The Poincare conjecture and intrigues around it

Few mathematical theories so excited the public, far from abstract geometric reasoning, like this one. The Poincare conjecture, put forward in 1887 by the French mathematician Henri Poincaré, has been haunting the scientists of different countries for more than a hundred years. It interested not only geometers, but also physicists, and even ... special services. Therefore, such a sensation was caused by a message that the secret of the hypothesis, over which so many bright minds had been puzzling, was finally solved, and Poincare's theorem was proved. The fact that the scientist, the Russian mathematician Grigory Perelman, who proved the theorem to the theorem, in 2006 rejected the oil to the fire of the people's interest in the field of the Fields mathematical award (and the accompanying million dollars). In no way did the scientist react to the awarding of his Millennium Prize by the Clay Institute of Mathematics.

However, - the reader will ask, far from mathematics, - why such an interest is caused precisely by the Poincare conjecture? And why such huge money is paid for its proof? For this, albeit in the most general terms, it is necessary to characterize what this hypothesis represents in the framework of such a field of mathematics as topology. Imagine a weakly inflated balloon. If it is crushed, then it can be given to it different forms: a cube, an oval sphere and even forms of people and animals. But all this variety of geometric forms can turn into one universal form - a ball. The only thing that the ball can not turn into without a break is in the form with a hole, for example, in a bagel.

The Poincaré conjecture asserted that all objects that do not have a through hole have one base - a sphere. But the bodies that have a hole (mathematicians call them torus, but for us let it be a "bagel") are compatible with each other, but not with solid bodies. For example, if we are blinded from plasticine cat, we can crush it in a ball and out of it to blind, without using tears, a hedgehog or a rail. If we are blind to a bagel, we can deform it into a "figure-eight" or a mug, but the ball will not succeed. The torus and the sphere are incompatible - they are not homomorphic in the mathematical language.

It is noteworthy that the proof of this theory was interested not so much in mathematics as in astrophysics. If Poincare's theory is applicable to all material bodies in the universe, then why not imagine for a moment that it is also true of the universe itself? And what if all matter originated from a small, one-dimensional point and is now unfolding into a multidimensional sphere? And where are its boundaries? And what is beyond the borders? And what if we find the mechanism of the Universe rolling back to the starting point? Since in the proof of his hypothesis the author himself made a mistake, many mathematicians and physicists, having fallen under the spell of the hypothesis of Poincare, began to work selflessly to prove it. Several of them - DG Whitehead, Bing, K. Papakiriakopoulos, S. Smale, M. Friedman - put their lives on the proof of the Poincare theory.

But as a result of the laurels went to the little-known St. Petersburg scientist Perelman, although formally - in the pages of peer-reviewed journals - his proof never saw the light. Grigory Yakovich's work was posted on arXiv.org in 2002, but nevertheless produced the effect of an exploding bomb in the scientific world. Since an eccentric mathematician did not even bother to "polish" his evidence, some scientists decided to intercept the laurels of the discoverer. Thus, the Chinese mathematicians Huai-dong Cao and Xiping Zhu named Perelman's evidence intermediate, and supplemented it. However, the award of the Millennium Prize to the Russian mathematician (even though he refused to receive it) put all the points on "i": Poincaré's conjecture was proved precisely by Perelman. When journalists finally managed to interview an ingenious mathematician, when asked why he refused a $ 1 million prize, a strange answer was made: "If I own the universe, then why would I need a million?"