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Strengthening the Linkages Between the Sciences and the Mathematical Sciences
APPENDIXES

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Strengthening the Linkages Between the Sciences and the Mathematical Sciences
A Ten Case Studies of Math/Science Interactions
1 Modeling Weather Systems Using Weakly Nonlinear, Unstable Baroclinic Waves
Joseph Pedlosky
Department of Physical Geography
Woods Hole Oceanographic Institution
The pioneering work of Jule Charney and Eric Eady in the late 1940s showed how the development of large-scale disturbances in atmospheric flow patterns that were associated with emerging weather systems could be explained as a natural instability of the largely westerly, zonal (west to east) winds in Earth's atmosphere. Using a linear, small-amplitude perturbation theory, they were able to explain the basic energy source of the weather wave in terms of the ability of the perturbations to tap into the potential energy distribution in the initial, basic flow on which the disturbances grew and from which they efficiently extracted energy. The linear theory gave reasonably good predictions for the spatial scale of the instabilities and their initial rate of growth. What was lacking in these models was an explanation of what limited the growth of the disturbances as they drained energy from the basic flow, at what amplitude saturation would occur, and what dynamical state would follow that saturation.
Some years ago, I became interested in developing the theory for weakly nonlinear, unstable baroclinic waves as models of weather systems in the atmosphere and eddies in the ocean. The weakly nonlinear theories existing at the time followed the work of Stuart and Watson (for the Orr-Sommerfeld problem) or the Malkus and Veronis approach (for convection). In each of these cases the threshold for instability is determined by a balance of energy release by the unstable mode against internal energy dissipation. That means that the marginally unstable wave already has to have a structure that extracts energy from the mean state. Therefore, a relatively straightforward perturbation expansion around that starting point could give the equation for the evolution of the amplitude of the disturbance in the form suggested by Landau, i.e.
where s is the linear growth rate and N is the fruit of the perturbation calculation (the Landau constant).
However, in the geophysical problem the threshold for instability is determined by overcoming inviscid (adiabatic) constraints associated with what is called potential vorticity conservation. That means the marginal wave (around which a perturbation expansion is started) neither dissipates nor extracts energy. In the absence of an energy-releasing structure in the marginal wave, it is impossible to calculate the effect of the perturbation on the mean and hence to get at the Landau constant.
Fortunately for me, a colleague at Chicago was deep at work on the problem of resonant interactions of capillary waves, and it occurred to me that the method of multiple time scales he was using would allow me to calculate the phase shifts in the wave required for energy extraction

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in an implicit way that would subsequently be determined by the evolution itself. This led to a new form of the evolution equation, which has a lot in common with the following equation:
where r is a measure of the very weak dissipation in the evolving wave. Actually the geophysical problem is a bit more complicated but not much more so (the last term is really a spatial integral over a function that satisfies a partial differential equation that is first order in time but whose amplitude is O(¦A¦2). Hence the full system is third order in time.
By another piece of good luck I spent the following summer visiting the Geophysical Fluid Dynamics (GFD) program at Woods Hole, which has always attracted a fair number of mathematicians interested in physical problems, for example, Lou Howard, Joe Keller, and Ed Spiegel (who is at the intersection of math and astrophysics). One of them recognized that the equation I had derived might be put in the form of the Lorenz equations (again a slight oversimplification of the situation—the partial differential equation again). This turned out to be true. It was the first time that I know of that the Lorenz equations were derived as a systematic asymptotic approximation to weakly nonlinear theory instead of an arbitrary, low-order, truncation of a Fourier expansion of a strongly nonlinear problem (where it usually fails as a solution to the original problem when the nonlinearity is important enough to be interesting—this is the part of the Lorenz theory of convection nobody mentions).
Another mathematician happened to be visiting the Massachusetts Institute of Technology and asked me after a seminar I gave there whether my results fit the Feigenbaum scenario for the map problem. My immediate reaction was, “Who is Feigenbaum?” With some further help I was able to take advantage of all of the work done by Feigenbaum and others. Indeed, perhaps not surprisingly, the map theory was a good conceptual organizer for the results of the baroclinic instability problem.
I think that body of work, which evolved further with the help of lots of other people (among them John Hart, Patrice Klein, and Arthur Loesch), was fundamentally enriched at the start by chance interactions with people who knew a lot more math than I do. That happened because certain venues (e.g., the Woods Hole GFD program) had patiently established an atmosphere where unprogrammed interactions of that type could take place if only the participants (1) met frequently enough to know who to go to for help, (2) had established some common language, and (3) shared an interest in the boundary regions between math and (in this case) geoscience.

