PRIZE NUMBERS. In work recognized with a Fields Medal, a heart-shaped curve encircles the melting region in a crystal representation of a random surface.Kenyon and Okounkov

Grigori Perelman electrified the mathematical world 3 years ago with his claim to have solved one of the most famous problems in mathematics (SN: 6/14/03, p. 378: http://www.sciencenews.org/articles/20030614/bob10.asp). The Russian mathematician has now been awarded a Fields Medal, the highest honor in mathematics, given every 4 years to up to four recipients age 40 or under.

Fields Medals, presented this week at the International Congress of Mathematicians in Madrid, also went to Wendelin Werner of the University of Paris-Sud in Orsay, Andrei Okounkov of Princeton University, and Terence Tao of the University of California, Los Angeles.

Perelman's work provides proofs of two major questions in topology, the mathematical theory of shapes. The century-old Poincaré conjecture states that if every loop embedded in a compact three-dimensional shape can be pulled down to a single point, then the shape is a three-dimensional sphere, albeit possibly a deformed one. The more recent but even more far-reaching Thurston geometrization conjecture states that every three-dimensional shape can be broken down into chunks and that each chunk possesses one of eight simple geometric structures.

In 3 years of scrutiny, mathematicians have uncovered no major flaws in Perelman's proofs of the two conjectures. Researchers have now written more than 1,000 pages elucidating his ideas. If his Poincaré proof goes unrefuted for another 2 years, Perelman may be eligible for a $1 million prize offered by the Clay Mathematics Institute in Cambridge, Mass., for a proof of the conjecture. However, Perelman has dropped out of the mathematical scene, and when tracked down in St. Petersburg, Russia, he refused to accept the award.

Werner, another of this year's Fields Medal winners, has made seminal advances in areas including percolation theory, an abstract model of how a fluid or gas filters through a semiporous material such as a sponge. At a critical degree of a material's porosity, the fluid inside forms a pattern of clusters that looks roughly the same at any scale larger than very small ones. Werner and his collaborators calculated such a pattern's fractal dimension, a measurement of the jaggedness of the clusters' boundary. The researchers also related these patterns to the trajectory of a Brownian path—the random route taken by, say, a pollen grain in a glass of water—to prove a decades-old conjecture.

Werner's work is "a watershed in the interaction between mathematics and physics," says Charles Newman, director of New York University's Courant Institute of Mathematical Sciences.

The other two winners' work also forged new links between physics and mathematics. For instance, Okounkov has used ideas from geometry and probability to characterize random shapes, such as the ones that form as tiny blocks melt off the corners of a cubical crystal. Okounkov linked the distribution of blocks of different sizes to the ways in which a whole number can be partitioned into a sum of smaller numbers. Using this connection, he and Richard Kenyon of the University of British Columbia in Vancouver showed that the two-dimensional shadow of such a crystal always has a distinctive shape and lies inside a simple curve.

Okounkov possesses "an unusual intuition that makes him able to find unexpected bridges between seemingly distant theories," says Enrico Arbarello, a mathematician at the University of Rome.

Tao's interests include prime numbers and quantum physics. In 2004, he and Ben Green of the University of Bristol in England proved that even though primes are few and far between, their sequence contains arithmetic progressions—sequences of numbers that differ by a fixed amount—of every possible length (SN: 4/24/04, p. 260: Available to subscribers at http://www.sciencenews.org/articles/20040424/fob4.asp). Tao has also made contributions to the study of wave maps, which are related to Einstein's equations of general relativity, and nonlinear Schrödinger equations, which describe the behavior of light in a fiberoptic cable and other quantum phenomena.

"Some of his work gives one a feeling of awe that a human mind can master such complexity," says Princeton mathematician Charles Fefferman. "And some makes one wonder how so many researchers could have missed something so seemingly obvious."

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