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Spheres in Disguise Solid proof offered for famous conjecture

A Russian mathematician may have finallycracked one of the most famous problemsin mathematics: the Poincaré conjecture, aquestion about the shapes of three-dimen-sional spaces. If his work is correct, it willmake him eligible for a $1 million prizefrom the Clay Mathematics Institute inCambridge, Mass., which has declared theconjecture one of the seven most impor-tant mathematical problems of the new mil-lennium.

More than 100 mathematicians packed alecture hall at the State University of NewYork at Stony Brook this week to hear Grig-ori Perelman of the Steklov MathematicalInstitute in St. Petersburg, Russia, describehis work. Last week, Perelman told an equallyattentive audience at the MassachusettsInstitute of Technology (MIT) that he hasproven the conjecture together with abroader problem called the Thurstongeometrization conjecture. This second prob-lem proposes that any three-dimensionalspace can be chopped in a standard way intopieces, each of which has a simple geomet-ric structure.

Perelman has posted two papers abouthis research on the Internet (http://xxx.lanl.gov/abs/math.DG/0303109 andhttp://xxx.lanl.gov/abs/math.DG/0211159).Mathematicians are now scrutinizing everyline of the work to verify its correctness.

“We’re all waiting with bated breath,” saysYair Minsky of Stony Brook, who attendedthe lecture there.

The Poincaré conjecture belongs to thefield of topology, which studies propertiesthat are preserved when a shape is stretchedor twisted without tearing. Topologicallyspeaking, the surfaces of a doughnut and ofa coffee cup are the same, but they’re dif-ferent from the surface of a ball.

Mathematicians have established crite-ria for distinguishing among types of sur-faces. For example, consider a loop of string

lying on a closed surface. More than a cen-tury ago, mathematicians proved that ifevery such loop can be shrunk to a singlepoint without leaving the surface, the objectis a sphere. On a doughnut, by contrast, aloop that encircles the hole can’t be shrunkto a point.

The traditional, or two-dimensional,sphere is the set of all points in three-dimensional space that are a given distancefrom a fixed center. Mathematicians alsostudy what they call the three-dimensionalsphere—the set of all points a given dis-tance from a center in four-dimensionalspace. French mathematician Henri Poin-caré conjectured 99 years ago that, just asin the case of surfaces, any closed three-dimensional space in which loops can betightened to a single point is really a three-dimensional sphere.

In the intervening years, dozens of math-ematicians have put forth mistaken proofs,often very subtly in error. For that reason,mathematicians are hesitant to declare theconjecture settled until Perelman’s proofhas been thoroughly checked. But theyagree that, unlike most previous attempts,Perelman’s papers contain a wealth ofimportant ideas that will be valuable evenif his work turns out to fall short of provingthe full Poincaré conjecture.

“The first paper is already amazing,” saysJeff Viaclovsky of MIT. “It’s a major break-through.” —E. KLARREICH

Fig-Wasp Upset Classic partnership isn’tso tidy after all

Textbooks that marvel over an extremeexample of the buddy system—fig speciesthat supposedly each pair up with a lonepollinating wasp species—may need rewrit-ing, according to a new genetic analysis.

In four out of eight fig species tested inPanama, genetic markers reveal that thesupposedly single type of wasp living inthe flower turns out to be two species,reports Drude Molbo of the SmithsonianTropical Research Institute (STRI) basedin Balboa, Panama. Fig partnerships withmultiple wasps may turn out to be “rou-tine,” Molbo and her colleagues suggest inan upcoming Proceedings of the NationalAcademy of Sciences. They also have evi-dence of a single wasp species teaming upwith different figs.

Another pollination biologist, OllePellmyr of the University of Idaho inMoscow, welcomes the new study as “nicework.” The old idea that, except for a fewoddballs, each of the world’s 800 fig specieshas an exclusive partnership with a wasphas been “dogma,” he says.

Pellmyr points out that biologists have

long used fig wasps to study big questions,such as sex ratios, cheating in partnerships,and formation of new species. Molbo’scoauthor Allen Herre, also of STRI, saysthat the team’s findings will require somerethinking across a wide range of work,including his own.

The wasps, usually only a few millime-ters long, make epic flights of up to 20 kilo-meters to find the right species of fig inbloom. The female wriggles into the flask-shaped flower, lays eggs, and dies there.Her offspring hatch and mate inside thefig flower. Each daughter then sets off tofind a new fig plant in which to lay eggs.When she arrives inside a flower, shedeposits her natal fig’s pollen.

Scientists have known that several femalewasps can converge on the same flower. Tosort out batches of offspring, Molbo iden-tified DNA markers that distinguish the off-spring of these females. As she analyzedpopulations in hundreds of fig flowers, somecombinations of markers never showed up.The researchers began to suspect that thefigs held pairs of wasp species.

To check their results, they turned to Car-los Machado, now of the University of Ari-zona in Tucson. He has identified DNAmarkers not from the cell nucleus, as Molbodoes, but from mitochondria, the cell pow-erhouses. The mitochondrial markers dis-played the same patterns.

“Drude was looking for one thing andfound something very surprising and dif-ferent,” says Herre.

One of the wasp pairs working the samefig species seems to have evolved from ashared ancestor within the past few millionyears, says Herre. Pellmyr highlights thisfinding as a possible example of a speciesthat split despite close quarters (SN:7/21/01, p. 42).

Rethinking wasps and figs may rock

SCIENCENEWSThis Week

ROOM IN BLOOM Inside the developinggreen fruit of a Panamanian fig, a waspgeneration hatches and mates. Inset showsmale (wingless, left) and female (right).

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