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2 Mixing in the Oceans and Chaos Theory
Larry Pratt
Department of Physical Oceanography
Woods Hole Oceanographic Institution
One of the central problems in physical oceanography is understanding the transport and mixing of chemical and physical properties such as heat, salinity, ice, and nutrients. These quantities generally are carried from one place to the next by relatively narrow and intense current systems such as the Gulf Stream and Kuroshio. However, there also is a great deal of mixing that takes place across the edges of these currents, causing a slower transport of properties into the surrounding fluid. Understanding how this mixing occurs has been a challenge for many years.
A relatively new approach to this problem has recently been developed within the branch of applied mathematics known as “dynamical systems,” a branch concerned, in part, with the science of chaos. Applied mathematician Chris Jones of Brown University and I have been working together over the past 5 years to develop the new approach and use it to gain new insights about ocean mixing.
We first met in 1992, when Chris was visiting the Woods Hole Oceanographic Institution (WHOI) with his colleague from Caltech, Steve Wiggins. The two were working on a project supported by the Office of Naval Research (ONR), and their program manager, Reza Malek-Madani, had encouraged them to find out more about physical oceanography.
Hence their visit to WHOI. They spent the day talking to WHOI staff, including me, about research problems involving ocean waves, currents, and mixing. This was unfamiliar territory to Chris, so we never would have run into one another had Malek-Madani not served as a matchmaker.
Over the next months, Chris and his mathematician colleagues continued their crash course in physical oceanography by meeting researchers from different institutions. With funds from ONR, Chris was able to arrange a 2-day workshop in 1993 to get all the mathematicians and oceanographers together at a beautiful inn near Little Compton, Rhode Island. Each researcher presented his interests to the gathering and there was plenty of time for casual discussion.
The workshop helped participants to begin crossing into each others' cultures. About a year later, Chris and I formulated the conceptual basis for what became a long-term collaboration.
Chris recalls the meeting as “a critical event in the process while not seeming so at the time.” For one thing, his initial discussions with me were far away from the transport issues that became the basis of our subsequent collaboration. And it was other researchers, including Wiggins, who were presenting material on transport. Yet Chris and I ended up collaborating on the transport issues.
This is good example of the long “spin-up” time often required for such work to get under way. The initial education stage cannot be rushed.
In 1993, the idea of a collaboration became real. Since then, a core group of four or five oceanographers, an equal number of applied mathematicians, and various students and postdoctoral scholars has continued to work together. Workshops at Caltech, Woods Hole, and the University of Minnesota have been a primary means of information exchange. The

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workshops do in fact take work. For one thing, most of the oceanographers in the group approach mathematics somewhat as a hobby. So it takes patient and articulate colleagues, including Chris and Steve Wiggins, to bring the ocean scientists up to speed on the relevant developments in mathematics and chaos theory.
Our workshops are occasionally attended by very well known researchers, but their participation has not catalyzed any collaborations that I know of. There is an important sociological point to be made here: it is not enough for people to be smart, they also have to be able to get along.
My work with Chris has centered on applying dynamical systems techniques to understand mixing and transport processes in the ocean. There is nothing new about applying dynamical systems methodology to the understanding of mixing, but that had not previously been done using a particular technique, lobe dynamics. The method involves calculating both stable and unstable manifolds in models of meandering ocean currents and recirculations.
That particular course of action began about a year after the Little Compton workshop in 1992. I had been struggling to understand a particular mixing problem that arises in connection with the Gulf Stream. Chris suggested using lobe dynamics, under the hunch that it might be better able to describe the Gulf Stream mixing phenomena than other techniques. The method allows quantification of a new transport process, and Chris suspected it also might be applicable for mathematically describing a variety of meandering ocean currents.
He was right, but applying lobe dynamics to the Gulf Stream problem would take some adjustments. For one thing, our Gulf Stream models remained incomplete (lacking, for example, analytically specified vector fields and infinite time). What might have seemed like a headache at first, ended up by catalyzing a benefit. This led to further developments, such as generalized definitions of manifolds over finite time. So while oceanographers were getting new tools to do their science, mathematicians ended up with new directions to take their math.
Postdoctoral scholars, assistant professors, and graduate students also have played key roles in the ongoing collaboration. They're also human carriers of any valuable cultural hybridization that comes out of such collaborations. That's why it is important for these participants to receive professional recognition for their work.
The recognition issue poses special challenges in these kinds of collaborations because the cross-disciplinary content of the resulting papers does not fit neatly within traditional disciplinary boundaries. Young physical oceanographers publishing in Physica D, for example, will be read mostly by applied mathematicians but not by the oceanographers in their own field. This is more than just a theoretical hardship. It has been a real problem for one of the younger team members working with us.
The professional culture of mathematics has been more accommodating when it comes to credit due. Applied mathematicians do in fact receive credit for publishing in, say, the Journal of Physical Oceanography, since the work is, as a result, perceived by their colleagues as relevant.
The general problem of reforming the criteria for professional recognition to accommodate interdisciplinary collaborations remains unsolved. The strongest resistance I have encountered in getting our work recognized has been within my own field, physical oceanography. The field has a large contingent of observationalists, who must devote vast amounts of time to planning scientific cruises, designing instruments, and processing data. For them, mathematics is not even a hobby, and they view mathematicians who work on

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oceanography with suspicion. It is sometimes difficult to get this important part of the field to learn enough about dynamical systems to appreciate the value of our work. The challenge, however, makes the eventual acceptance of our work that much more satisfying.
There are no prescriptions or templates for interdisciplinary collaborations. They are adventures in problem solving. As Chris points out, there is an extra sociological dimension involved in bridging disciplines. As in any successful partnership, both sides need to be equally committed to the collective goal. A lopsided distribution of interest can poison the effort.
The culture of mathematicians may be able accommodate the flexibility needed for successful collaborations between its own members and the other scientific disciplines. But Chris says that mathematicians willing to work on two levels at the same time will be crucial players. First, they should work on problems of short-term interest to the collaborating applied scientist. That will help to win the attention and respect of the scientists by demonstrating how unfamiliar mathematical techniques can help them do more. Secondly, the mathematician should glean from the work some deeper mathematical problems that promise to further develop mathematical theory and to suggest future directions, enriching the profession.
The collaboration that developed between me, a physical oceanographer, and Chris, an applied mathematician, owes a lot to the presence of a third party outside of either discipline—the Office of Naval Research. Even back in 1992, ONR had discussed the possibility of creating a research initiative to bring the minds and skills of mathematicians and oceanographers together. The promise of such an initiative was a considerable incentive to us, not merely because it offered funding but also because it could mean recognition of the work.
Ironically, the initiative for such interdisciplinary work did not become a reality until late 1998, when ONR began funding projects under its new Departmental Research Initiative. But even during the years when the initiative remained merely an idea, it served as a catalyst. “It seemed for many years that we were working to chase something that always escaped us just as we reached it,” Chris recalls. The potential for the ONR research initiative encouraged us would-be collaborators to begin the process of melding our research cultures. Yet because the initiative was not yet real, and ONR was not sending anyone checks, this critical getting-to-know-each-other phase unfolded casually. Talk of an initiative, in Chris's words, “gave us long-term hope without short-term pressure.”

