By Stephen H. Wildstrom At first glance, the notion of a conference on "The Mathematical Sciences' Role in Homeland Security" looks like a shameless attempt to channel some of the mighty stream of security funding that has been rolling out of Washington since September. 11. "When I got the e-mail invitation," says Howard Schmidt, vice-chair of the President's Critical Infrastructure Protection Board, "I thought at first it was a joke."
No joke. In fact, the Apr. 26-27 workshop sponsored by the National Research Council's Board on Mathematical Sciences & Their Applications was deadly serious. It turns out that many of the diffuse and complex problems of homeland security are deeply mathematical in nature, and even some of the science's most abstruse branches, such as topology and high-dimension geometry, can be brought to bear on security problems.
The most obvious application is the encryption techniques used to protect data from prying eyes. But encryption issues are so well known that they went largely unmentioned. Even crypto specialist David Wagner of the University of California at Berkeley devoted much of his presentation to other topics, from the mathematics of power-grid reliability to the design of "inherently self-stable systems."
DATA EXTREMES. Not surprisingly, much of the discussion focused on the use of advanced statistical techniques to deal with two almost opposite problems. First, the growing use of cameras and other surveillance techniques is overwhelming analysts with more data than they can hope to make use of. Mathematicians can help by developing data-mining techniques that help spot patterns in an ocean of seemingly random information.
At the other extreme, epidemiologists chasing, say, an outbreak of anthrax must figure out whether they're dealing with a terrorist attack or a random, natural event based on extremely scanty evidence. A situation that presents a large number of variables and a small number of data points is very poorly handled by traditional statistical analysis.
Michael H. Freedman of Microsoft Research pointed out that techniques developed in the field of high-dimensional geometry are relevant here. Geometers have found ways that a space with a large number of dimensions can be approximated using a much smaller dimensional field of numbers that are far easier to work with. In statistics, this is generally analogous to reducing the number of variables.
WORKING BACKWARDS. Freedman, a winner of the most prestigious prize in mathematics, the Fields Medal, added a touch of levity to an otherwise serious session by describing "the general world view of mathematicians." On his way from Seattle to Washington, he was selected for a secondary search at SeaTac airport. He pulled out his itinerary and said he was on his way to a conference on mathematics and national security. "The guard was very skeptical," Freedman said. "She asked, 'Are you a mathematician?' I said 'yes." She replied, 'Then God help us.'"
Alexander H. Levis, chief scientist for the Air Force, suggested that mathematicians might find ways to apply to domestic security the statistical techniques that the military has developed for analyzing threats. One, using an approach called a Bayesian inference network, works backwards from a set of possible events to assign probabilities to the potential actions that opponents might take.
In the end, mathematicians don't suffer illusions that math alone is going to make the nation significantly more secure. Workshop organizer Jennifer Chayes, director of Microsoft Research's Theory Group, says she chose the topic for the annual workshop simply because it seemed natural and relevant. For example, Freedman's research specialty, the mathematics of quantum computing, could one day enable the solution of problems that today remain dauntingly complex.
Practical applications remain, at best, years away. "I don't think," Freedman says, "that we can stop terrorism in time by building quantum computers." All in all, however, the conference put the relationship between math and national security front and center. Wildstrom is Technology & You columnist for BusinessWeek