B
Workshop Agenda and Presentations

Agenda

March 25, 1998

8:00 a.m.

Chairman's welcome and introductions—Thomas Budinger

8:30 a.m.

Introductory remarks—Phillip Griffiths

8:45 a.m.

The Elucidation and Quantification of Transport and Mixing Processes in the Ocean by Dynamical Systems Techniques, Christopher Jones and Lawrence Pratt

9:45 a.m.

How a Physicist and a Mathematician Got Together and Did Something Useful in Brain Imaging—Lawrence Shepp and Jay Stein

10:45 a.m.

Break

11:00 a.m.

Panel Discussion: Multidisciplinary Research and Training in the Mathematical Sciences: Successes and Failures—Michael Tabor, Avner Friedman, Alan Newell, Nancy Sung, and Mary Wheeler

12.45 p.m.

Lunch

1:45 p.m.

Blue Lasers: Materials Growth, Characterization, and Computational Physics—David Bour and Chris Van de Walle

2:45 p.m.

Coping with Complex Surfaces: An Interface between Mathematics and Condensed Matter Physics—Jack Douglas and Fern Hunt

3:45 p.m.

Break

4:00 p.m.

Numerical Simulation of Subsurface Flow and Reactive Transport—Todd Arbogast and Mary F. Wheeler

NOTE: The workshop “Exploring the Interface Between Other Sciences and the Mathematical Sciences” was held March 25–26, 1999, at the National Academy of Sciences, Washington, D.C.



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Strengthening the Linkages Between the Sciences and the Mathematical Sciences B Workshop Agenda and Presentations Agenda March 25, 1998 8:00 a.m. Chairman's welcome and introductions—Thomas Budinger 8:30 a.m. Introductory remarks—Phillip Griffiths 8:45 a.m. The Elucidation and Quantification of Transport and Mixing Processes in the Ocean by Dynamical Systems Techniques, Christopher Jones and Lawrence Pratt 9:45 a.m. How a Physicist and a Mathematician Got Together and Did Something Useful in Brain Imaging—Lawrence Shepp and Jay Stein 10:45 a.m. Break 11:00 a.m. Panel Discussion: Multidisciplinary Research and Training in the Mathematical Sciences: Successes and Failures—Michael Tabor, Avner Friedman, Alan Newell, Nancy Sung, and Mary Wheeler 12.45 p.m. Lunch 1:45 p.m. Blue Lasers: Materials Growth, Characterization, and Computational Physics—David Bour and Chris Van de Walle 2:45 p.m. Coping with Complex Surfaces: An Interface between Mathematics and Condensed Matter Physics—Jack Douglas and Fern Hunt 3:45 p.m. Break 4:00 p.m. Numerical Simulation of Subsurface Flow and Reactive Transport—Todd Arbogast and Mary F. Wheeler NOTE: The workshop “Exploring the Interface Between Other Sciences and the Mathematical Sciences” was held March 25–26, 1999, at the National Academy of Sciences, Washington, D.C.

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Strengthening the Linkages Between the Sciences and the Mathematical Sciences 5:00 p.m. Wavelets: A Synthesis of Ideas in Harmonic Analysis and Subband Filtering That Happened Serendipitously—Ingrid Daubechies and Martin Vetterli 6:00 p.m. General discussion 6:15 p.m. Adjourn for day March 26, 1998 8:15 a.m. Language and Dynamical Systems: A View from the Bridge—Robert Berwick and Partha Niyogi 9:15 a.m. Protein Folding Class Predictions—Temple Smith and James White 10:15 a.m. Economics in Infinite Dimensional Spaces—Robert Anderson1 and William Zame 11:15 a.m. Break 11:30 a.m. Roundtable Discussion: What Helps and What Hinders Collaboration between Fields in Academics and Industry? James Phillips, Suzanne Withers, and Margaret Wright 12:45 p.m. Closing remarks 1:00 p.m. Adjourn SUMMARY OF PRESENTATIONS AND DISCUSSIONS Chairman's Welcome Thomas Budinger After a general welcome and introductions, the Chair of the Committee on Strengthening the Linkages Between the Sciences and the Mathematical Sciences, Thomas Budinger, explained that the purpose of the workshop was not so much to look at interesting problems in science and mathematics but to consider how successful collaborations between science and mathematics came about, what factors contributed to their success, and what factors inhibited them. He thanked the presenters for their contributions and opened the workshop presentations. 1    Anderson was unable to attend due to illness.

