Which gives a proof of the Poincaré conjecture. The Poincaré Hypothesis and the Origin of the Universe

Poincare conjecture put forward at the beginning of the 20th century. French mathematician Henri Poincare. To formulate it, we give

Definition. Topological space X is called simply connected if it is path-connected and any continuous mapping
X circles into space X can continue to continuous display
the whole circle
. It is not difficult to see that the sphere is simply connected at n 2.

The Poincaré hypothesis. Every closed, simply connected 3-manifold is homeomorphic to a 3-sphere.

Analogues of the Poincaré conjecture concerning manifolds of dimension 4 or more are proved. Moreover, a topological classification in general of all closed simply connected four-dimensional manifolds is obtained.

It is interesting: Almost 100 years ago, Poincaré established that the two-dimensional sphere is simply connected and suggested that the three-dimensional sphere is also simply connected.

In other words, the Poincaré conjecture states that every simply connected closed 3-manifold is homeomorphic to a 3-sphere. The conjecture was formulated by Poincaré in 1904. The generalized Poincaré conjecture states that for any n every manifold of dimension n is homotopy equivalent to a sphere of dimension n if and only if it is homeomorphic to it. For clarification, the following picture is used: if you wrap an apple with a rubber band, then, in principle, by pulling the tape together, you can squeeze the apple into a point. If you wrap a donut with the same tape (a pie with a hole in the middle), then you cannot squeeze it into a point without tearing either the donut or rubber. In this context, the apple is called a "singly connected" figure, but the donut is not simply connected.

Jules Henri Poincaré discovered the special theory of relativity at the same time as Einstein (1905) and is recognized as one of the the greatest mathematicians throughout the history of mankind.

The Poincaré hypothesis remained unproven throughout the twentieth century. In the mathematical world, it has acquired a status similar to that of Fermat's Last Theorem.

For the proof of the Poincaré conjecture Clay awarded a million dollar prize, which may seem surprising, since we are talking about a very private, uninteresting fact. In fact, for mathematicians, it is not so much the properties of the three-dimensional surface that are important, but the fact that the proof itself is difficult. In this problem, in a concentrated form, what could not be proved with the help of previously available ideas and methods of geometry and topology is formulated. It allows you to sort of look at a deeper level, into that layer of tasks that can be solved only with the help of the ideas of the “new generation”. As in the situation with Fermat's theorem, it turned out that the Poincare conjecture is special case A much more general statement about the geometric properties of arbitrary three-dimensional surfaces is Thurston's Geometrization Conjecture. Therefore, the efforts of mathematicians were directed not to solving this particular case, but to building a new mathematical approach that can cope with such problems.

Russian mathematician Grigory Perelman, employee of the Laboratory of Geometry and Topology of the St. V.A. Steklov, claims that he proved the Poincaré conjecture, that is, he solved one of the most famous unsolved mathematical problems. Unusual was the way that Perelman chose to publish his evidence. Instead of publishing it in a solid scientific journal, which, by the way, was a prerequisite for awarding a million dollar prize, Perelman posted his work on one of the Internet archives. Although the proof took only 61 pages, it created a sensation in the scientific world.

The scientific world applauded the genius, promising mountains of gold and honorary titles. The American Clay Institute of Mathematics was ready to award him a $1 million award. No one doubted that the World Congress of Mathematicians would call Perelman the winner. By the way, as you know, mathematicians are not among the scientists awarded Nobel Prize. Evil tongues claim that this fact is not accidental. Indeed, according to rumors, it was the mathematician who fell out of favor with the famous Swede Alfred Nobel, having beaten off his beloved girl in his youth. Meanwhile, the Russian genius refused a million, without publishing his discovery in specialized publications, and resigned from the Mathematical Institute. Steklov RAS, went into seclusion and, at the award ceremony, which was presented by the King of Spain Juan Carlos I, did not appear. He did not react in any way to the message about the award and the invitation to receive it, but as acquaintances say: the genius "went into the forests" to pick mushrooms near St. Petersburg.

