winds and from flooding due to high tides. Forecasts of temperature and humidity are used to minimize the economic cost and maximize the benefits of complying with environmental air quality regulations. From this example of weather forecasts, it becomes evident that there is not a sharp distinction between the technology base and direct applications.

The following discussion of the technology base does not focus narrowly on modeling and simulation. There is a fundamental significance to the mathematical way of thinking. Briefly, mathematics provides methods for organizing and structuring knowledge so that, when applied to technology, it allows scientists and engineers to produce systematic, reproducible, and transmittable knowledge. Analysis, design, modeling, simulation, and implementation then become possible as efficient, well-structured activities.

Similarly, the distinction between short- and long-term applications cannot be drawn unambiguously. In a series of cases, one finds the same subjects moving back and forth between theory and applications, and becoming richer with each transition, as the following examples illustrate:

  • Nearly integrable systems of differential equations and fiber optics;

  • Geometrical optics and the asymptotic solutions of differential equations;

  • Fourier analysis, group symmetries, and special functions;

  • Covariant differential geometry and elastic deformations; and

  • Fourier analysis on finite groups and the fast-Fourier-transform (FFT) algorithm.

The ability of the mathematical sciences to deliver short-term results with any degree of consistency depends crucially on support for their long-term development. Conversely, the more fundamental areas of the mathematical sciences are continually invigorated by interaction with applications. It is an almost universal experience that once an application succeeds, further progress depends on the development of new, and often fundamental, theories. This report endorses the principle that short-term applications and fundamental theories are virtually

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