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Strengthening the Linkages Between the Sciences and the Mathematical Sciences Introductory Remarks Phillip Griffiths, Institute for Advanced Study Griffiths described the genesis of this study in his own experiences. As provost at Duke University, he was struck when reviewing tenure files by the amount of mathematics being used throughout the university—not just in the physics department, but in unexpected places like the business school, the medical center, the engineering school, and the environment school. It seemed a sizable fraction of the university faculty were using mathematics in a substantive form in their research, but there was a low level of interaction between the mathematics department and these other departments. When he was chairing the NRC's Board on Mathematical Sciences, the board produced several reports on the interface between mathematics and specific areas of science, such as medical imaging. In reviewing those reports, he noted both the great potential use of mathematics in interesting scientific problems and the large number of interesting mathematics problems that were arising out of such applications. These two experiences were important factors in his early discussions with NRC staff to scope this study. Griffiths also noted that science can now be considered to have three components: experimentation/observation, theory, and modeling/simulation. Each feeds into the other, and two of the three components clearly involve mathematics. Finally, Griffiths expressed his hopes for the outcome of this study. He hoped to see a final report that included substantive accounts of interactions between mathematics and the other sciences and that drew from those accounts implications for the mathematics and other scientific communities. The Elucidation and Quantification of Transport and Mixing Processes in the Ocean by Dynamical Systems Techniques Lawrence Pratt, Woods Hole Oceanographic Institute Christopher Jones, Brown University Pratt, a physical oceanographer, and Jones, a mathematician, worked together on mixing and exchange in ocean current systems. The questions in oceanography included how mixing and exchange occur in major current systems such as the Gulf Stream and how current trajectories reduce to the physical features of the current. The resulting mathematical problems concerned how to understand dynamical systems with insufficient data to perform statistical analysis. The approach taken was to represent the velocity distribution in the current by a model, which led to an inverse problem: How much Lagrangian dynamics is needed to understand the entire velocity field? Results included a student thesis on Melnikoff functions and other mathematical spin-offs. Pratt and Jones met when the Office of Naval Research (ONR) program officer funding Jones recognized the connection between his mathematics and problems in physical oceanography. Jones visited Woods Hole and was introduced to the sort of data on currents collected by physical oceanographers and the sort of dynamical systems they wished to understand. This first meeting led to an informal workshop, held at an inn on the coast of Rhode Island. In this relaxed setting, mathematicians and oceanographers met, with the goal of setting an agenda for collaboration. The motivation was a potential new ONR funding program for such

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Strengthening the Linkages Between the Sciences and the Mathematical Sciences work. This workshop did not result in any concrete work—no papers, no reports—but is remembered by Jones as a key event in the collaboration. Although no formal agenda for collaboration resulted, the researchers began to understand each other's languages, and the mathematicians began to formulate “big pictures” of what the main issues were in understanding the dynamics of ocean mixing and exchange. Despite the fact that the ONR funding program had not yet materialized, the mathematicians got down to work. They were hampered by the lack of appropriate data on currents and by the lack of an explicit analytical expression for the velocity field. Success required that the mathematicians become accustomed to dealing in the imperfect world of physical oceanographers. As worked geared up, a postdoctoral fellow in mathematics was assigned to the project full-time, with funding from ONR. This provided the intense engagement necessary to really move the project out of its initial stages. It took another meeting at Woods Hole before the concrete research agenda began to form and the really interesting mathematics inherent in these oceanographic problems began to become clear. Regular meetings between the oceanographers and mathematicians at various venues furthered the work. It took a period of several years for the work to reach this point. In summarizing their observations about the collaboration, Jones and Pratt made several points. Jones noted that such interdisciplinary collaborations bring tremendous excitement to the field of mathematics, enriching mathematical investigations by the new problems that arise. The mathematicians involved needed to be patient and willing to work on two levels—first at the level of the application and only later at the second level of genuinely new and interesting mathematical problems. Many events without concrete results were, in retrospect, very critical, as they were part of a larger process that led to successful communication and collaboration between the researchers. Finally, Jones and Pratt raised the issue of incentives for collaborations. They felt that the long time needed to begin such a collaboration, a period that might not produce publications or other easily recognized results, was a disincentive, particularly for researchers in the early stages of their career. Research results were also not always readily published in journals relevant to both disciplines—not many mathematicians read the Journal of Physical Oceanography. Left open were some questions. Is it fair to get students or postdoctoral fellows involved in this area? Will they end up falling between two disciplines, with no job? Technical Bibliography Dutkiewicz, S. 1997. Intergyre Exchange: A Process Study. PhD dissertation, Graduate School of Oceanography, University of Rhode Island. 196 pp. Flierl, G.R., P. Malanotte-Rizolli, and N.J. Zabusky. 1987. Nonlinear waves and coherent vortex structures in barotropic β-plane jets . J. Phys. Oceanogr. 17:1408–1438. Pratt, L.J., M.S. Lozier, and N. Beliakova. 1995. Parcel trajectories in quasigeostrophic jets: Neutral modes. J. Phys. Oceanogr. 25:1451–1466. Ridgway, K.R., and J.S. Godfrey. 1994. Mass and heat budgets in the East Australian current: A direct approach. J. Geophys. Res. 99(C2):3231–3248.

