Anglican Samizdat

March 22, 2010

Poincaré Conjecture

Filed under: Mathematics — David Jenkins @ 8:53 pm

As Einstein observed, pure mathematics is, in its way, the poetry of logical ideas. To find a concise elegant solution to a mathematical problem is satisfying, exciting and intrinsically worthwhile; I’m sure God likes pure mathematics.

The Poincaré Conjecture has bothered mathematicians for almost 100 years; it has been solved by a somewhat eccentric Russian mathematician:

The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture. The citation for the award reads:

The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman.

The Poincaré conjecture is one of the seven Millennium Prize Problems established by CMI in 2000. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude.

Formulated in 1904 by the French mathematician Henri Poincaré, the conjecture is fundamental to achieving an understanding of three-dimensional shapes (compact manifolds). The simplest of these shapes is the three-dimensional sphere. It is contained in four-dimensional space, and is defined as the set of points at a fixed distance from a given point, just as the two-dimensional sphere (skin of an orange or surface of the earth) is defined as the set of points in three-dimensional space at a fixed distance from a given point (the center).

Since we cannot directly visualize objects in n-dimensional space, Poincaré asked whether there is a test for recognizing when a shape is the three-sphere by performing measurements and other operations inside the shape. The goal was to recognize all three-spheres even though they may be highly distorted. Poincaré found the right test (simple connectivity). However, no one before Perelman was able to show that the test guaranteed that the given shape was in fact a three-sphere.


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