Few mathematical theories have excited the public so far from abstract geometric reasoning as this. The Poincaré hypothesis, put forward in 1887 by the French mathematician Henri Poincaré, has been harassing scientists from different countries for more than a hundred years. Not only geometers, but also physicists became interested in her, and even ... special services. Therefore, such a sensation was caused by the message that the secret of the hypothesis, over which so many bright minds were puzzling, was finally revealed, and the Poincare theorem was proved. The fact that the scientist who proved the theorem - the Russian mathematician Grigory Perelman - in 2006 refused the award awarded to him by the Fields mathematical prize (and the accompanying million dollars) added fuel to the fire of public interest. The scientist did not react in any way to the award of his Millennium Prize by the Clay Mathematical Institute.
However, asks a reader far from mathematics, why is Poincare’s hypothesis so interesting? And why do they pay such huge money for her proof? For this, albeit in the most general terms, it is necessary to characterize what this hypothesis is within the framework of such a field of mathematics as topology. Imagine a slightly inflated balloon. If it is crumpled, then you can give it different forms: a cube, an oval sphere and even the shapes of people and animals. But all this variety of geometric shapes can turn into one universal shape - a ball. The only thing a ball cannot break into without breaks is a shape with a hole, for example, a bagel.
The Poincaré hypothesis claimed that all objects that do not have a through hole have one base - a ball. But bodies with a hole (mathematicians call them a torus, but for us let it be a “donut”) are compatible with each other, but not with solid bodies. For example, if we blindfold a cat from plasticine, we can make it into a ball and blind it out without using tears, a hedgehog or rail. If we blind a bagel, we can deform it into a figure eight or a mug, but it will not succeed in turning it into a ball. The torus and the sphere are incompatible - in a mathematical language they are not homeomorphic.
It is noteworthy that the proof of this theory was not so much interested in mathematics as in astrophysics. If the Poincare theory is applicable to all material bodies in the Universe, then why not imagine for a moment that it is also true regarding the Universe itself? But what if all the matter arose from a small, one-dimensional point and now unfolds into a multidimensional sphere? And where are its borders? And what is beyond the borders? And what if we find the mechanism for the Universe to roll back to its starting point? Since the author himself made a mistake in the proof of his hypothesis, many mathematicians and physicists, falling under the spell of the Poincare hypothesis, began to work selflessly to prove it. Several of them - D.G. Whitehead, Bing, K. Papakiriakopoulos, S. Smale, M. Friedman - put their lives to prove the Poincare theory.
But as a result, laurels went to the little-known St. Petersburg scientist Perelman, although formally - on the pages of peer-reviewed journals - his proof never saw the light of day. The work of Grigory Yakovich was posted on arXiv.org in 2002, but, nevertheless, produced the effect of an exploding bomb in the scientific world. Since the eccentric mathematician did not even bother to “polish” his evidence, some scientists decided to intercept the laurels of the discoverer. So, the Chinese mathematicians Huaidong Cao and Siping Zhu called Perelman's evidence intermediate, and supplemented it. However, the award of the Millennium Prize to Russian mathematicians (even though he refused to receive it) put all the points over the “i”: Poincare’s hypothesis was proved precisely by Perelman. When journalists nevertheless managed to interview the brilliant mathematician, when asked why he refused the prize of one million dollars, a strange answer sounded: “If I own the Universe, then why do I need a million?”