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Strengthening the Linkages Between the Sciences and the Mathematical Sciences How a Physicist and a Mathematician Got Together and Did Something Useful in Brain Imaging Lawrence Shepp, Rutgers University Jay Stein, Hologic, Inc. Shepp, a mathematician, and Stein, a physicist, collaborated to invent what was then known as the fourth-generation CAT scanner—a practical scanner with 600 photomultiplier detectors. The basic principle behind what is now known as CT scanning is the collection of simple X-ray images of the body from many different angles. Mathematical techniques allow one to reconstruct from these many images a single, more detailed image providing information on both bone structure and soft tissue. The basic problem facing Shepp and Stein was the artifacts generated by a system of many imperfectly balanced, nonidentical X-ray detectors. What artifacts would be generated? Stein presented a brief overview of the science and mathematics employed in their work. Using mathematics to come to an understanding of these artifacts, they were able to design a practical system and enable CT scanning with better resolution of soft tissue. Shepp raised the question of why a mathematician would want to get involved in an applied problem. In his own case, he was motivated when his child was diagnosed with a brain tumor. It took such a great personal motivation for him to turn from more abstract work to this applied problem. He described pressure from mentors in mathematics not to work on medical applications—they felt it was a mistake that would damage his career. Shepp attributes such attitudes to a phenomenon he refers to as tribal bonding—people banding together in groups for purposes of security. Mathematics, like all the disciplines, suffers from such bonding, in his view, which makes it difficult for researchers, especially young researchers, to move into interdisciplinary collaborations. Shepp noted, however, the great rewards of interdisciplinary collaborations: “If you take something head on, like Fermat's last theorem, you have very little chance of success and it takes enormous effort. If you work in an interdisciplinary field where nobody has been working before and bringing your own methods from mathematics, you are in a tremendous position. It is really easy to make a big splash.” Shepp felt his recognition, despite the warnings of his colleagues, was a result of rather than in spite of his interdisciplinary work. He described this work as “the high point in . . . my scientific life, and no question about it. This interaction was really fantastic.” Technical Bibliography Shepp, L., and J.A. Stein. 1977. Simulated artifacts in computerized tomography, Reconstructive Tomography in Diagnostics Radiology and Nuclear Medicine, M.M. Ter-Pogossian, ed. University Park Press, Baltimore, Md. Shepp, L., J.A. Stein, and B.F. Logan. 1977. A variational problem for random young tableaux. Adv. Math. 26:206–222.

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Strengthening the Linkages Between the Sciences and the Mathematical Sciences Panel Discussion: Multidisciplinary Research and Training in the Mathematical Sciences: Success and Failures Michael Tabor, University of Arizona Avner Friedman, University of Minnesota Alan Newell, University of Warwick Nancy Sung, Burroughs Wellcome Fund Mary F. Wheeler, University of Texas at Austin Tabor opened by speaking briefly of some of the observations he has made while developing interdisciplinary applied mathematics programs. First, he cautioned against what he termed fraud—repackaging existing research programs under the rubric “interdisciplinary” without actually developing a significant interdisciplinary interaction. Those interactions, he noted, took time and commitment to nurture. A program in biomathematics he helped organize took three years before solid enough relationships existed between the biologists and mathematicians to allow meaningful collaborations to begin—and in those 3 years, there were very few tangible benefits to be shown along the way. On the question of whether young researchers should be encouraged to engage in interdisciplinary pursuits, he noted that his current program does engage graduate students successfully in interdisciplinary work. He noted that such students require good dual mentors from both disciplines in order to obtain sufficiently solid training in both disciplines, but that such dual mentoring brings faculty partners from different departments into closer, more fruitful relationships. For mathematicians, he felt it was important to engage in what he termed “service mathematics”—assisting scientists with problems that involve little or no new development of mathematics and thus do not further the mathematician's own research career—in order to open doors to other disciplines. This was a means of testing the waters and finding out what the open research questions were in other fields that might be of interest to mathematics. Finally, he raised the difficulties inherent in the review of interdisciplinary proposals and questioned whether mechanisms currently exist at funding agencies to properly review truly interdisciplinary work. Friedman discussed his experiences establishing and running the Minnesota Center for Industrial Mathematics. He saw the purposes of the center as threefold: (1) education, which in practice means courses for students, (2) research, which is carried out by students with mentors from both the university and industry, and (3) training, which students obtain through industrial internships. The center was born when the mathematics department at the University of Minnesota garnered the support of university administrators for additional faculty slots for such a center. The department provided two faculty slots for every one created by the university. As Friedman described it, obtaining administration support for a center for industrial mathematics was straightforward, but his colleagues in the mathematics department required more convincing, especially when the proposal for a PhD program was made. Two points were needed to convince the mathematics faculty. First, they had to be convinced that the PhD program would be built on solid, traditional mathematical training that did not cut corners to accommodate the need to spend time in applications. Second, supporters of the center raised the argument that to fail to embrace applications and interdisciplinary collaborations would eventually erode the program as the number of graduate students declined below a critical level. These two points eventually won the day, and Friedman reported that graduates of the center's programs are very successful in obtaining interesting, competitive industrial positions. Friedman

