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Aripoutre for a warm-up:
“The problem of the millennium,” solved by a Russian mathematical genius, is related to the origin of the Universe.
THE GAME OF THE MIND
Not so long ago, mathematics did not promise either fame or fortune to its “priests.” They weren't even given the Nobel Prize. There isn't such a nomination. Because, according to a very popular legend, Nobel's wife once cheated on him with a mathematician. And in revenge, the rich man deprived the entire brotherhood of their respect and prize money.
The situation changed in 2000. The Clay Mathematics Institute, a private mathematical institute, selected seven of the most difficult problems. And promised to pay one million dollars for solving each. Mathematicians were looked at with respect. In 2001, the film “A Beautiful Mind,” starring a mathematician, was even released.
Today, only those far removed from civilization are unaware that one of the promised millions – the very first – has already been awarded. The prize was awarded to a Russian citizen, a resident of St. Petersburg, Grigory Perelman, for solving the Poincaré conjecture, which, thanks to his efforts, became a theorem. Our dear 44-year-old bearded man has outdone the whole world. And now he continues to keep it – the world – in suspense. Because it is unknown whether the mathematician will honestly accept the well-deserved million dollars or refuse. Progressive society in many countries is naturally worried. At least, newspapers on all continents are chronicling the financial and mathematical intrigue.
And against the backdrop of these fascinating activities – guessing and dividing other people's money – the meaning of Perelman's achievement was somehow lost. Jim Carlson, President of the Clay Institute, of course, stated at one time that the purpose of the prize fund was not so much to find answers as to try to raise the prestige of mathematical science and interest young people in it. But what is the essence?
THE POINCARÉ CONJECTURE – WHAT IS IT?
The riddle solved by the Russian genius touches on the foundations of a branch of mathematics called topology. It – topology – is often called “geometry on a rubber sheet.” It deals with the properties of geometric shapes that are preserved if the shape is stretched, twisted, or bent. In other words, deformed without breaks, cuts, or gluing.
Topology is important for mathematical physics because it allows us to understand the properties of space. Or to assess it without being able to look at the shape of this space from the outside. For example, our Universe.
Explaining the Poincaré conjecture, they start like this: imagine a two-dimensional sphere – take a rubber disk and stretch it onto a ball. So that the circumference of the disk is gathered in one point. In the same way, for example, you can pull a sports backpack with a cord. As a result, you get a sphere: for us – three-dimensional, but from the point of view of mathematics – only two-dimensional.
Then they suggest stretching the same disk onto a donut. It seems to work. But the edges of the disk will converge into a circle, which can no longer be pulled into a point – it will cut the donut.
Then comes something inaccessible to the imagination of an ordinary person. Because you need to imagine a three-dimensional sphere – and namely, a ball stretched onto something that goes into another dimension.
As another Russian mathematician, Vladimir Uspensky, wrote in his popular book, “unlike two-dimensional spheres, three-dimensional spheres are inaccessible to our direct observation, and it is as difficult for us to imagine them as it was for Vasily Ivanovich from the famous anecdote to imagine a square trinomial.”
So, according to the Poincaré conjecture, a three-dimensional sphere is the only three-dimensional thing whose surface can be pulled into one point by some hypothetical “hyperscord.”
Jules Henri Poincaré suggested this in 1904. Now Perelman has convinced everyone who understands that the French topologist was right. And turned his hypothesis into a theorem.
The proof helps to understand what shape our Universe has. And allows us to reasonably assume that it is that very three-dimensional sphere. But if the Universe is the only “figure” that can be pulled into a point, then, perhaps, it can also be stretched out of a point. This serves as indirect confirmation of the Big Bang theory, which states that the Universe originated from a point.
It turns out that Perelman, together with Poincaré, upset the so-called creationists – supporters of the divine origin of the universe. And they poured water on the mill of physicists-materialists. So, all the evidence is yet to come, man is the main god, he at least tries to understand, solve, prove, create, destroy, give life, take life. Although the theory of the divine origin of the universe is very convenient, it does not need to be proven, you just need to insist on it, because how he created it, man cannot understand:)