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3 Wavelets: A Case Study of Interaction Between Mathematics and the Other Sciences
Ingrid Daubechies
Department of Mathematics
Princeton University
During the 1980s, an often-haphazard play starring mathematicians and scientists unfolded on many stages concurrently. Out of it emerged a versatile new mathematical tool—wavelets, which are being used by everyone from theoretical physicists and neuroscientists to electrical engineers and image-processing experts.
One of the early actors in this play is Alex Grossmann, a theoretical physicist at the Centre Nationale de la Recherche Scientifique, France, with a specialty in quantum mechanics. Through a common friend, he met Jean Morlet, a geophysicist at the oil company Elf Acquitaine, who was interested in finding better ways of extracting information about Earth from the echoes of seismic waves. At the time, Morlet was looking for alternatives to the workhorse mathematical technique of Fourier transformation for analyzing seismic data. The transformation was proving inadequate for combining the detail he desired at high frequencies with keeping track, simultaneously, of low-frequency behavior. Morlet intuited that there ought to be a different type of transform that would be able to reveal finer features in the data.
Grossmann recognized a connection between Morlet's proposed technique and his own work focusing on the coherent states of quantum systems. Together they hammered out the mathematics behind the first “wavelet transforms.” Wavelet transforms are mathematical tools that enable researchers to decompose complex mathematical functions (or electronic signals, or operators) into simpler, easy-to-compute building blocks. There are other such tools out there, but not with the scope wavelets have. Wavelets are good building blocks to describe phenomena at different scales. Thus, sharply localized wavelets describe or characterize fine-scale detail of something like acoustic signals or neurophysiological data, while the more spread-out wavelet building blocks describe coarser features of the phenomena.
The connection Grossmann recognized between his work in quantum mechanics and Morlet's work in geophysics resided in the dualities inherent in the data of both fields. Quantum mechanics experts routinely confront the duality of a particle's position and momentum, which they only can determine precisely one at a time. To Grossmann, that was akin to the geophysical duality of time and frequency analysis of seismic and acoustic signals.
The link between these dualities and wavelet transforms subsequently emerged from the quantum mechanical concept of coherent states. Using these states as elementary building blocks, physicists can construct arbitrary state functions and operators even though each coherent state remains well localized around specified momentum and position values. A standard coherent state decomposition corresponds to the kind of windowed Fourier transform Morlet was familiar with. But a variant on this construction corresponded to the new transform Morlet had intuited ought to be possible, namely, a wavelet transform.
Word of this new way to process data began spreading, and more mathematicians began joining in. Yves Meyer, a renowned harmonic analyst, was one of them. He heard about Grossmann and Morlet's work while he was standing in line at a photocopy machine. He recognized their wavelet reconstruction formula as a rephrasing of an earlier construction by the mathematician Alberto Calderon, of the University of Chicago. That insight, in turn, moved Meyer to read an early wavelet paper jointly authored by Grossmann and me. And that input

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contributed to Meyer's own construction of a very beautiful and surprising wavelet basis. At the time, we would have bet, wrongly, that such a construction could not exist.
More scientists also began entering the fray. Stephane Mallat, a vision researcher at the University of Pennsylvania, heard about Yves Meyer's basis from an old friend of his, who had been a graduate student of Meyer's. Mallat realized that the hierarchical view prevalent in vision theory might feed back into wavelet theory and lead to a different understanding of a wavelet's construction. That hunch led to a collaboration with Meyer that yielded multiresolution analysis, a much better and more transparent tool for understanding most wavelet bases.
As a scientist who needed to understand and represent images in an algorithmic way, Mallat was motivated to relate the wavelet bases constructions with practical algorithms. It turned out later that those same algorithms already were in the hands of electrical engineers, who were using them in digital signal processing. Using these algorithms as a point of departure, I developed a series of different bases constructions in which the wavelets are defined through the algorithm itself. And that led to especially efficient wavelet-based computational techniques.
In the whole chain of events, which is only partially outlined here, several accidental encounters proved critical to the outcome—a mathematical framework for data analysis with widespread practical application whose development depended heavily on the input of scientists.
Many interesting questions arise in connection with this success story. What technical precursors provided entry for the idea that eventually became wavelet theory? What made these contacts between mathematicians and other scientists and engineers work so well? What sociological or other special factors, if any, played a role? Could this synergy have happened sooner? And what is happening now—are collaboration and cross-fertilization across disciplinary boundaries still taking place?
One of the reasons wavelet theory took off is because most wavelet transforms are associated with fast algorithms. That means that wavelet-based models of even real-life, large-size practical examples become computable. The situation is similar to the Fourier transform, which mathematicians and theoretical physicists have used for a very long time. The technique became a practical tool for scientists only after researchers made it more computable by developing the variant aptly known as fast Fourier transform. The same applies for wavelets. Although their mathematical content is useful regardless of the speed of algorithms for computing with them, most of their concrete applications would not exist if we didn't have a fast algorithm.
Because of its association with wavelet theory, the algorithm itself has been even more widely applied than ever. And that has cycled back into the mathematical development of wavelet theory. For example, wavelets led to a new mathematical understanding of the subband filtering algorithms first developed by electrical engineers. That new perspective, in turn, led to the construction of novel filtering algorithms designed to optimize certain mathematical properties of the associated wavelets. What's more, all of this attention to the original subband filtering algorithms ended up revealing many more applications for the algorithms than would have seemed natural from the filter point of view alone.
Wavelet theory has many more technical roots. Among them is approximation theory. As one builds up a function from its wavelet components, starting with the coarse-scale ones and then adding finer and finer scales, the construction is very similar to typical arguments in approximation theory, using, for example, successive approximations by splines. Aspects of vision theory and computer-aided geometric design also feed into the emergence of wavelets.

