A Distinctively West Coast MII Workshop at ClaremontSeptember 23, 1999
Figure 1. The unemployment rate among recipients of U.S. doctorates in mathematics has dropped from its peak in the mid-1990s, while the fraction of employed new doctorates who are working in business, industry, or government has risen for most of the decade. (Data from reports of the Mathematical Sciences Data Committee, published in the Notices of the American Mathematical Society.)
The third SIAM Regional Workshop on Mathematics in Industry was held June 16-19 in Claremont, California, under the joint sponsorship of the Claremont Graduate University and Harvey Mudd College, with support from the National Science Foundation. Appropriately, the workshop coincided with the celebration of the 25th anniversary of the Claremont Colleges' Math Clinics, one of the earliest programs created to challenge teams of undergraduate and graduate mathematics students to solve industrial problems.
Although the regional workshop series was certainly conceived by SIAM as a traveling tent meeting to recruit new converts, the Claremont event was more a reaffirmation of faith among true believers in industrial mathematics. Fully half of the workshop speakers were from industry---entertainment, satellite, finance, and bio-engineering---and university speakers described ten distinct programs in industrial mathematics.
Marketing Mathematics in Hollywood
Industrial mathematics has become a marketable commodity on its own in the businesses represented at the Claremont meeting, a happy contrast with some of the smokestack industries, where mathematics often competes for shelf space with the traditional engineering disciplines. Like New Zealand kiwi fruit in the corner grocery store or New York bagels in Des Moines, mathematics is a sought-after item in such upstart markets as the digital visual effects (VFX) business in Hollywood.
Indeed, mathematics is more than a commodity to the whiz kids of VFX. Through lingo like height map, reinterpolatation, convolution, linear transformation, implicit function, Lindenmayer structure, Voronoi diagram, and anti-aliasing, it is deeply embedded in their speech.
Mathematics provides an intellectual framework as well as a vocabulary. VFX shots can be complex, either because of the volume of a long sequence of scenes or because of the intricacy of a single scene, such as the 275 layers of simulated images needed to show the Titanic leaving the pier on its maiden voyage. Such complexity requires a procedural or algorithmic approach, the stuff of mathematical thinking in general, if not always of Bourbaki in particular.
Wook, who is vice president of Intertainment, Inc., referred simultaneously to the role of mathematics in stimulating science fiction and in realizing such stories on the screen: "Mathematics and science create the intellectual fodder for story tellers and the production challenges for digital artists. Twenty-five years of SIGGRAPH papers are nothing but math. Computer graphics is a tool to exploit mathematics, and iterative application of symbolic processing is the foundation of animated visualization." (SIGGRAPH is the Association for Computing Machinery's Special Interest Group on Computer Graphics and Interactive Techniques. Its meetings are the primary marketplace for jobs in the digital effects industry.)
VFX places other unusual demands on mathematics. The digital simulation of water waves, for example, can call for some of the standard tools of computational fluid dynamics. As David Wasson of ARETE Entertainment warned, however, "You only need enough math to make it look right. You throw away as much of the science as possible while still getting a good picture." Even conservation of mass may go by the boards, at least until observers begin to notice that blobs of water are disappearing into thin air. "The beauty of computational graphics," Wasson added, "is that you don't have to validate." Artists are the ultimate authorities.
But the change in authority doesn't make the problems easier. Jerry Tessendorf of Cinesite Visual Effects, for example, described the technical difficulties that arise in simulating light striking individual water droplets. Digital simulation of a foggy day in London town is not yet feasible.
Mathematics is both the entertainment and the tool in the mass-market fractal visualization software produced by Ben Weiss of MetaCreations. His dynamic, easily manipulated visualizations of Mandelbrot and Julia sets put 32-bit mathematics, a far cry from the stereotyped, chalk-covered variety, into the hands of anyone with a little curiosity and a PC.
Satellites and Bio-engineering
The business of launching satellites is neither as new as digital special effects nor as old as rust belt manufacturing, and its relations to industrial mathematics are in a similarly intermediate state. At government-supported organizations like the Jet Propulsion Laboratory, aerospace researchers seem to be rediscovering mathematics as they strive to navigate the solar system and to manage and analyze the vast amounts of telemetered data collected on scientific satellite missions. At the same time, commercial concerns are using mathematics to deliver cheaper and more reliable communications satellites. Both public and private operators in outer space need mathematics to compress and filter signals and to control space craft. The latter can require such precision that a variation of less than 0.001g in a planet's gravitation is important.
Data, both their acquisition and their analysis, are fundamental to the exploding business of genetically engineered therapies. David Galas of the Keck Institute explained the key paradigm of this new kind of biology: "Information is not knowledge." Modeling and simulation are required for the construction of what he called "a knowledge tomography," to be used in extracting understanding in many dimensions from rapidly increasing masses of genetic data.
The design and testing of devices like heart valves create another market for mathematics. Some of the challenges, such as computing the three-dimensional, fluid-driven motion of an artificial heart-valve leaflet, look like familiar but tough problems from computational solid and fluid mechanics. Others, like estimating the working life of a heart valve when failures are (thankfully) few and far between, are more novel because the sparse failure data render reliable prediction difficult.
