research in data collection, storage, management, and automated large-scale analysis based on modeling and machine learning . . . [and] continued investment in important core areas such as high performance computing, scalable systems and networking, software creation and evolution, and algorithms.11

The report goes on to point out that five broad themes cut across its recommendations, including the need for capabilities to exploit ever-increasing amounts of data, to improve cybersecurity, and to better ensure privacy. As noted above, progress on these fronts will require strong mathematical sciences research.

More generally, the mathematical sciences play a critical role in any science or engineering field that involves network structures. In the early years of U.S. telecommunications, very few networks existed, and each was under the control of a single entity (such as AT&T). For more than 50 years in this environment, relatively simple mathematical models of call traffic were extremely useful for network management and planning. The world is completely different today: It is full of diverse and overlapping networks, including the Internet, the World Wide Web, wireless networks operated by multiple service providers, and social networks, as well as networks arising in scientific and engineering applications, such as networks that describe intra- and intercellular processes in biology. Modern technologies built around the Internet are consumers of mathematics: for example, many innovations at Google, such as search, learning, and trend discovery, are based on the mathematical sciences.

In settings ranging from the Internet to transportation networks to global financial markets, interactions happen in the context of a complex network.12 The most striking feature of these networks is their size and global reach: They are built and operated by people and agents of diverse goals and interests. Much of today’s technology depends on our ability to successfully build and maintain systems used by such diverse set of users, ensuring that participants cooperate despite their diverse goals and interests. Such large and decentralized networks provide amazing new opportunities for cooperation, but they also present large challenges. Despite the enormous amount of data collected by and about today’s networks, fundamental mathematical science questions about the nature, structure, evolution, and security of networks remain that are of great interest for the government, for innovation-driven businesses, and for the general public.

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11 Ibid., p. xiii.

12 The remainder of this section draws from pages 5, 6, and 10 of Institute for Pure and Applied Mathematics, 2012, Report from the Workshop on Future Directions in Mathematics. IPAM, Los Angeles.



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