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Simple as it may sound, however, the open-mindedness of human beings may have been the most important factor of all. Yves Meyer, for one, told me that his first reaction to the work of Grossmann and Morlet was, “So what! We [harmonic analysts] knew all this a long time ago!” That might have hampered the ascent of wavelet theory. But he looked again and saw that Grossman and Morlet had done something different and interesting. He built on that difference to eventually formulate his basis construction.
The open-mindedness of engineers has played an equally important role in the development of wavelet theory. Consider Martin Vetterli at Berkeley. Like Meyer, he overcame his first reaction, shared by many electrical engineers, that wavelets seemed but a rediscovery of their own subband filtering algorithms; when he looked at the wavelet papers more closely, he saw there was more.
Vetterli is now a human bridge between mathematicians and scientists and engineers. Skilled at explaining engineering issues to mathematicians, he also enjoys learning about mathematical ideas from mathematicians. What's more, engineers like Vetterli are the ones who tend to usher ideas like those from wavelet theory several steps closer to the “true” applications and who point out to mathematicians interesting mathematical developments by engineers.
Even when mathematical ideas turn out to be useful for specific scientific and engineering problems, they usually require further development to make them practical. In image compression, for example, thresholding (which in wavelet language translates into setting to zero all wavelet coefficients below a threshold) may be mathematically equivalent to more sophisticated approximation techniques. But in practice, engineers squeeze out much better results by using smarter procedures of bit allocation. Practical knowledge of this kind sometimes feeds all the way back to the mathematics of wavelet theory; in this particular case, the engineering approach to obtain a better coding led to interesting new approximation results. There is a kind of poetic justice to the way engineers' “smart tricks,” as Vetterli calls them, turn out to be useful mathematical ideas in their own right that feed back into theoretical advances.
Interestingly, almost all the players in the early development and application of wavelet theory had one or more changes of field or subfield during their scientific lives. Exposure to different fields may be a key to becoming open to ideas from other fields. Or it may be that researchers who choose to develop expertise in more than one field also are the ones more inclined to listen to and explore ideas from other fields.
Technology, most notably easy access to computers, also was important in the development of wavelet theory in the 1980s. I probably would not have taken the path I did take had I been in my home lab in Brussels. At that time and place, access to computer facilities was not as straightforward as here. But I was visiting the Courant Institute at New York University, where I found a computer terminal on my desk.
The potential payoff in knowledge and technology was yet another incentive that accelerated the development of wavelet theory. That it might lead to new insight about vision inspired both Yves Mallat and me. When I wrote my first paper on the bases I constructed, I included a table of coefficients and a description of the algorithm. It laid out more clearly than usual, in a mathematics paper, how the construction could be used in practice. That opened up the technique even to those uninterested in slogging through the mathematical analysis of how I derived the particular numbers, or how I proved the mathematical properties of the corresponding wavelets.

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Including a ready-to-use table of numbers rather than just a description of how these numbers could be derived should one choose to do so, was a feature borrowed from engineering articles. This was not a customary way of communicating in mathematical circles. So I am fortunate that Communications in Pure and Applied Mathematics, the math journal that published the paper, did not flinch at including such a lengthy table. It was a valuable tool. It is what inspired engineers and other scientists to try out the construction, even though the paper appeared in a journal that many of them would not routinely check.
When it comes down to it, those who communicate well to diverse audiences are the ones who have the most impact. The most influential papers by mathematicians in this area are written in a more transparent, more readable style than mathematics papers often are. In my own case, I was trained as a physicist even though I became a professional mathematician. That background has helped me communicate my own work in ways accessible to many scientists.
The culture of mathematicians often leads to work that appears opaque to those outside the culture. Many mathematicians write their journal papers in an extremely terse style. That succeeds in getting a maximal number of results onto limited journal pages, but it often keeps people outside the subfield from understanding the research. The lesson is simple: mathematicians who hope to collaborate with other scientists successfully must try to write their journal articles, and communicate in general, with their would-be collaborators in mind,
The synthesis of minds, ideas, and research that yielded wavelet theory in the 1980s probably could have happened sooner. Many conceptual precursors, including powerful mathematical ideas in harmonic analysis, had been developed in the last 40 years. But they remained largely confined to a small community. Similarly, the algorithms that underlie the fast wavelet transform were known almost only to mostly electrical engineers. Until the wavelet synthesis took place—almost by accident—it was not clear these ideas could be useful outside their own communities. The question never even came up, because the would-be suitors were completely unaware of one another. Had there been better mechanisms to promote interdisciplinary contacts, the wavelet synthesis might have happened at least a decade sooner.
In summary: the interdisciplinary drama that led to wavelet theory worked because it involved mathematicians who were open to other scientists and capable of explaining their work and ideas to nonmathematicians, and because of the involvement of engineers who were interested in interacting with mathematicians and had a similar open-mindedness. I think the most important lesson for mathematicians, however, is that we must value and cultivate this type of interdisciplinary contact just as we value intradisciplinary contacts between colleagues in different subfields of mathematics. Contacts like those have a knack for sparking new discoveries.