Scientists believe that the 38-year-old Russian mathematician Grigory Perelman proposed the correct solution to the Poincaré problem. Keith Devlin, Professor of Mathematics at Stanford University, announced this at the Exeter Science Festival (Great Britain).

The problem (also called the problem or hypothesis) of Poincaré is one of the seven most important mathematical problems, for solving each of which Clay Mathematics Institute appointed a prize of one million dollars. This is what attracted such wide attention to the results obtained by Grigory Perelman, an employee of the Laboratory of Mathematical Physics St. Petersburg branch of the Steklov Institute of Mathematics.

Scientists around the world learned about Perelman's achievements from two preprints (articles that precede a full-fledged scientific publication) posted by the author in November 2002 and March 2003 on the site of the archive of preliminary works Los Alamos Science Laboratory.

According to the rules adopted by the Clay Institute's Scientific Advisory Board, a new hypothesis must be published in a specialized journal with an "international reputation". In addition, according to the rules of the Institute, the decision on the payment of the prize is ultimately made by the "mathematical community": the proof must not be refuted within two years after publication. The verification of each proof is done by mathematicians in different countries peace.

Poincaré problem

The Poincare problem belongs to the field of the so-called topology of manifolds - spaces arranged in a special way and having different dimensions. Two-dimensional manifolds can be visualized, for example, on the example of the surface of three-dimensional bodies - a sphere (the surface of a ball) or a torus (the surface of a donut).

It is easy to imagine what will happen to a balloon if it is deformed (bent, twisted, pulled, squeezed, pinched, deflated or inflated). It is clear that with all the above deformations, the ball will change its shape over a wide range. However, we will never be able to turn the ball into a donut (or vice versa) without breaking the continuity of its surface, that is, without breaking it. In this case, topologists say that the sphere (ball) is not homeomorphic to the torus (donut). This means that these surfaces cannot be mapped to one another. talking plain language, sphere and torus are different in their topological properties. And the surface of a balloon, with all its various deformations, is homeomorphic to a sphere, as well as the surface of a lifebuoy is to a torus. In other words, any closed two-dimensional surface without through holes has the same topological properties as a two-dimensional sphere.

The Poincaré problem states the same for three-dimensional manifolds (for two-dimensional manifolds such as the sphere, this proposition was proved as early as the 19th century). As the French mathematician noted, one of the most important properties of a two-dimensional sphere is that any closed loop (for example, a lasso) lying on it can be contracted to one point without leaving the surface. For a torus, this is not always true: a loop passing through its hole will shrink to a point either when the torus is broken, or when the loop itself is broken. In 1904, Poincaré conjectured that if a loop can contract to a point on a closed three-dimensional surface, then such a surface is homeomorphic to a three-dimensional sphere. The proof of this conjecture turned out to be an extremely difficult task.

Let's clarify right away: the formulation of the Poincaré problem we mentioned does not speak at all about a three-dimensional ball, which we can imagine without much difficulty, but about a three-dimensional sphere, that is, about the surface of a four-dimensional ball, which is already much more difficult to imagine. But in the late 1950s, it suddenly became clear that it was much easier to work with high-dimensional manifolds than with three- and four-dimensional ones. Obviously, the lack of visualization is far from the main difficulty that mathematicians face in their research.

A Poincaré-like problem for dimensions 5 and above was solved in 1960 by Stephen Smale, John Stallings, and Andrew Wallace. The approaches used by these scientists, however, turned out to be inapplicable to four-dimensional manifolds. For them, the Poincaré problem was only proven in 1981 by Michael Freedman. The three-dimensional case turned out to be the most difficult; his decision and offers Grigory Perelman.

It should be noted that Perelman has a rival. In April 2002, Martin Dunwoody, professor of mathematics at the British University of Southampton, proposed his own method for solving the Poincaré problem and is now awaiting a verdict from the Clay Institute.

