The most significant scientific achievement of 2006 was, according to the issue of Science magazine that came out today, the solution of a 100-year-old mathematical problem that had baffled some of the best minds of the 20th century. Russian Mathematician Grigory Perelman solved this problem in a series of papers he circulated in 2003, and this year the mathematical community officially recognized his solution by offering him the Fields Medal, the most important prize in mathematics.
The Poincaré conjecture, as the problem is known, belongs to the field of math called topology. But the methods that led to Perelman’s proof took hints from the physical world, and may have implications for research in theoretical physics, experts say.
The conjecture was stated in the early 1900s by the French mathematician Henri Poincaré. It concerns the shape of three-dimensional spheres — spheres that have three dimensions, as opposed to the surface of an ordinary sphere, which only has two: latitude and longitude. Mathematically, one way to construct a three-dimensional sphere is by taking the boundary of a four-dimensional ball, which fills out a solid region in four-dimensional space.
This is the analogue of taking the two dimensional sphere as the boundary of a three-dimensional ball. Some cosmologists — starting with Albert Einstein — have long suspected that the universe we live in might be a three-dimensional sphere; some historians have hinted that this view might be traced as far back as to Dante’s Divine Comedy, written in the 1300s, when Dante declared that above the space of the heavens was just one point, which he called God.
Poincaré believed that the three-dimensional sphere plays a special role among all three-dimensional spaces, a class that includes the boundary spaces of more general four-dimensional objects (other than a ball). In a sphere of two dimensions or more, any loop can be continuously shrunk to a point. For example, one can shrink loops continuously on a two-dimensional sphere; by contrast, one can’t do that on other two-dimensional objects such as the surface of a doughnut. The Poincaré conjecture says that the ability to shrink loops can be used as a test that a space is a sphere, since the three-dimensional sphere is the only three-dimensional space that has this property.
Throughout the 20th century, hundreds of mathematicians have tried solving the Poincaré conjecture, using dozens of different approaches. In the early 1980s, Richard Hamilton, now at Columbia University, in New York, introduced a new line of attack. His idea was to take a space that satisfies the loop-shrinking test and deform it in a “natural” way so that the end result would be a space that’s uniformly curved everywhere. From there, standard mathematical techniques would imply that the space was a three-dimensional sphere.
The “natural” way of deforming a space found by Hamilton was what he called the Ricci flow. As it turned out, the equations that defined the Ricci flow were strikingly similar to those that govern the propagation of heat, as embodied in the so-called heat equation. “The Ricci flow dissipates curvature just like the heat equation dissipates heat,” explains David Morrison, a mathematician and physicist at the University of California, Santa Barbara. This similarity helped develop a theory of the Ricci flow based in part on the machinery that had worked for the heat equation, which physicists and mathematicians had studied since the early 1800s.
Much work was done along these lines by Hamilton himself and others, producing new results about geometry in any number of dimensions. But large gaps in Hamilton’s program remained, due to technical difficulties in estimating the behavior of the Ricci flow. The breakthrough came in 2003, when Perelman, of the Steklov Mathematical Institute in St. Petersburg, solved those difficulties.
Perelman’s solution relied in part on the intuition he got from another connection between math and physics. In fact, physicists had discovered the Ricci flow independently from Hamilton in the early 1980s , in the context of quantum theory. In quantum theory, the evolution of a system follows formulas invented by Richard Feynman which take into account all possible ways a physical system can evolve — for example, the infinitely many paths a particle can take in its motion. But in practice, physicists need to make predictions based on a finite number of calculations.
Daniel Friedan, now at Rutgers, the State University of New Jersey, found out that in certain contexts the infinite number of ways a system can evolve can be reduced to finite calculations by following the Ricci flow on a multidimensional space. (The context he was studying later became popular among theoretical physicists because of its relevance to string theory, the proposed fundamental theory of nature that would replace elementary particles with tiny vibrating strings. )
At the time when Hamilton and Friedan discovered the Ricci flow, there wasn’t much communication between physicists and mathematicians, Morrison says. “They arrived at this notion completely independently.” But the communication between the fields has partially improved in recent years, he says. “In Perelman’s first preprint, he seems really inspired by the occurrence.” And since Perelman’s proof appeared, there has been renewed interest to investigate the applications of the Ricci flow to string theory, he says.
Perelman’s proof was the first mathematical discovery to be recognized by Science magazine as the Breakthrough of the Year. The Poincaré conjecture also carries a $1 million prize, as one of the Millennium Problems singled out by the Clay Mathematical Foundation. However, Perelman has reportedly expressed little interest in the prize, just as he has turned down the Fields medal last August.
(This was the article I wrote for Inside Science News Service, and is reproduced here courtesy of the American Institute of Physics.)