This type of unconventional clock is commonly used to illustrate problems in modular arith- metic. This particular clock has 7 “hours”; each “hour” position holds a set of numbers that are congruent modulo 7, or separated by multiples of 7. For one particularly elusive set of equations, earlier mathematicians located an ingenious but labor-intensive solution method in modular arithmetic. Richard Taylor’s work finds a shortcut in geometry.
the Times reported that another mathe- equation required counting one’s matician had found a mistake in the new way around the “clock” (see figure proof. In the spotlight and under pres- at right). In proving the conjecture, sure, Wiles called on Taylor, his former Taylor essentially mapped the direc- doctoral student, for help. Together, they tions for a massive shortcut. But the published the corrected proof in 1995, and proof was, as Taylor puts it, “a nice Wiles won the Fermat Prize for mathe- by-product” of his main work on the matics that year. Taylor won the same connection between arithmetic and prize in 2001, and last year shared the geometry. prestigious, million-dollar Shaw Prize for Taylor’s Shaw Prize co-recipient, mathematician Robert Langlands, had “It just astounded me been the first to realize, back in the 1970s, that instead of counting the number of that there should be a solutions for these equations, it would be from undergraduate calculus to advanced possible to get the same answer using graduate courses, his primary research is connection between geometry. Specifically, the shortcut lay in work on a kind of foreign-language dic- the symmetry of hyperbolic space, one of tionary that translates between these two these two things.” the mathematical concepts illustrated by mathematical domains—between the the artist M.C. Escher. In this weird, language of clocks and that of saddles. proving the 1963 Sato-Tate Conjecture, warped world, pairs of parallel lines Taylor describes the dictionary this way: which said that one could draw a curve break all the laws of Euclidean geometry “It says, ‘Here’s an equation. I’m going to that would predict the number of solu- and bend away from one another, and give you a problem about symmetry that tions for any equation describing an ellip- lines that, to the human eye, look wildly has the same behavior.’ ” He and other tic curve. di≠erent are actually the same length. number theorists are busily filling in more The conjecture itself was a brilliant in- The analogy commonly made is to a entries to illustrate patterns and perhaps sight, but applying it is labor-inten- horse’s saddle: viewed from above, a sad- solve still-unidentified mathematical sive—it relies on modular arithmetic, a dle appears as an oval, but its circumfer- mysteries. “We’ve done A and B so far,” he domain of mathematics that studies the ence is unexpectedly long because it says. “All the way up to Z is left.” relationships between numbers, often il- curves downward along the sides of the elizabeth gudrais lustrated using clock-like figures with horse’s body and upward toward the unusual numbers of hours (i.e., not 12). horse’s head and tail. richard taylor e-mail address: Counting the number of solutions for an Although Taylor teaches everything [email protected]
ANNALS OF DE-MINING Man, Mongoose, and Machine
tanding outside a Sri Lankan machines are too expensive. Dogs weigh harness to keep the mongoose under con- army base in the spring of 2007, enough to trigger mines, injuring or kill- trol and a video camera to record its find- SThrishantha Nanayakkara mapped ing themselves and their handlers. ings. Although the mongoose walks a few an entire minefield without once Nanayakkara, a visiting scholar at the feet ahead, the robot with its eight metal setting foot in it. Nanayakkara held a re- School of Engineering and Applied Sci- legs sets the pace. During the test run, the mote control and periodically made a note ences and a 2008-09 Radcli≠e Institute pair went back and forth across a 10-by- on his computer. A mongoose hitched to a fellow, picked an indigenous mongoose 10-meter plot, stopping whenever the robot did most of the work. for its temperament, size (roughly 2.5 mongoose detected a mine, which it indi- This unorthodox de-mining team kilograms, light enough to step on a mine cated by sitting up (as it was trained to avoids most of the traditional pitfalls. without detonating it), and sense of smell do). In a morning’s work, the mongoose Metal detectors give too many false (able to detect explosives three meters found every mine. alarms, fooled by bullets or other debris. away). He equipped his robot (roughly a The land mines in Sri Lanka—and Hand-held ground-penetrating radar meter long and half a meter wide) with a other war-torn nations—are both physi-