Experts believe that the solution of the Poincaré problem will make it possible to take a serious step in the mathematical description physical processes in complex three-dimensional objects and will give a new impetus to the development of computer topology. The method proposed by Grigory Perelman will lead to the discovery of a new direction in geometry and topology. A Petersburg mathematician may well qualify for the Fields Prize (an analogue of the Nobel Prize, which is not awarded in mathematics).

Meanwhile, some find the behavior of Grigory Perelman strange. Here is what the British newspaper The Guardian writes: “Most likely, Perelman’s approach to solving the Poincaré problem is correct. But not everything is so simple. Perelman does not provide evidence that the work was published as a full-fledged scientific publication(preprints do not count as such). And this is necessary if a person wants to receive an award from the Clay Institute. Besides, he doesn't show any interest in money at all."

Apparently, for Grigory Perelman, as for a real scientist, money is not the main thing. For solving any of the so-called "millennium problems" a true mathematician will sell his soul to the devil.

Chinese mathematicians have published a complete proof of the Poincaré conjecture, formulated in 1904, the Xinhua news agency reports. The hypothesis concerning the classification of multidimensional surfaces (more precisely, manifolds) was one of the "millennium problems", for the solution of each of which the American Clay Institute offered a million dollar award.

According to Poincaré, any closed three-dimensional "surface without holes" (a simply connected manifold) is equivalent to a three-dimensional sphere, that is, the surface of a four-dimensional ball. Poincare himself, the author of the mathematical apparatus of Einstein's theory, presented the first justification, but later discovered an error in his own reasoning. The hypothesis in this formulation was proved in 2003 by the Russian mathematician Grigory Perelman, whose 70-page work is still being checked by experts. Other cases (dimensions four and higher) were considered earlier.

According to the authors, the new 300-page article in the Asian Journal of Mathematics is not independent and relies primarily on Perelman's results. Zhu Xiping and Cao Huaidong claim that they have now eliminated a number of difficulties, the ways to overcome which Perelman had only just outlined. It is known that Shing-Tun Yau also participated in the work on the proof, whose topological works (in particular, the theory of Calabi-Yau manifolds) are considered key for modern theory strings. The new work, experts say, will also require a lengthy recheck.

Aleksandrov A.D., Netsvetaev N.Yu. Geometry. Moscow: Nauka, 1990

Appendix to abstract 2:

What is the essence of the Poincaré theorem

1. Sofya proved it to E, and here it is also RED ....
2. The bottom line is that the universe is not a sphere, but a donut
3. The meaning of the Poincare conjecture in its original formulation is that for any three-dimensional body without holes there is a transformation that will allow it to turn into a ball without cutting and gluing. If this seems obvious, then what if the space is not three-dimensional, but contains ten or eleven dimensions (that is, we are talking about a generalized formulation of the Poincaré hypothesis, which Perelman proved)
4. can't tell in 2 words
5. In 1900, Poincaré conjectured that a three-dimensional manifold with all homology groups like that of a sphere is homeomorphic to a sphere. In 1904, he also found a counterexample, now called the Poincaré sphere, and formulated the final version of his conjecture. Attempts to prove the Poincaré conjecture led to numerous advances in the topology of manifolds.

   Proof of the generalized Poincaré conjecture for n #10878; 5 was obtained in the early 1960-1970s almost simultaneously by Smale, independently and by other methods by Stallings (Eng.) (for n #10878; 7, his proof was extended to the cases n = 5 and 6 by Zeeman (Eng.)) . The proof is much more hard case n = 4 was obtained only in 1982 by Friedman. It follows from Novikov's theorem on the topological invariance of Pontryagin's characteristic classes that there exist homotopically equivalent but not homeomorphic manifolds in high dimensions.