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Strengthening the Linkages Between the Sciences and the Mathematical Sciences noted three essential elements in starting up such a program. First, the individual(s) pursuing the program must be committed, as establishing an interdisciplinary program or collaboration requires significant time and energy. Second, resources in the form of time release for faculty are necessary. Finally, it is essential to success that any such program have the strong support of the chair of the mathematics department. Sung described programs the Burroughs Wellcome Fund supports to bring scientists and mathematicians from other disciplines into problems with biological applications. In her experience, the Fund has two effective ways of bringing about interdisciplinary collaborations. The first is to encourage direct interactions between biologists and researchers from other disciplines. Institutions supported by the fund often use the mechanism of consortia to bring researchers together. The second method is to fund postdoctoral researchers or graduate students working at the research interface—these young researchers serve as a link between their faculty mentors and thus as a link between two fields. The fund is attempting to measure its success in breaking down research barriers by tracking the careers of the individuals it funds—what sort of jobs are they taking, what research are they publishing, and what journals are they publishing in? The programs were as yet too new to have definitive answers to these questions. Newell spoke from his experience setting up interdisciplinary mathematics programs at Arizona and Warwick. He began by pointing out that strong disciplinary departments were necessary to nucleate good interdisciplinary centers. Researchers then had to be persuaded to leave the comfort of their own discipline's culture to interact with another discipline. He noted that the challenges of such interactions attracted high-quality students and, in his experience, raised the quality of students applying to the mathematics program. To support those students, it was necessary to establish an environment to encourage and nurture interactions. At Warwick, this was achieved by setting aside one afternoon a week for interdisciplinary speakers and seminars. Resources were also important—extra time was needed to establish solid interdisciplinary work. Thus administration support was also key, but generally easy to obtain as universities are dependent on their mathematics departments to provide training to large numbers of undergraduates from many departments. By taking responsibility for meeting undergraduate educational needs in mathematics—including any remedial work necessary for entering students—departments can raise their value to the university and be in an even stronger bargaining position with the administration of resources. Wheeler described another model for interdisciplinary work used at the Texas Institute for Computational and Applied Mathematics (TICAM). Faculty involved in TICAM have appointments in their home departments and are affiliated with the institute. The institute involves the departments of mathematics, computer science, chemistry, physics, geology, astronomy, biology, and engineering. It has very strong ties with local industrial concerns. Students work in groups on applied problems identified by the industrial partners. The real-life problems motivate students and give them practical experience. Many have gone on to jobs at well-respected industrial firms such as Bell Labs, Shell, and Exxon. The question-and-answer period revolved around whether it is best to provide students with interdisciplinary training or to provide them with solid disciplinary training and foster the traits—good communication, professional respect and curiosity, summed up by one participant as “entrepreneurial attitude”—that will enable them to use their training to take on interdisciplinary tasks. There was no consensus on this issue. Some participants' experience suggested that solid disciplinary training was important for both industrial and academic jobs,

