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A Solution to the Poincare Conjecture May Be Near
Remember all the hours you spent working on your Math 210 proof portfolio in the hopes of earning a respectable grade? The Clay Mathematics Institute has published a portfolio of seven mathematics problems for the mathematical community. Rather than offering a good grade, however, the institute offers $1 million for the solution of just one of these problems. Recent work of Grigori Perelman, a Russian mathematician working at the Steklov Institute of Mathematics, seems to give a solution to one of these problems, the so-called Poincaré conjecture, a problem in topology first posed in 1904.
The Poincaré conjecture deals with the structure of three-dimensional space. To understand what it says, let’s first think about a two-dimensional analogy. Imagine that we take a piece of string tied in a loop with a slip-knot and arrange it on the surface of a basketball. No matter how the string is laid out, we can make the loop disappear by pulling it through the slip-knot. Because it possesses this property, the surface of the basketball is called “simply connected.”
However, if we perform the same experiment on the surface of a bagel, this is no longer true. For instance, if the string winds around the hole in the bagel, the loop will never disappear when pulled on. In fact, an old theorem in topology tells us that the surface of the basketball, known to topologists as the two-dimensional sphere, is the only surface that is simply connected. The Poincaré conjecture says, plainly enough, that the three-dimensional sphere, the set of points in four-dimensional space that are a constant distance from the origin, is the only simply connected three-dimensional space. This conjecture is, in many respects, the beginning point of any study of three-dimensional spaces.
Though the statement of the Poincaré conjecture is simple enough, a proof has eluded mathematicians for the past hundred years in spite of intensive efforts by many talented mathematicians. In the late 1970’s, William Thurston of Princeton University made an exceptionally bold conjecture about the geometry of all three-dimensional spaces. It was immediately recognized that if Thurston’s conjecture were true, then the Poincaré conjecture must be true as well.
Perelman’s work is exciting for it seems to give a proof of Thurston’s much more general conjecture. Several teams of mathematicians from around the world are working to determine whether Perelman’s proof contains any flaws, and at this time, it is too early to state definitively that the proof is correct. However, Perelman’s ideas have been called “highly original and of deep insight” by experts and are already being used to investigate related areas of mathematics. Perelman completed his papers while working in relative seclusion and entered the spotlight only last year when he gave a few talks on his work in the United States. Curiously, the publicity-shy Perelman seems uninterested in submitting his papers for publication, a requirement if he is to lay claim to the prize money offered by the Clay Institute. You may read more about the Clay Mathematics Institute and its list of seven problems here.