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approximate a “lowest energy level.” This procedure worked fairly well for simple knots and links but was useless for more complex ones. Just because you are unable to twist a plastic ribbon representation of one knot to be like another knot does not mean the knots aren't still identical.
Once again I went to a mathematics department—this time at Berkeley—and sought out a topologist. The result was the same as at Chicago. I could not formulate the questions to him in a way that he understood.
I decided to write to three mathematicians who had worked with DNA and to ask them for help. I hoped that their knowledge of biology would bridge the gap that existed between me and the mathematicians. James White answered my letter and said that topologists could indeed solve my problems. He introduced me to the world of topological invariants and to topologists—including Ken Millet, Lou Kauffman, Vaughan Jones, and De Witt Sumners—who worked with them. This started some productive collaborations in which we explained to the biologists how to use invariants and used them ourselves to study DNA structure and transformations. It was a two-way street. The mathematicians were fascinated by the rich variety and essentiality of DNA topology and were often surprised at the forms they adopted when viewed by electron microscopy.
In one paper I went too far in the use of topology, but it brought me to a new way of looking at DNA. We had found something very interesting about the effects of catenating two rings on the linking number of each ring and had interpreted the results in terms of surface topology. The data were correct but the interpretation was not useful. We received a dozen letters from around the world pointing this out and suggesting various alternative interpretations. I had by that time become director of the Program in Mathematics and Molecular Biology (PMMB), and I invited all of our critics to a workshop at a meeting of PMMB in Santa Fe, New Mexico.
At that workshop, most of us concluded that surface topology was great for describing some biological processes but that another formulation was simpler and more useful in describing our results with supercoiling. One of the most vociferous critics of the use of surface topology to describe supercoiling was a Russian, Alex Vologodskii, who was an expert in the use of computer methods to simulate DNA. At the Santa Fe meetings, Alex and I realized that our approaches were nicely complementary. I was an experimentalist who was uncovering the roles and uses of DNA topology and he was a theoretician who could, via simulation, test out the underlying bases for our results. We started a collaboration that continues to this day. A number of important results have emerged that would not have been possible had each of us been working alone. Let me give just one example.
A vexing but important problem in biology is the conformations of supercoiled DNA. The DNA in all organisms is supercoiled, and proper supercoiling is vital to function. Indeed, drugs that block topoisomerases, which introduce and remove supercoils, are among the most widely used antibacterial and anticancer agents. The development of these new drugs was a direct consequence of the basic studies on topoisomerase mechanism and structure. DNA must be relatively large to be supercoiled, and this precluded the use of high-resolution structural methods, such as X-ray crystallography. Instead, we found that a combined approach—biophysical measurements and simulation—was very powerful. Because of the excellent agreement between theory and experiments, we could calculate what we could not measure. As a result, we could specify all the important conformational properties of DNA and

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how they varied with DNA length, supercoiling density, and the concentration of mono-, di-, and trivalent neutralizing ions. The problem that had seemed intractable was solved.
Program in Mathematics and Molecular Biology
Thus far this narrative has focused on my own work. In 1986, DeLill Nasser, an administrator from NSF who was involved in science and technology centers (STCs), called to encourage me to think about a center for mathematics and biology. I asked Sylvia Spengler at Berkeley if she might be interested in leading such a facility with me. She loved the idea and we set out to find like-minded mathematicians and biologists. Jim White readily joined, as did Jim Wang, Lou Kauffman, Vaughan Jones, and De Witt Sumners. We went to DeLill and said we were ready to submit our application to NSF, but she wisely counseled us to broaden out beyond DNA topology and geometry. In the next phase we brought Eric Lander, Mike Waterman, Gene Lawler, and Mike Levitt into the group. These were the founding members of PMMB, and NSF funded our program.
Our main job was to bridge the barriers between the two disciplines. To do so we held a workshop at Santa Fe like the one described above, where the audience was an equal mix of mathematicians and biologists. We had didactic lectures about mathematics for the biologists and about biology for the mathematicians. There were also cutting-edge lectures where the only rule was to try to be clear to both disciplines. We also left a lot of time for informal discussions. It worked remarkably well. In a survey we found that fully 35 percent of the participants in this one conference had formed an interdisciplinary collaboration.
But this was just one aspect of PMMB. Another was our matchmaker function, getting the right mathematicians and biologists together. Mathematicians and biologists, even those on the same campus, usually do not know each other, but we in PMMB had a lot of experience in interdisciplinary work and we had contacts in both communities. We invited biologists who were looking for a mathematician collaborator to ask us for a potential match. Likewise, mathematicians who were thinking of making the plunge into biology were invited to talk to us. We then tried to make appropriate matches or bring them to Santa Fe to find their own mates. I have no doubt that several years of work could have been saved if I had from the beginning been able to find the appropriate mathematical collaborators, and I did not want others to repeat that experience.
The whole arrangement would have little future unless we brought in young people. We did so through our fellowship programs. There were PMMB fellows in the members' laboratories and offices and a larger national fellowship program to extend our reach. The latter has been greatly expanded by funds from Burroughs Wellcome Foundation. Our requirement is just that the work we support is high quality and interdisciplinary. We also bring the young people to Santa Fe and to our retreats, so that they become part of a community.1
1
The current members of PMMB are Bonnie Berger (Massachusetts Institute of Technology), Pat Brown (Stanford University), Carlos Bustamante (University of California, Berkeley), Nicholas Cozzarelli (University of California, Berkeley), David Eisenberg (University of California, Los Angeles), Vaughan Jones (University of California, Berkeley), Lou Kauffman (University of Chicago), Eric Lander (Massachusetts Institute of Technology), Mike Levitt (Stanford University), Wilma Olson (Rutgers University), Terry Speed (University of California, Berkeley), Sylvia Spengler (Lawrence Berkeley National Laboratory), De Witt Sumners (Florida State University), Elizabeth