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Strengthening the Linkages Between the Sciences and the Mathematical Sciences while others felt it should be possible to tolerate gaps in disciplinary training for students interested in interdisciplinary problems. Blue Lasers: Materials Growth, Characterization, and Computational Physics David Bour and Chris Van de Walle, Xerox Palo Alto Research Center Bour and Van de Walle collaborate at Xerox PARC on the development of blue semiconductor lasers. These devices are sought because the shorter wavelength of blue light can provide higher storage density on DVD devices and, in combination with red and green semiconductor lasers already available, enable full-color LED displays. The problems facing Bour, the engineer, and Van de Walle, the theoretician, were essentially materials science problems. They needed to optimize the properties of the materials used in the semiconductors, then optimize their interfaces as they are layered in the semiconductor structure. The most efficient path to breakthroughs in these areas was a combination of experiment and modeling. An understanding of the physical phenomena underlying the materials properties, obtained through theory, guided the choice of materials and material growth mechanisms employed in experiments intended to yield a successful device. The empirical approach, which is instinctive to engineers, was a barrier to Bour's acceptance of the contributions that Van de Walle could make to the project. Building trust took time and effort for the two researchers. They noted that it was necessary to learn how to present results to each other in a way that each could understand. They noted, however, that PARC has an environment that nurtures such interactions and that inherently values the contributions modeling can make to applications, and this encouraged their relationship. Technical Bibliography McCluskey, M.D., C.G. Van de Walle, C.P. Master, L.T. Romano, and N.M. Johnson. 1998. Large band gap bowing of InxGa1-xN alloys. Appl. Phys. Lett. 72:2725. Neugebauer, J., and C.G. Van de Walle. 1994. Atomic geometry and electronic structure of native defects in GaN. Phys. Rev. B 50:8067. Van de Walle, C.G., and J. Neugebauer. 1997. Small valence-band offsets at GaN/InGaN heterojunctions. Appl. Phys. Lett. 70:2577. Coping with Complex Surfaces: An Interface between Mathematics and Condensed Matter Physics Jack Douglas and Fern Hunt, National Institute of Standards and Technology Douglas and Hunt described their collaboration on problems of polymer dynamics and complex geometry of polymers. Douglas, a theoretical materials scientist, works in the area of phase transitions, polymers, membranes, and percolation theory—areas requiring the application of statistical physics. Douglas was motivated to meet and know his mathematical colleagues at

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Strengthening the Linkages Between the Sciences and the Mathematical Sciences the National Institute of Standards and Technology (NIST). As he put it, his research interest requires so many different areas of expertise—pattern formation problems, dynamics of phase separation, free-boundary problems such as dewetting—and one person can't know everything. As Hunt described it, Douglas “patrols the halls” and makes it a point to meet new researchers. He sought her out because aspects of their research interests overlap. Hunt, a mathematician, has interest in the area of invariant measures and the ergodic theory of dynamical systems. Hunt's position at NIST calls for her to spend a significant portion of her time providing service mathematics to scientists and engineers at the Institute. She described her work with Douglas as very interesting and rewarding because it led to genuinely new mathematical problems for her to work on, in the area of boundary behavior of harmonic functions. Hunt and Douglas both identified barriers to their collaboration. Hunt felt that there was increasing pressure at NIST to produce research results in the short term. This worked against establishing the rapport and relationship necessary to enable her collaboration with Douglas. Time pressures also came into play when it became apparent that some problems would be furthered only by large-scale computation—and there was insufficient time to devote to both the computations and the mathematics. Douglas posed as a barrier the difficulty in reading mathematical literature. He felt that mathematicians needed to spend more time developing the context of their research problems and explaining the motivating factor for pursuing the solution to a given mathematical problem. On the positive side, Hunt noted that scientists at NIST were very supportive of mathematician colleagues and their research. This, she felt, contributed to a positive atmosphere for collaboration. Technical Bibliography Bernal, J., J.F. Douglas, and F.Y. Hunt. 1995. Probabilistic computation of Poiseville flow velocity fields. J. Math. Phys. 36:23-86. Numerical Simulation of Subsurface Flow and Reactive Transport Todd Arbogast and Mary Wheeler, University of Texas at Austin Arbogast (Department of Mathematics) and Wheeler (Departments of Aerospace, Engineering, Engineering Mechanics, and Petroleum and Geosystems Engineering) discussed ongoing work at the TICAM Center for Subsurface Modeling. The center uses high-performance parallel processing as a tool to model the behavior of fluids in permeable geologic formations such as petroleum and natural gas reservoirs, groundwater aquifers and aquitards, and shallow water bodies such as bays and estuaries. The research revolves around practical tasks such as contamination cleanup. In addition to the obviously challenging engineering problems such tasks pose, they also pose difficult mathematical and modeling problems. These problems involve highly nonlinear systems of equations that exhibit hysteresis. The equations exist on scales from the minute to the very large. Localized events provide singularities in the equations modeling the larger system. Large sets of data require challenging parallel computations.

