Archives - Mathematics: Page 14
Author: paul carson (Thu Aug 24, 2006 7:06 pm)
Title: Solving the Poincare Conjecture
The solution of the Poincaré conjecture, that has frustrated mathematicians for more than a century was accomplished by the Russian mathematician Grigory Perelman. Perelman who rejected the Fields Medal the equivalent of the Nobel Prize in the field of mathematics, if his proof survives two years of continued scrutiny, he will be first in line to receive or share a $1 million award.
We would like to know what the conjecture is, but it deals with realms far beyond the comprehension for most people. The field is topology, the branch of mathematics that studies the fundamental properties of shapes and space. A topologist considers a cigar and a rabbit’s head the equivalent of a sphere because all these shapes can be deformed into each other. By contrast, a donut and a coffee mug each has one hole and can’t be deformed into a sphere.
In 1904, the French mathematician Henri Poincaré conjectured that the same kind of thinking would also apply in a higher dimension, where it is impossible even to visualize what objects might look like. Arduous work over decades has proved the conjecture holds for virtually all higher dimensions. But no one had proved it true for spheres that have three-dimensional surfaces, not merely the two we see every day. Dr. Perelman’s papers topped out at 22 and 39 pages and met complaints that they were too sketchy.
After posting a few short papers on the Internet and making a tour lecture in the United States, Dr. Perelman disappeared back into the Russian woods in the spring of 2003, leaving the world’s mathematicians to pick up the pieces and decide if he was right.
Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics. As a result there is a growing feeling, a cautious optimism that they have finally achieved a landmark not just of mathematics, but of human thought.
The understanding of three-dimensional space brought about by Poincaré’s conjecture could be one of the major pillars of math in the 21st century.
What impact this will have on the practical world is uncertain. We should celebrate the dedication, some might call it obsession, that led Dr. Perelman to labor for years to solve a problem that once appeared insoluble
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