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. Fueling Innovation and Discovery describes recent advances in the mathematical sciences and advances enabled by mathematical sciences research. It is geared toward general readers who would like to know more about ongoing advances in the mathematical sciences and how these advances are changing our understanding of the world, creating new technologies, and transforming industries. Although the mathematical sciences are pervasive, they are often invoked without an explicit awareness of their presence. Prepared as part of the study on the Mathematical Sciences in 2025, a broad assessment of the current state of the mathematical sciences in the United States, Fueling Innovation and Discovery presents mathematical sciences advances in an engaging way. The report describes the contributions that mathematical sciences research has made to advance our understanding of the universe and the human genome. It also explores how the mathematical sciences are contributing to healthcare and national security, and the importance of mathematical knowledge and training to a range of industries, such as information technology and entertainment. Fueling Innovation and Discovery will be of use to policy makers, researchers, business leaders, students, and others interested in learning more about the deep connections between the mathematical sciences and every other aspect of the modern world. To function well in a technologically advanced society, every educated person should be familiar with multiple aspects of the mathematical sciences. Table of ContentsFront MatterIntroductionCompressed Sensing / Through the KaleidoscopeEigenvectors / From the Mathematical Sciences to . . . an IPOMathematical Simulations / When the Lab Isn't Big EnoughBayesian Inference / Not an Enigma AnymoreDiffusion Tensor Imaging / A New View of the BrainFast Multipole Method / A Long-Term PayoffCellular Automata / Sublimely ComplexGraph Spectra / Sparsest Cuts in Minimum TimeBioinformatics / Interpreting the Human GenomeGeometry and Physics / Endlessly IntertwinedProbability and Statistical Physics / Connecting Microscopic andMacroscopic