Introduction

The mathematical sciences are part of everyday life. Modern communication, transportation, science, engineering, technology, medicine, manufacturing, security, and finance all depend on the mathematical sciences, which consist of mathematics, statistics, operations research, and theoretical computer science. In addition, there are very mathematical people working in theoretical areas of most fields of science and engineering who also contribute to the mathematical sciences. There is a healthy continuum between research in the mathematical sciences, which may or may not be pursued with an application in mind, and the range of applications to which mathematical science advances contribute. To function well in a technologically advanced society, every educated person should be familiar with multiple aspects of the mathematical sciences.

Although the mathematical sciences are pervasive, they are often invoked without an explicit awareness of their presence. For example, in the everyday operation of making a cell phone call, the mathematical sciences are essential in every step: We enter numbers in the decimal system, which are converted into sequences of bits (zeros and ones); next comes conversion to an electromagnetic signal; after an available receiver is located, the signal is transmitted and (finally) converted into the sound of our voice. Wireless technology uses techniques called “error correcting codes,” “linear



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Introduction T he mathematical sciences are part of everyday life. Modern communication, transportation, science, engineering, technol- ogy, medicine, manufacturing, security, and finance all depend on the mathematical sciences, which consist of mathematics, statistics, operations research, and theoretical computer science. In addition, there are very mathematical people working in theoretical areas of most fields of science and engineering who also contribute to the mathematical sciences. There is a healthy continuum between research in the mathematical sciences, which may or may not be pursued with an application in mind, and the range of applications to which math- ematical science advances contribute. To function well in a technologi- cally advanced society, every educated person should be familiar with multiple aspects of the mathematical sciences. Although the mathematical sciences are pervasive, they are often invoked without an explicit awareness of their presence. For example, in the everyday operation of making a cell phone call, the mathematical sciences are essential in every step: We enter numbers in the decimal system, which are converted into sequences of bits (zeros and ones); next comes conversion to an electromagnetic signal; after an available receiver is located, the signal is transmitted and (finally) converted into the sound of our voice. Wireless technology uses techniques called “error correcting codes,” “linear in the 21st Century The Mathematical Sciences 1

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and nonlinear filtering,” “hypothesis testing,” “spatial multiplexing,” “statistical waveform or parameter estimation,” and these are built on tools of the mathematical sciences, such as matrix analysis, linear algebra, algebra, random matrices, graphical models, and so on. More generally, the mathematical sciences contribute to modern life whenever data must be analyzed or when computational modeling and simulation is used to enable design and analysis of systems or exploration of “what-if” scenarios. The emergence of truly massive data sets across most fields of science and engineering, and in business, government, and national security, increases the need for new tools from the mathematical sciences. Because the mathematical sciences are independent of a particular scientific context, they can facilitate the translation of advances from one discipline to another. The mathematical sciences provide a language—numbers, symbols, graphs, and diagrams—for expressing ideas in everyday life as well as in science, engineering, medicine, business, and the arts. Mathematical symbols, which are more universal than Chinese, English, or Arabic, allow communication across communities with completely dissimilar spoken and written languages. The stories told here describe a number of recent advances made possible by research in the mathematical sciences. FUELING innovation and discovery 2