate students admitted to the course are required to initiate research projects under the close guidance of the staff on topics connected to the particular theme of the summer course. The program is guided by a core of senior faculty, whose presence ensures longterm continuity. This core is
^{11} 
Further information on this program can be found at <http://www.whoi.edu/education/dept/#GFD>. 
supplemented by a flow of long and shortterm visitors who contribute special insights to each summer's theme.
The program has been influential in establishing the value of a mathematical foundation for research in geophysical fluid dynamics. It has a history of involving both mathematicians and fluid dynamicists. This longterm involvement has broken down barriers between the two areas, as vocabulary and perspectives are shared and mutually appreciated, and has been vital to the program's success. At the same time the flow of students through the program has populated the field of geophysical fluid dynamics with a cohort that has learned the benefits of interaction between mathematicians and scientists. The program has also served to educate senior faculty.
The summer course has several important components that have led to successful mathscience collaboration, foremost among them the graduate student component. The educational aspect of the program is the key to its longterm influence on both mathematical science and other sciences. Longterm support for this initiative and its summerlong duration have allowed communication barriers between the disciplines to be overcome. The infrastructural support provided by the Woods Hole Oceanographic Institution enables the course, provides the proximity that fosters research interactions, and allows participants to readily interact with Woods Hole scientists, all of which magnify the benefits of the program.
SUCCESS FACTORS
The programs cited above by the committee are by far not the only programs, nor have they been scientifically chosen. Nonetheless they point to a number of factors that seem to be a regular feature of successful crossdisciplinary research. The committee believes that these factors can be usefully applied to other efforts to create linkages led by scientists or mathematicians:^{12}

The programs bring collaborators together for an extended period of time or for repeated interactions over a long period of time. The Courant Institute, the Institute for Mathematics and Its Applications (IMA), the Mathematical Sciences Research Institute (MSRI), the National Center for Supercomputing Applications (NCSA), the University of Michigan Mathematics Department, and the Woods Hole programs all provide researchers with enough time together, which is a fundamental part of their strategy. These periods of time vary in length, and some positions are resident and some are not. Some programs allow researchers to choose their length of stay (a full year or a half year, for example). While the programs often sponsor seminars and symposia, thereby reaching a larger audience and providing short, onetime contact between researchers from various disciplines, clearly it is from those researchers who spend prolonged periods that the strongest and most fruitful research interactions spring.

Many programs have a formal educational component. The Carnegie Mellon, Courant, IMA, Spelman, University of Arizona, University of Chicago, University of Michigan, and Woods Hole programs all have a formal curriculum. This has a twofold impact: it trains a new generation of researchers in crossdisciplinary thinking and it challenges the established

researchers who teach the courses to broaden their own ideas. Virtually the entire community of U.S. geophysical fluid dynamics researchers can trace its roots to researchers educated at the Woods Hole summer program, testifying to the longterm impact such programs can have.

Strong leadership is needed to set up a successful program. The Courant, IMA, Spelman, University of Chicago, and University of Michigan programs all show the importance of a visionary leader in establishing a successful crossdisciplinary research program, and the NCSA program shows the importance of continued leadership in sustaining excellence. Sometimes this leadership came from a small group of investigators, sometimes from a program officer at a funding agency or from the management of a research center, and sometimes from a department head. These people were able to appreciate the importance of creating a crossdisciplinary interface and to communicate it to all the players involved, giving them a rationale to expend the resources necessary to achieve success.

Mentoring lowers the barriers to successful crossdisciplinary research. This is a component of all the programs but is seen most clearly in the Carnegie Mellon, Spelman, University of Arizona, and Woods Hole programs. Active mentoring of students helps assure that they receive appropriate training and identifies employment opportunities. The presence of good role models on the faculty makes it easier for students to consider crossdisciplinary work. Something as simple as faculty attitude, as reflected in attendance at crossdisciplinary seminars and willingness to teach crossdisciplinary courses, delivers a strong message to students about the importance of crossdisciplinary training and breaks down the psychological barriers to crossing the lines between disciplines.
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