avoidance systems that may one day lead to automated driving. Whereas statistics has long been a key element of medical trials, now the mathematical sciences are involved in drug design and in modeling new ways for drugs to be delivered to tumors, and they will be essential in making inferences in circumstances that do not allow double-blind, randomized clinical trials. The financial sector, which once relied on statistics to design portfolios that minimized risk for a given level of return, now makes use of statistics, machine learning, stochastic modeling, optimization, and the new science of networks in pricing and designing securities and in assessing risk.
What is most striking, however, is the number of new industries that the mathematical sciences are a part of, often as a key enabler. The encryption industry makes use of number theory to make Internet commerce possible. The “search” industry relies on ideas from the mathematical sciences to make the Internet’s vast resources of information searchable. The social networking industry makes use of graph theory and machine learning. The animation and computer game industry makes use of techniques as diverse as differential geometry and partial differential equations. The biotech industry heavily uses the mathematical sciences in modeling the action of drugs, searching genomes for genes relevant to human disease or relevant to bioengineered organisms, and discovering new drugs and understanding how they might act. The imaging industry uses ideas from differential geometry and signal processing to procure minimally invasive medical and industrial images and, within medicine, adds methods from inverse problems to design targeted radiation therapies and is moving to incorporate the new field of computational anatomy to enable remote surgery. The online advertising industry uses ideas from game theory and discrete mathematics to price and bid on online ads and methods from statistics and machine learning to decide how to target those ads. The marketing industry now employs sophisticated statistical and machine learning techniques to target customers and to choose locations for new stores. The credit card industry uses a variety of methods to detect fraud and denial-of-service attacks. Political campaigns now make use of complex models of the electorate, and election-night predictions rely on integrating these models with exit polls. The semiconductor industry uses optimization to design computer chips and in simulating the manufacture and behavior of designer materials. The mathematical sciences are now present in almost every industry, and the range of mathematical sciences being used would have been unimaginable a generation ago.
This point is driven home by the following list of case studies assembled for the SIAM report Mathematics in Industry.13 This list is just illustrative, but its breadth is striking:
13 Society for Industrial and Applied Mathematics, 2012, Mathematics in Industry. SIAM, Philadelphia, Pa.