Bruce Stromberg of the Beckman Laser Institute argued "for the need to recruit mathematicians to produce functioning clinical devices." His new technology exploits intensity-modulated laser beams for noninvasive diagnosis and therapy. For each application of such new tools, modeling and simulation are essential to the design of the best combinations of forward measurement and inverse reconstruction.
The Healthy State of Industrial Mathematics
In a comment that clearly applied to more than just his employer, Gary Green of Aerospace Corporation restated the value of mathematics to industry: "The disciplined thinking which characterizes mathematics is valued by the engineers and physical scientists with whom we work." Indeed, an unspoken version of that assumption was the premise for many of the interactions at the workshop. In the same spirit, the breadth of industrial presentations at the workshop demonstrated conclusively that "industrial mathematics" is a wide umbrella that covers government and all sorts of businesses and industries, both novel and familiar.
Of the half of the workshop presenters who came from industry, most described the science and technology behind their problems and outlined their fundamental questions. A few posed mathematical problems directly in their presentations at the workshop. Likewise, the mathematicians in the audience listened with experienced interdisciplinary ears. Both sides knew how to conduct a fruitful conversation across disciplinary fences.
There was also ample evidence of fundamental intellectual contributions from mathematicians. A team at the University of Delaware Industrial Study Group, for example, had studied the extraction of water from paper pulp. The team made a substantial advance for the sponsoring paper company by introducing a moving boundary, thereby adding a second dimension in a key part of what had been just a one-dimensional model. In another success story, a student team from the previous academic year's Harvey Mudd Math Clinic had used simulation to significantly improve the reliability of the calculation of the optimum amount of propellant needed for the commercial telecommunication satellites launched by the Loral Corporation.
Reports of the short working life of PhD mathematicians in Hollywood's digital effects business elicited concern among workshop participants. Several speakers argued that these well-trained individuals often either burn out or move to management positions after just a few years. The VFX market, they suggested, may be better served by younger individuals with less advanced training.
In counterpoint, Jerry Purcell, president of Momentum Data Systems, a developer of digital signal processing components, suggested that executives need to monitor workloads so that their technical staff can remain productive over much longer periods. No one mentioned the possibility that talented managers are in short supply or that the shift from technical work to business now seen in Hollywood might be just an accelerated reflection of the same pattern in other technology-based industries or in the law.
The workshop participants saw that university programs in industrial mathematics come in many forms: clinics and projects, in which a student team solves an industrial sponsor's problem over an academic year; courses that import industrial problems for as little as a week or as long as a semester; industrial-academic workshops like the Claremont meeting itself; Oxford-style study groups, in which graduate students and experienced faculty gather for an intensive week of hammering big cracks into a handful of industrial problems; open-ended problem-solving courses; long-term, collaborative research involving faculty, graduate students, and industrial scientists; and NSF-sponsored Research Experience for Undergraduates programs, which let teams of undergraduates spend the summer solving genuine problems proposed by their industrial sponsors.
Undergraduates are beneficiaries of industrial clinics and projects at Worcester Polytechnic Institute and the University of Colorado at Denver, in addition to the clinics operated by the Claremont Colleges. Bachelor's-level work with industry is particularly appealing both because it reaches a large audience and because it provides a feasibility argument for industrial programs at the graduate level, where students have even more mathematical tools at their disposal.
Workshop participants learned of new degree programs at Utah State and Mexico's Autonomous Technical University (UTAM). They also heard of the continuing success of pioneers like North Carolina State's long-term collaborations with industry and the European Study Groups that were first sponsored by Oxford University and later by the European Consortium for Mathematics in Industry (ECMI). The University of Delaware and the University of Calgary each reported on their own study group experiences; both follow the Oxford/ECMI model.
These and other initiatives may be having some effect. The employment data shown in Figure 1 suggest that attitudes about working in industry may be changing among new mathematics doctorates. Even though overall unemployment rates have dropped in the last two years from the devastating double-digit rates experienced earlier in the decade, the rate of nonacademic employment continues to grow, exceeding 35% in the last two years.
A range of educational issues remain. Some, like the proper balance between teaching fundamental mathematics and ancillary interpersonal and interdisciplinary skills, can be settled only on a campus-by-campus basis.
Others are more profound. To cite one troubling example, the Claremont workshop included descriptions of ten different academic programs in industrial mathematics. The institutions ranged from relatively small schools like Claremont, Harvey Mudd, and WPI through larger state universities like Delaware, Utah State, and North Carolina State. But many of the western state colleges and universities, institutions that confer relatively large numbers of BS and MS degrees in mathematics, were inexplicably absent from this regional workshop.
Ultimately, such questions remained unanswered, but the workshop participants went home worn out, wiser, and with even more certainty of the value of their work in industrial mathematics, thanks to the organizing efforts of Ellis Cumberbatch of the Claremont Graduate University and Bob Borrelli of Harvey Mudd College.
Paul Davis is a professor of mathematical sciences at Worcester Polytechnic Institute.