Report Documents Successful Interactions Between Science and Mathematics

March 15, 2000

The good news from Washington, SIAM president Gilbert Strang reports in this issue's column, includes "exceptional" support for the National Science Foundation's Division of Mathematical Sciences, in the form of a 22.5% increase in the 2001 budget. The increase is the largest requested for any division in the Mathematical and Physical Sciences Directorate. Along with the impressive show of support, Strang points out, comes a responsibility: NSF director Rita Colwell, MPS head Robert Eisenstein, and DMS director Philippe Tondeur, as they work to set priorities for the years ahead, need to hear from the mathematics community about outstanding successes and opportunities in research.

Describing the impact of mathematical research to an audience composed in part of non-mathematicians is a critically important, but admittedly difficult task. Fortunately, those willing to take it on can benefit from a model document, written by Alexandre Chorin of the University of California, Berkeley, and Margaret Wright of Bell Laboratories, Lucent Technologies. Chorin and Wright's report has been described by Tondeur and others as just the sort of information needed by decision makers at NSF; Donald J. Lewis, Tondeur's predecessor at NSF and the person who initiated the report, describes it as "exciting reading."

The report, titled "Mathematics and Science," can be accessed at Chorin and Wright point out that they did not set out to write an exhaustive survey of the interactions between mathematics and science. Rather, they were seeking "to present examples of scientific advances made possible by a close interaction between science and mathematics"; the validity of the conclusions drawn, they write, "should transcend the examples." Their examples, chosen (and labeled) in a way that "emphasizes the ubiquity and centrality of mathematics from the point of view of science," are:

Combustion; Cosmology; Finance; Functional Magnetic Resonance Imaging; Hybrid System Theory and Air Traffic Management; Internet Analysis, Reliability, and Security; Materials Science; Mixing in the Oceans and Atmosphere; Physiology; Diagnosis Using Variational Probabilistic Inference; Iterative Control of Nuclear Spins; and Moving Boundaries and Interfaces.

Cutting across the examples (which are presented with the help of experts working in each area) are several themes that will be familiar to readers of SIAM News: modeling, complexity and size, uncertainty, multiple scales, computation, and large data sets.

Chorin and Wright are especially effective in establishing motivation for each of their examples. In the "Hybrid System Theory and Air Traffic Management" section, for example, they point out that aging air traffic control equipment is already contributing to "ground holds and airborne delays in flights due to congestion." The displays used by controllers to look at radar tracks and flight information "are the very same ones that were installed in the early 1970s, and they fail regularly. The computer systems that calculate radar tracks and store flight plans were designed in the 1980s, using software written in 1972." Even with new equipment, "the Federal Aviation Administration admits that any significant improvement will require that many of the basic practices of air traffic control be automated."

Key to the success of the Chorin/Wright report is its accessibility to a non-mathematical audience. "After all," they write in introducing the "Mixing in the Oceans and Atmosphere" example, "children who make chocolate milk from a powder quickly learn that the longer and more energetically they stir, the more evenly the chocolate powder is spread and dissolved in the milk. While that common-sense lesson is valid, the oceans and atmosphere are, in some sense, less vigorously stirred, so that the mixing is incomplete."

Many readers will find the report inspiring for the abundance of mathematical challenges it presents. From the "Functional Magnetic Resonance Imaging" section: Two approaches have been taken to the problem of noisy data. Mathematically, "the goal is to develop a mathematical model that accurately relates the data to parameters of interest, but this remains a daunting task. Substantial progress has been made by successively estimating and correcting for each effect known to cause noise" (e.g., gradient mis-timings, receiver drift, subject head motion). "After these corrections, the images are reconstructed by a fast Fourier transform and then the (still unexplained) voxel-wise trend over time is removed. Finally, statistical methods such as t-tests are used to assess the effect of the experimental paradigm."

These are arbitrarily chosen quotes from a detailed, thoughtful presentation. Readers are encouraged to read the report, think about examples from research familiar to them, and write them up and offer them for consideration. Ultimately, what the NSF officials need is a convincing, balanced view of successes and opportunities in the mathematical sciences. In his column, the SIAM president volunteers to be the initial recipient of such messages. SIAM hopes that readers will take him up on the offer!

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