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Strengthening the Linkages Between the Sciences and the Mathematical Sciences The TICAM Center for Subsurface Modeling is composed of what Wheeler described as a close-knit team of faculty and research scientists. Expertise represented includes applied mathematics; engineering; physical, chemical, and geological sciences; and computer science. Participating students earn a degree in computational and applied mathematics. The center came about when the petroleum industry was downsizing and seeking to outsource much of its research. The university was able to capitalize on that, establishing an industrial affiliates program with the center. Dues for affiliation provided monies for foreign travel and other expenses not readily covered under federal research grants. Also, center investigators found that the industrial funding made it easier to attract the federal grants that provide the bulk of the center's resources. Center researchers, both faculty and student, are very actively involved in interactions with the industrial affiliates. Workshops, meetings, poster sessions, speaker series, and affiliate visits to the center provide substantial opportunity for interaction. The industrial affiliates look to the center to provide the basic research needed to support practical applications. Wheeler noted that the continuity of funding which the center had been able to achieve was important. This allowed research interactions to develop over time. The ability to support foreign travel and significant amounts of conference travel also contributed to the center's reputation and success. Finally, the intensive interactions with industry gave students training in presenting their work to a diverse audience. Technical Bibliography Arbogast, T., and M.F. Wheeler. 1995. A characteristics-mixed finite element method for advection dominated transport problems. SIAM J. Numer. Anal. 32:404–424. Arbogast, T., M.F. Wheeler, and I. Yotov. 1996. Logically rectangular mixed methods for flow in irregular, heterogeneous domains. Pp. 621–628 in Computational Methods in Water Resources XI, Vol. 1, A. Aldama et al., eds. Computational Mechanics Publications, Southampton. Arbogast, T., M.F. Wheeler, and N.-Y. Zhang. 1996. A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media. SIAM J. Numer. Anal. 33:1669–1687. Wavelets: A Synthesis of Ideas in Harmonic Analysis and Subband Filtering That Happened Serendipitously Ingrid Daubechies, Princeton University Martin Vetterli, University of California, Berkeley Wavelet theory has its roots in harmonic analysis and the development of the Fourier transform. Wavelet expansions provide a better description of systems that contain phenomena occurring at multiple scales, from the very coarse to the very fine, than previously available algorithms. They have found widespread application in areas such as vision theory and signal compression. In particular, their application by electrical engineers to achieve better image compression produced new mathematical insights that led to new results in approximation theory. Vetterli described the development of wavelets as a two-way street in which insights and

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Strengthening the Linkages Between the Sciences and the Mathematical Sciences advances flowed back to both the mathematicians and the scientists involved in their development and use. Daubechies and Vetterli outlined the history of wavelet theory. Its roots lie in several fields, and influence can be traced to harmonic analysis, standard coherent-state decompositions used in quantum mechanics, approximation theory, vision theory, and computer-aided geometric design. The synthesis of these roots into wavelet theory occurred over a relatively short period, 1982 to 1988, and involved encounters and interactions among a number of researchers from different fields. Daubechies traced it as follows. A common friend introduced Alex Grossman, a theoretical physicist, to Jean Morlet, a geophysicist, who was working out new transforms that would allow better localization at high frequencies. Grossman used his expertise in coherent states to help Morlet put his work on a more sound mathematical basis, and the result was the first crude wavelet transform. Yves Meyer, a harmonic analyst, heard about Grossman and Morlet's work while standing in line for a photocopy machine and recognized that it was a rephrasing of earlier work by Calderon. Meyer contacted Grossman, and their work was advanced another step. Work by Meyer, Grossman, and Daubechies led Meyer to a new wavelet basis set, which the researchers would intuitively have thought couldn't exist. That basis was picked up by Stephane Mallat, a vision researcher, who heard about it from an old friend who was a graduate student of Meyer. The multilevel view prevalent in vision theory led to a different understanding of the basis construction, and a collaboration of Mallat and Meyer resulted in multiresolution analysis, a better tool for understanding most wavelet bases. Mallat had tried to relate the wavelet bases construction to practical algorithms that existed in electrical engineering. Using these algorithms as a point of departure, Daubechies developed a series of different bases constructions in which the wavelets were defined through the algorithm itself. For reasons he couldn't recall, Vetterli read Daubechies' paper, and his work with Daubechies to generalize the bases for use in signal compression led to yet more mathematical insights and fruitful research for both parties. Daubechies and Vetterli emphasized several factors which they thought made the collaborations work. Serendipitous interactions triggered many of the key collaborations, but the wavelets problem provided topics of interest to both sides of the interaction, which kept all the contributors engaged. Openness on the part of the researchers—openness to looking at a problem from the perspective of another field, openness to talking to researchers in other fields, openness to those who were not expert in one's own field—was crucial. Daubechies, trained initially as a physicist, took possible applications seriously when writing up her results. She included a table of coefficients and a description of the algorithm so that the construction could be used even by those who could not or would not piece through the mathematical analysis in the remainder of the paper. She provided the reader with more context than usual in mathematics publications, trying to help the nonmathematical reader to understand the implications of her work. This was particularly important when Vetterli, the electrical engineer, read her paper—he commented that Daubechies had done an excellent job of explaining some rather esoteric points. Vetterli commented that the U.S. tenure system could work against such interactions. He spent a year learning harmonic analysis and thus probably had fewer papers on his publication list when he came up for tenure review. His department chairman counseled him against writing a book on wavelets, since it would be only one line on his publication list. He also felt that, in general, the engineering and mathematics communities had different cultures, reward systems, and ways of approaching problems, which often made interactions difficult. Formal incentives