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In addition to the interdisciplinary research conference at Santa Fe, PMMB sponsors smaller workshops, short courses, and symposia at universities and professional society meetings. These activities result in text materials and collaborations between PMMB members. The number of highly qualified fellowship applicants has tripled since the PMMB's inception, indicating that the combined field of mathematics and biology is growing rapidly.
Thompson (University of Washington), Ignacio Tinoco (University of California, Berkeley), Jim Wang (Harvard University), Mike Waterman (University of Southern California), and Jim White (University of California, Los Angeles).

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8 Challenges, Barriers, and Rewards in the Transition from Computer Scientist to Computational Biologist
Dannie Durand
Department of Molecular Biology
Princeton University
Metal, glass, and many other materials get stronger when they're heated up and allowed to cool slowly. The process, called annealing, works by enabling the materials' molecular structures to assume more stable arrangements than faster cooling rates would allow. In effect, annealing lets the materials “compute” an internal structure corresponding to an energy minimum. Simulated annealing is a computational technique corresponding to the physical one, which means it is a marriage of mathematics and physics. It has become a widely used heuristic method to solve combinatorial optimization problems that are too large to be solved exactly. The method has been particularly useful for optimizing electronic VLSI (very large scale integrated) circuit layout, which includes challenges like distributing individual components on a chip so that the total length of wire connecting them is minimized.
My own doctoral thesis in 1990 focused on the challenge of making simulated annealing calculations faster using parallel computers. After completing my doctorate, I worked for several years on a variety of scheduling problems in parallel computation in an industrial research laboratory before making the transition to another area—computational biology—where mathematics and science come together in a rich intersection.
I had been interested in computation and biology for some time, but my participation in the 1994–1995 Special Year on Mathematical Biology (hosted by the Center for Discrete Mathematics and Theoretical Computer Science, or DIMACS, at Rutgers University) helped me to begin research in this area. This special year focused on computational and mathematical problems that arise from the wealth of protein and DNA sequence data that has become available since the 1970s. Taken together, the tutorial, workshops, and seminars associated with the special year provided a comprehensive background in almost all areas of computational molecular biology. I also met many of the leading researchers in the field, became familiar with their published work, and learned about the field's important open problems.
My introduction to the field's players led to a collaboration with Lee Silver, a molecular biologist from Princeton. This was another event that was crucial for my transition from simulated annealing studies to research in computational biology. Silver and I met as part of the DIMACS Special Year, where a number of Princeton biologists presented computational problems that had arisen in their research. I became interested in his work because the biological system he described—a gene that promotes itself at the expense of the organism that carries it and the other genes in that organism—was bizarre and compelling. We began to meet weekly to discuss the problem.
Our initial collaboration addressed issues in the evolution and population dynamics of the t haplotype, a mutant chromosomal region in mice that violates Mendel's laws and causes sterility in males. Our first results on t haplotype (or just t) population dynamics were based on Monte Carlo simulations. They overturned the long-standing hypothesis that the low-level persistence of t is due to a balance between loss of t due to stochastic effects in small populations and reintroduction of t through migration. This work appeared in the journal Genetics in 1997.

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As I learned more about this biological system, I saw that there were several interesting mathematical aspects as well. Two papers on the evolution of the t haplotype—one based on Monte Carlo simulation, the other on Markov chain analyses—are currently in preparation as a result. The main contribution of these papers is in the field of theoretical biology. The work also suggests new experimental hypotheses that may now be tested through laboratory and field work.
I am now working on a second project with Silver and one of his graduate students, Ilya Ruvinsky, that studies rearrangements in the mouse genome. Ruvinsky initially suggested this second project, which has a substantial bioinformatics component. Neither one of us has the expertise to pursue the project alone, and our collaboration would probably not have come to pass had I not been regularly attending Silver's lab meetings. If successful, this research will add to biology by shedding light on the evolution of mammalian genomes. It could contribute to computer science in the areas of Internet information retrieval, data mining, and data visualization.
Silver and I have an effective collaboration because each of us is willing to go to considerable lengths to understand the other's field and culture. He has enough computational expertise to understand what I have to contribute. He can also identify computational problems that arise in his laboratory and has a realistic sense of what it would take to solve them. I understand enough of the molecular biology to see just how computational methods can help solve problems and also understand the limits of those methods.
My switch to research in computational biology has been the most important decision in my professional career. My research has become much more rewarding and I have become far more productive. But there also are barriers to research in areas like computational biology that intimately combine mathematics and science. One barrier centers on how research in computational biology should be evaluated and rewarded. Can a computer scientist in a computer science department receive tenure if she publishes a significant part of her research in biology journals? Her colleagues may not value such work. Even if they do, they probably do not have the expertise to distinguish top quality work in computational biology from lesser efforts. Alternatively, will a biology department be willing to hire a professor with a PhD degree in computer science and little or no formal training in biology?
Another challenge to interdisciplinary programs is the physical separation of collaborators. On the one hand, researchers in an interdisciplinary program need to interact. For this to work, they need to be sitting in the same building and sharing the coffee machine, secretarial staff, etc. On the other hand, researchers also need to spend time in their own departments to stay up to date on new developments in their own fields. I, for one, select the problems to work on based on their biological importance. To solve these problems, I frequently need to draw on expertise from subareas in computer science other than my own. I can be most effective, therefore, if I spend a substantial part of my time sitting in a computer science department whose faculty members are interested in a fairly broad spectrum of areas.
It takes more than individuals with an interest in merging math and science before anyone actually does the interdisciplinary research that interest suggests. It requires a commitment to interdisciplinary research all the way from the national level, where funding agencies decide where to put their money, down to the university level and the level of individual departments. It requires infrastructure to support activities in both the home departments of collaborators and the interdisciplinary research center (or other venue) supporting the collaborative work. And it requires a mechanism by which to reward research activities that are specifically interdisciplinary.

