highly nonlinear, high-dimensional (meaning tens of millions of unknowns) equations and on statistical analysis of past observations integrated with freshly collected information about atmosphere and ocean conditions.

•   To surf the Internet, Bob turns to a search engine, which performs rapid searches using a sophisticated mathematical algorithm. The earliest Web search techniques treated the interconnections of the Web as a matrix (a two-dimensional data array), but modern search methods have become much more sophisticated, incorporating protection from hacking and manipulation by outsiders. Effective Web search relies more than ever on sophisticated strategies derived from the mathematical sciences.

•   Alice is plagued by unwanted e-mails from irrelevant people who want to cheat her or sell her items she does not want. A common solution to this problem is a spam filter, which tries to detect unwanted or fraudulent e-mail using information and probability theory. A major underlying tool is machine learning, in which features of “legitimate” e-mails (as assessed by humans) are used to train an algorithm that classifies incoming e-mails as legitimate or as spam.

•   When Alice needs to attend a meeting next month in Shanghai, China, both the schedule of available flights and the price she will pay for her ticket are almost certain to be determined using optimization (by the airlines).

•   Bob uses his cell phone almost constantly—a feature of modern life enabled, for better or worse, by new developments in mathematical and statistical information theory that involve wireless signal encoding, transmission, and processing, and by some highly ingenious algorithms that route the calls.

•   Alice’s office building consumes energy to run electric lights, telephone landlines, a local computer network, running water, heating, and cooling. Mathematical optimization and statistical techniques are used to plan for efficient energy delivery based on information about expected energy consumption and estimated safety factors to protect against unusual events such as power outages. Because of concerns about excessive energy usage, Alice’s utility companies are investing in new mathematical and statistical techniques for planning, monitoring, and controlling future energy systems.

•   When Bob and Alice need medical or dental attention, they explicitly benefit from sophisticated applications of the mathematical sciences. Everyone has heard of X-rays, CT scans, and MRIs, but few people realize that modern medical and dental image analysis and interpretation depend on complicated mathematical concepts, such

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