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Strengthening the Linkages Between the Sciences and the Mathematical Sciences and rewards for interdisciplinary work, he felt, would foster more interactions and successful research collaborations. Technical Bibliography Daubechies, I. 1988. Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41:906–966. Daubechies, I. 1992. Ten Lectures on Wavelets. Philadelphia, Pa.: Society for Industrial and Applied Mathematics. Vetterli, M., and C. Herley. 1992. Wavelets and filter banks: theory and design. IEEE Transactions on Signal Processing 40:2207–2232. Vetterli, M., and J. Kovacevic. 1995. Wavelets and Subband Coding. Englewood Cliffs, N.J.: Prentice Hall. Language and Dynamical Systems: A View from the Bridge Robert Berwick, Massachusetts Institute of Technology Partha Niyogi, Bell Laboratories Niyogi and Berwick worked together on problems of modeling the development of human language using dynamical systems. Such work is one approach in the attempt to understand, for example, how modern English developed from old English. This work occurred under the auspices of the Center for Biological and Computational Learning at the Massachusetts Institute of Technology. This NSF-funded center had the aim of providing an explicit infrastructure for bringing together researchers from different disciplines. Graduate students and postdoctoral researchers at the center were required to have two advisors—one from mathematics or computer science and one from brain or cognitive science. Berwick used the analogy of the coffeepot to describe the center. A coffeepot provides a nucleation site, where people meet and bump into each other, begin to talk, and learn from each other. The center performed such a function for these two research communities at MIT. It enabled young researchers like Niyogi to pursue an interdisciplinary interest for which they might not otherwise find support and mentoring. Technical Bibliography Niyogi, P., and R.C. Berwick. 1996. A language learning model for finite parameter spaces. Cognition 61:161–193. Brent, Michael R., ed. 1997. Computational Approaches to Language Acquisition. Cambridge Mass.: MIT Press.

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Strengthening the Linkages Between the Sciences and the Mathematical Sciences Protein Folding Class Predictions Temple Smith, Boston University James White, TASC, Inc. Smith and White collaborated on research that sought to model protein folding using Markov models. Smith trained as a physicist and is now part of the Molecular Engineering and Research Center (MERC) at Boston University. White, an engineer, described his training as having a strong mathematical component. They felt that in their collaboration, Smith functioned as a molecular biologist and White as a mathematician. They agreed that the value of their collaboration was that each partner brought a different skill and perspective to the task, which changed the way each thought about the research problem. Their collaboration was aided, they felt, by the relative similarity of their training— they felt they had more language in common than two researchers actually trained in molecular biology and mathematics would have had. White's company encouraged its employees to have contact with academic research centers. MERC—with its multidisciplinary faculty and graduate students drawn from computer science, chemistry, physics, electrical engineering, biology, and medicine—provided a venue for their collaboration to unfold. White and Smith were joint mentors for students at the center. Smith commented on the difficulty of obtaining funding for interdisciplinary research. He noted, however, that while 20 years ago his proposals elicited questions about whether physics had anything to contribute to biology, that attitude is changing. Technical Bibliography White, J.V., C.M. Stultz, and T.F. Smith. 1994. Protein classification by stochastic modeling and optimal filtering of amacid sequences. Math. Biosc. 119:35–75. Economics in Infinite Dimensional Spaces William Zame, University of California, Los Angeles Robert Anderson, University of California, Berkeley2 Anderson and Zame have collaborated on problems of commodities trading. The problems they have focused on concern continuous trading over a time period, which resolves to a problem involving an infinite number of commodities. Anderson trained as a mathematician, but his involvement in economics problems began as early as his graduate thesis. Zame also trained as a mathematician but came to economics much later in his career, when he had already held several faculty positions. Zame came to economics through a personal contact. Near the end of a sabbatical leave at the University of California, Los Angeles, a colleague introduced him to problems in economics by taking him to the university's economics department and to a meeting of the econometrics society. He made another contact there, whom he later looked up when attending a mathematics conference in her home city. Over dinner, she described the problems she was 2    Zame presented for both himself and Anderson, as Anderson was unable to attend due to illness.