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9. Crossbreeding Mathematics and Theoretical Chemistry: Two Tales with Different Endings
R. Stephen Berry
Department of Chemistry
University of Chicago
Spectra of Nonrigid Molecules
The spectra of molecules—the pattern of wavelengths the molecules absorb or emit—reveal to us the structures of molecules and how they rotate, vibrate, and change their shapes. Interpreting molecular spectra is not always straightforward, and sophisticated mathematical tools frequently come into play when chemists extract the hidden messages of such spectra. Nonrigid molecules, whose atoms may exchange sites on time scales approaching the fast time scales of atomic vibrations in molecules, have spectra that pose some of the greatest challenges to interpretation. The effective symmetry of such a molecule appears higher than just that of the rigid ball-and-stick object we would normally use to represent it. When the atoms rapidly exchange places in the molecule, the result looks just like the original molecule (so long as identical atoms are unlabeled and hence indistinguishable) but is rotated in space from the original orientation.
Even as the atoms are switching their positions, the entire molecular framework also rotates at a comparable rate. The problem is much like that of how a falling cat rights itself as it falls. How do those atomic permutations interact with that normal, “rigid-body” rotation of the molecule and how does that influence the spectrum? And how can chemists infer from the pattern of the spectrum how the atoms move when they exchange places? One natural path to the solution is in the mathematics of group theory.
The part of group theory most relevant here is the theory of representations of continuous groups (Lie groups). More particularly, subgroups of the rotation group in three dimensions provide a natural way to describe the motions of nonrigid molecules. My chemist colleagues and I set off in that direction and soon bumped into some mathematical difficulties—specifically, the need to handle infinite, discrete groups (discrete subgroups of Lie groups).
We needed help. So I, as the leader of the research group, a chemistry professor, approached a member of the resident math faculty and explained our difficulty. The next day, the mathematician came back to say that he had found the problem interesting. More than that, he said it had led him to a conjecture in number theory. He then described the problem to a local number theorist, who proved the conjecture the next evening and wrote out his proof. Copies were sent to the other mathematician and, kindly enough, to the group of chemists whose search for help sparked this bit of innovation in number theory. The sad and ironic ending—or at least pause—for the chemists in this story is that the problem in group representation theory whose solution would enable them to interpret spectra went unsolved and remains so.
This commingling of scientists and mathematicians started out with promise, but it withered. Even a cursory postmortem reveals some of the pitfalls and challenges that members of these two professions confront when they aim to collaborate. One key factor is probably that the mathematicians and the theoretical chemists never came back together after the number theorist found the solution to the conjecture. There was neither a sufficiently strong collaborative sensibility on either side nor enough motivation for the mathematicians to return to the original problem. Although the mathematicians found the problem in number theory to be

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interesting enough at first, they soon deemed it more or less a dead end. Had the problem seemed to the mathematicians to have better long-term prospects, they would probably have wanted to pursue the connection the theoretical chemists had brought up.
One obvious lesson for scientists looking to work with mathematicians is that their would-be collaborators are more likely to become committed stakeholders in a project if they have reason to believe they will have something to gain in the long term. That might have happened eventually in this case, but the interaction disintegrated too soon to find out. There's a lesson in that too. Unless someone works at nurturing the nascent linkage between scientists and mathematicians, it is likely to fall apart.
Finite-Time Thermodynamics
Now for some better news. An undergraduate who had received a bachelor's degree with a major in mathematics went to graduate school in a chemistry department, intending to study theoretical chemistry. He fortuitously joined a group just at a time when a new problem had arisen that was tailor-made for collaborative research with mathematicians. The problem was in the area of thermodynamics. No one in the group had worked in this area before, but they took on the problem anyway. The question they asked themselves was this: Could they extend methods of thermodynamics to systems constrained to operate in finite times or at nonzero rates? That it was a math-heavy challenge was clear to everyone in the group. Fortunately, the requisite mathematics, especially differential geometry, was familiar and friendly territory to the new grad student. With his background in the intellectual mix, the group worked productively to show how a sophisticated use of mathematics could lead to new insights in theoretical chemistry, and in nonequilibrium thermodynamics in particular. One thing the group was able to do was articulate and then prove an existence theorem (with necessary and sufficient conditions) for quantities analogous to traditional thermodynamic potentials but for systems and processes with time constraints. In doing this, they solved the first major problem they set out to solve. The group also developed a procedure for generating these potential-like quantities via Legendre-Cartan transformations, a modern extension of the Legendre transformation widely used in natural sciences. In addition, they applied more mathematics—in this case, optimal control theory—to find the pathways that yield or best approach the optimum performances of thermodynamic processes, the performances given by the changes of the limiting potentials. The research turned out to be remarkably successful. The scientific knowledge of the professor and postdoctoral associates combined well with the graduate student's ability to learn and master new mathematics quickly. What had been merely a problem in thermodynamics grew into an entire subfield. It has even led to some new mathematics, an example of which concerns the properties of surfaces of constant but nonzero curvature. The then-student has become one of those human bridges whose hybrid knowledge and interests spawn interdisciplinary research. He became a faculty member of a Department of Mathematics, where he currently studies problems on the border between mathematics and the natural sciences.
The collective moral of these two vignettes is simple. Mixing science and math can be a formula for discovery in both mathematics and science. Achieving a successful collaboration, however, requires not only a serendipitous coincidence of ideas, interests, and recognition of relevance but also plenty of perseverance.

