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.