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Strengthening the Linkages Between the Sciences and the Mathematical Sciences working on. The conversation became a minitutorial in economics, and by the end Zame had found interesting problems to work on. From there his training in economics was furthered by formal programs. He spent 18 months at an NSF-funded program in mathematical economics at the Institute for Mathematics and its Applications, in Minnesota. Next, he spent time at the Mathematical Sciences Research Institute in a similar program. Soon he was receiving job offers from economics departments. Zame identified several factors that enabled his transition to mathematical economics. One was time—he was able to spend several sabbaticals in economics departments, interacting and learning from colleagues there. Another was good mentoring. He felt he had received much patient mentoring from economists as he learned the ropes and made mistakes—such as “solving” a problem with a mathematical model that allowed negative consumption. Zame felt that the typical academic career path in economics hampered the movement of mathematicians into faculty positions in economics. Economics departments generally do not offer postdoctoral research positions. If a department is interested in a promising mathematician, it typically must offer him or her a tenure-track position—meaning the department has committed to the individual for 6 years, rather than the 2- or 3-year period typical for a postdoctoral fellow. Zame felt many departments were unwilling to commit such a large amount of resources to a person who might or might not make the transition to a new discipline successfully. To foster more math/economics interdisciplinary research, Zame felt that investments should be made in means to allow mathematicians and economists to spend time together—such as conferences, seminars, and sabbatical programs. Technical Bibliography Anderson, R.M., and W.R. Zame. 1997. Edgeworth's conjecture with infinitely many commodities. Econometrica 65:225–274. Roundtable: What Helps and What Hinders Collaboration Between Fields in Academics and Industry? James Phillips, The Boeing Company Suzanne Withers, University of Washington Margaret Wright, Bell Laboratories The discussion identified several themes that had come from the various presentations: Time and patience were needed to develop individual interdisciplinary collaborations—to develop trust and rapport between collaborators, to understand each other's language sufficiently, to recognize the emerging questions underlying each research problem. Time is also needed to build up a culture more supportive of interdisciplinary interactions generally. Sometimes many small steps are required to see a result. Fruitful interdisciplinary interactions involved intellectual challenges in both disciplines. This was important to motivate both collaborators to invest the extra time and effort needed to undertake a collaboration.

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Strengthening the Linkages Between the Sciences and the Mathematical Sciences Mutual respect between collaborators and the ability to communicate problems and results were also key. Researchers often have stereotypical views about other disciplines that may prevent them from listening carefully to their colleagues in other fields, or even from listening at all. Differences in jargon between the disciplines can also make clear communication more difficult, and both parties must be willing to work to overcome them. Proper reward systems can make interdisciplinary collaborations easier. Industry generally has less trouble rewarding such work, provided it furthers company goals. Agreement on the need for formal training across disciplines varied. Some participants felt it was sufficient to learn enough language of the other discipline to communicate; others advocated more formal course work across disciplines. All agreed that good mentoring was necessary for both young and well-established researchers to make the move to interdisciplinary work. All agreed that face-to-face interactions were essential—one cannot become an interdisciplinary researcher by just reading up on another field. The question, Why encourage interdisciplinary interactions? was raised and then answered. Interdisciplinary research can be uniquely innovative and ground-breaking, leading to new approaches to research questions that would not have been generated in a monodisciplinary setting. The question, Why encourage math-science interactions over other interdisciplinary efforts? was also raised and answered. Mathematics brings qualities such as rigor, abstraction, and generality to a problem. It has the ability to transcend an application. It gives structure to a problem and can bring a unique way of understanding to an application. All the participants agreed that interdisciplinary research can be difficult, time-consuming, and frustrating. They also agreed that it can provide some of the most rewarding and challenging opportunities of a research career.

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