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10 Martingale Theory
John Lehoczky
Department of Statistics
Carnegie Mellon University
The mathematical term “martingale” has been traced to the French village of Martigues, whose inhabitants were said to have a passion for gambling. The mathematician Jean Ville first used the term to refer to a particular type of stochastic process, which he applied initially to model a sequence of fair games. The martingale theory that developed out of this initial work did not remain exclusively within the realm of mathematics. It also would provide insights in many scientific areas. Some of those, in turn, would lead to new developments in mathematics.
In the language of mathematics, a martingale is a sequence of pairs , in which the random variable Xn represents the cumulative winnings of a gambler after n trials and (a sigma field) represents the history of the gambler's fortunes up through the first n trials. The defining martingale property is given by the following equation:
from which one can derive
for all m > n. In words, the equations say that the cumulative winnings, on average, will remain unchanged over the sequence of future gambling trials.
Ville defined the martingale concept to model gambling. It carried little importance beyond that until the 1940s, when J.L. Doob and Paul Levy separately investigated the structure of martingale processes. They proved a variety of mathematical results, including limit theorems governing the processes' asymptotic behavior. More importantly, they generalized the concept of martingales to submartingales and supermartingales, which they linked to an entirely different branch of mathematics—the subharmonic and superharmonic functions arising in potential theory.
Doob worked out much of the interrelationship between martingale theory and potential theory, and the role that Brownian motion plays in the solution of certain partial differential equations. He summarized this research and provided extensive historical commentary in his 800-page book Classical Potential Theory and Its Probabilistic Counterpart, published in 1983.
That accelerated the diffusion of martingale theory into a wider range of mathematical areas and from there into scientific and technical areas as well. Consider the following vignette.
During World War II, a group of statisticians, including Abraham Wald, were working at Columbia University on applications of the mathematical sciences to military problems. One of those problems concerned the development of efficient sampling plans for inspecting materiel. Already in place were well-defined military standards for sampling materiel before shipping it to the battlefield. These standards specified such details as the number of items to be inspected and the actions to be taken for different test outcomes. For certain types of materiel like ammunition,

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the testing procedure was destructive so there was plenty of incentive to minimize the sample size. This is where martingale theory enters the picture.
Wald and his collaborators approached this sampling size issue by considering sampling plans with a variable number of units being tested. The number of units would depend on the earlier tests. For example, a sufficient number of early successes could allow one to infer that the shipment was of sufficient quality to accept the entire lot without wasting any further parts to testing.
Wald developed a particular testing procedure, now known as the Wald Sequential Probability Ratio Test (SPRT). In determining the performance of his procedure, he used martingale methodology, and his methods, in turn, made further contributions to martingale theory. The formalization of that concept eventually would influence potential theory through the notion of Brownian motion hitting the boundary of analytic sets. More importantly, Wald and Wolfowitz were able to prove the ability of the SPRT procedure to minimize the expected sample size for destructive inspections. This research would become the foundation of a new field of statistics, now known as sequential analysis.
Sequential analysis, in turn, evolved extensively and has proven valuable in many scientific and technical domains, including clinical trials in medicine. These techniques also have become important to designers of industrial experiments. Determining settings of an industrial process's different control parameters to optimize the output or yield is one important example.
Interestingly, the study of the performance of the SPRT also stimulated the development of another branch of mathematics—dynamic programming—which now falls within the larger framework of operations research. A seminal paper by Arrow, Blackwell, and Girshick, which offered an alternative proof of the SPRT's optimal performance, often is credited with starting the field of dynamic programming. Martingale theory also plays an important role in stochastic control theory.
Martingales have continued to serve scientists in a variety of ways that have circled back into the mathematical sciences. One example of this emerged in the 1980s from the work of Harrison, Kreps, and Pliska, who put martingale theory to work in financial mathematics. They followed research by others (among them Black, Scholes, Merton, and Samuelson) on the pricing of contingent claims.
The newer work developed the theory of continuous trading and revealed its connection to the absence of arbitrage possibilities. And this implied that discounted asset prices must behave like martingales. Then, by applying martingale theory developed many years earlier, it became possible to determine the arbitrage-free price of a contingent claim and how to reconstruct a portfolio to hedge it. The celebrated Black-Scholes formula is a special case of this methodology. Indeed, contingent claims prices often can be expressed as an expected value with respect to a “martingale measure”—a probability measure under which the discounted asset price process forms a martingale.
This particular marriage of mathematics and the financial world has had an important role in the creation of financial markets with a notional value today of trillions of dollars. More importantly, the connections between martingale theory and contingent claim pricing have raised a whole new set of problems in martingale theory for mathematicians to ponder.
Martingale theory continues to make marks in both mathematical and scientific arenas. The theory has been applied to counting process theory. That, in turn, has found use in the part

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of operations research known as queuing theory, as well as in survival analysis in biostatistics. What's more, Radon-Nikodym derivatives and likelihood ratios now can be examined in terms of martingales, which means martingale theory has earned an important place in mathematical statistics.
The interaction between mathematical science and the other sciences catalyzed by martingale theory clearly has borne much fruit for all parties involved.

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