fortunate consequence is that the average person may not appreciate the richness of the mathematical sciences.

The mathematical sciences include far more than numbers—they deal with geometrical figures, logical patterns, networks, randomness, and predictions from incomplete data, to name only a few topics. And the mathematical sciences are part of almost every aspect of everyday life.

Consider a typical man (Bob) and a typical woman (Alice) in a developed society such as the United States. Whether they know it or not, their lives depend intimately and deeply on the mathematical sciences; they are wrapped in an intricate and elegant net woven with strands from the mathematical sciences. Here are some examples. A remarkable fact is that these extremely varied applications depend crucially on the body of mathematical theory that has been developed over hundreds of years—on ingenious new uses of theoretical developments from long ago, but also on some very recent breakthroughs. Some of the pioneers of this body of theory were motivated by these applications; some by other applications that would seem completely unconnected with these; and in many cases by the pure desire to explore the fundamental structures of science and thought.

•   Bob is awakened by a radio clock and usually listens to the news. But he is unlikely to think twice, if at all, about how the radio can receive signals, remove noises, and produce pleasant sound, yet all of these tasks involve the mathematical and statistical methods of signal processing.

•   Alice may begin her day by watching the news in a recently purchased high-definition LCD television. To achieve the high-quality image that Alice takes for granted, many sophisticated steps are required that depend on the mathematical sciences: compression of digital signals, conversion from digital to analog and analog to digital, image analysis and enhancement, and LCD performance optimization.

•   Bob and Alice love movies like Toy Story, Avatar, and Terminator 3. A growing number of films feature characters and action scenes that are the result of calculations performed by computers on mathematical models of movements, expressions, and actions based on mathematical models. Obtaining a realistic impression of, say, the collapse of downtown Los Angeles, requires intricate mathematical characterizations of explosions and their aftermath, displayed through the application of high-end computational power to sophisticated mathematical insights about the fundamental equations governing fluids, solids, and heat.

•   If Bob’s plans for his day (or the next few days) take into account weather predictions, he is relying on the numerical solution of



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