Harvard Magazine 11 Above left: A trained mongoose sniffs the “Let’s say there are five land mines in engineers, he is perfecting at Harvard ground in search of mines. Behind the mon- Boston, and we don’t know where they what he once demonstrated in practice. goose is an impediment sensor, which tells the robots (right) if it has bumped into are. Immediately people restrict their Developed with a team of students at the something too solid to push out of the way. movements to the most essential errands. University of Moratuwa, the robot’s char- Above right: Two linked robots directly This worsens if we get the news that acteristics (such as size, speed, and sens- behind the sensor allow a human operator to direct the mongoose over the minefield with- somebody has been caught in a mine ing ability) have now been programmed out risk of personal injury. The robots move blast. The remaining four land mines can into a minefield simulation that includes by flexing their jointed bodies back and forth. freeze the economic activity of Boston.” obstacles like trees or rough terrain. By
In Sri Lanka, the land mines laid by both digitally tweaking the robot’s design, Na- COURTESY OF THRISHANTHA NANAYAKKARA cal obstacles and obstacles to economic the government and Tamil Tiger rebels nayakkara can determine how various me- growth. “To understand how de-mining number between 1.5 million and 3 million. chanical changes improve its e∞ciency. expands economic activity, it is good to Although Nanayakkara no longer Perfecting the machine’s locomotion, imagine the reverse,” Nanayakkara says. works directly with Sri Lankan military he believes, is the first step. He is working
One day, his son would speak to the world.
He wanted them to understand.
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12 September - October 2008 RIGHT NOW with students and professors at Harvard DEADLINE DOWNSIDE and MIT to develop a spring-loaded leg to help the robot move more easily over the soft dirt of the jungle floor. Prescription for Error? He trained his three mongooses mostly through trial and error: rewarding them for reacting to explosives held on a stick, n recent years, safety recalls of regulator. Rather, they are rooted in then covering the cage so that they would widely prescribed drugs like the pain- specific legislation intended to balance learn to identify their target by scent, Ikiller Vioxx have sent an unsettling the cost of ensuring public safety with rather than sight. At Harvard, he is taking message to consumers: Today’s super the need for prompt approval of benefi- a more systematic approach by collabo- cure may be tomorrow’s health hazard. cial medicines. In 1992, Congress passed rating with assistant professor of molecu- Many drug-industry critics believe the the Prescription Drug User Fee Act lar and cellular biology Naoshige Uchida, expanding financial influence of “big (PDUFA), a payment program under who is studying how rats learn to pick pharma” has compromised federal over- which pharmaceutical producers pay a out a single smell in a pungent environ- sight of new medicines, allowing unsafe “user fee” to the FDA to help cover the ment. Nanayakkara hopes that determin- drugs to reach the market and remain cost of reviewing new drugs, and the ing how another rodent’s brain sorts there for months or even years. At the FDA, in exchange, rules on applications smells will lead to improved training same time, patient groups and pharma- within a set period of time—12 months techniques that in turn will make his ceutical investors continue to blame the for “standard” reviews and six months for country a safer place to live. Food and Drug Administration (FDA) for “priority” reviews. Congress revised the “Land mines write o≠ fertile farmlands slowing innovation through bureaucratic bill in 1997, shortening the “standard re- and destroy social structures,” he says. ine∞ciency and over-regulation. view” time to 10 months, while maintain- “E∞cient removal of mines helps the According to Freed professor of gov- ing a 6-month deadline for “priority” ap- a≠ected communities to return to a nor- ernment Daniel Carpenter, an expert on plications. When the law passed, its mal, productive life.” paul gleason the history of the FDA, the flaws in the detractors charged that it created a dan- U.S.’s drug-review system do not stem gerous conflict of interest within the thrishantha nanayakkara e-mail address: from a single culprit—whether profit- drug-review process by making the FDA [email protected] driven manufacturer or inept government overly dependent on pharmaceutical BETTERBETTER RETURNSRETURNS Savings come back to you faster when you open a Term Share Certifi cate from HUECU, which features our highest savings rates, and a variety of term options from 6 months to 5 years. And when you open a Sixty+ Account, you’ll receive an extra .50% on all Term Share Certifi cates. Sign up today.
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Harvard Magazine 13