of expertise of a modest-sized committee. What appears below is just a sampling of advances, to give a flavor of what is going on, and it is not meant to be either comprehensive or proportionally representative. This chapter is aimed primarily at the mathematical sciences community, and so the examples here presume a base of mathematical or statistical knowledge. The topics covered range from the solution of a century-old problem by using techniques from one field of mathematics to solve a major problem in another, to the creation of what are essentially entirely new fields of study. The topics are, in order:

•   The Topology of Three-Dimensional Spaces

•   Uncertainty Quantification

•   The Mathematical Sciences and Social Networks

•   The Protein-Folding Problem and Computational Biology

•   The Fundamental Lemma

•   Primes in Arithmetic Progression

•   Hierarchical Modeling

•   Algorithms and Complexity

•   Inverse Problems: Visibility and Invisibility

•   The Interplay of Geometry and Theoretical Physics

•   New Frontiers in Statistical Inference

•   Economics and Business: Mechanism Design

•   Mathematical Sciences and Medicine

•   Compressed Sensing

THE TOPOLOGY OF THREE-DIMENSIONAL SPACES

The modest title of this section hides a tremendous accomplishment. The notion of space is central to the mathematical sciences, to the physical sciences, and to engineering. There are entire branches of theoretical mathematics devoted to studying spaces, with different branches focusing on different aspects of spaces or on spaces endowed with different characteristics or structures.1 For example, in topology one studies spaces without assuming any structure beyond the notion of coherence or continuity. By contrast, in geometry one studies spaces in which, first of all, one can differentiate, leading to notions such as tangent vectors, and, second, for which the notion of lengths and angles of tangent vectors are defined. These concepts were first introduced by Riemann in the 1860s in his thesis “The hypotheses that underlie geometry,” and the resulting structure is called a Riemannian metric. Intuitively, one can imagine that to a topologist spaces are made out of rubber or a substance like taffy, while

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1 Box 2-1 discusses the concept of mathematical structures.



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