diffeomorphisms. Shape analysis also comes into play in virtual surgery, where surgical outcomes are simulated on the computer before being tried on a patient, and in remote surgery for the battlefield. Here one needs to combine mathematical modeling techniques based on the differential equations describing tissue mechanics with shape description and visualization methods.
As our society is learning somewhat painfully, data must be protected. The need for privacy and security has given rise to the areas of privacy-preserving data mining and encrypted computation, where one wishes to be able to analyze a data set without compromising privacy, and to be able to do computations on an encrypted data set while it remains encrypted.
The mathematical sciences have a long history of interaction with other fields of science and engineering. This interaction provides tools and insights to help those other fields advance; at the same time, the efforts of those fields to push research frontiers routinely raise new challenges for the mathematical sciences themselves. One way of evaluating the state of the mathematical sciences is to examine the richness of this interplay. Some of the interactions between mathematics and physics are described in Chapter 2, but the range extends well beyond physics. A compelling illustration of how much other fields rely on the mathematical sciences arises from examining those fields’ own assessments of promising directions and identifying the directions that are dependent on parallel progress in the mathematical sciences. A number of such illustrations have been collected in Appendix D.
The role of the mathematical sciences in industry has a long history, going back to the days when the Egyptians used the 3-4-5 right triangle to restore boundaries of farms after the annual flooding of the Nile. That said, the recent period is one of remarkable growth and diversification. Even in old-line industries, the role of the mathematical sciences has expanded. For example, whereas the aviation industry has long used mathematics in the design of airplane wings and statistics in ensuring quality control in production, now the mathematical sciences are also crucial to GPS and navigation systems, to simulating the structural soundness of a design, and to optimizing the flow of production. Instead of being used just to streamline cars and model traffic flows, the mathematical sciences are also involved in the latest developments, such as design of automated vehicle detection and