   The proof of the original Poincaré conjecture (and the more general Trston conjecture) was found only in 2002 by Grigory Perelman. Subsequently, Perelman's proof was verified and presented in a twisted form by at least three groups of scientists. 1 The proof uses the Ricci flow with surgery and largely follows the plan outlined by Hamilton, who was also the first to use the Ricci flow.

6. who is this
7. Poincaré's theorem:
   Poincaré's vector field theorem
   Bendixson's Poincaré theorem
   Poincaré's theorem on the classification of homeomorphisms of the circle
   Poincaré's conjecture on the homotopy sphere
   Poincaré recurrence theorem

   What are you asking about?

8. In theory dynamic systems, the Poincaré theorem on the classification of homeomorphisms of the circle describes the possible types of reversible dynamics on the circle, depending on the rotation number p(f) of the iterated map f. Roughly speaking, it turns out that the dynamics of mapping iterations is to a certain extent similar to the dynamics of rotation through the corresponding angle.
   Namely, let a circle homeomorphism f be given. Then:
   1) The rotation number is rational if and only if f has periodic points. Moreover, the denominator of the rotation number is the period of any periodic point, and the cyclic order on the circle of points of any periodic orbit is the same as that of the points of the rotation orbit on p(f). Further, any trajectory tends to some periodic one both in forward and backward time (a- and -w limit trajectories can be different in this case).
   2) If the rotation number f is irrational, then two options are possible:
   i) either f has a dense orbit, in which case the homeomorphism f is conjugate to a rotation on p(f). In this case, all orbits of f are dense (since this is true for an irrational rotation);
   ii) either f has a Cantor invariant set C which is the unique minimal set of the system. In this case, all trajectories tend to C both in forward and backward time. Moreover, the mapping f is semi-adjoint to a rotation on p(f): for some mapping h of degree 1, p o f =R p (f) o h

   Moreover, the set C is exactly the set of growth points of h, in other words, from the topological point of view, h collapses the complement intervals to C.

9. the crux of the matter is $1 million
10. The fact that no one understands it except for 1 person
11. In foreign policy France..
12. Here Lka answered best of all http://otvet.mail.ru/question/24963208/
13. A brilliant mathematician, the Parisian professor Henri Poincaré was engaged in various areas of this science. Independently and independently of the work of Einstein in 1905, he put forward the main provisions of the Special Theory of Relativity. And he formulated his famous hypothesis back in 1904, so it took about a century to solve it.

    Poincaré was one of the founders of topology, the science of properties geometric shapes, which do not change under deformations that occur without discontinuities. For example, balloon can be easily deformed into a variety of shapes, as is done for children in the park. But you need to cut the ball in order to twist a donut (or, in geometric terms, a torus) out of it; there is no other way. And vice versa: take a rubber donut and try to turn it into a sphere. However, it still won't work. In terms of their topological properties, the surfaces of a sphere and a torus are incompatible, or non-homeomorphic. On the other hand, any surfaces without holes (closed surfaces), on the contrary, are homeomorphic and are capable of transforming into a sphere when deformed.

    If everything about the two-dimensional surfaces of the sphere and torus was decided back in the 19th century, for more multidimensional cases it took much more time. This, in fact, is the essence of the Poincare conjecture, which extends the regularity to multidimensional cases. Simplifying a little, the Poincaré conjecture says: Every simply connected closed n-dimensional manifold is homeomorphic to an n-dimensional sphere. It's funny that the variant with three-dimensional surfaces turned out to be the most difficult. In 1960 the conjecture was proved for dimensions 5 and above, in 1981 for n=4. The stumbling block was precisely three-dimensionality.

    Developing the ideas of William Tristen and Richard Hamilton, proposed by them in the 1980s, Grigory Perelman applied to three-dimensional surfaces special equation smooth evolution. And he was able to show that the original three-dimensional surface (if there are no discontinuities in it) will necessarily evolve into a three-dimensional sphere (this is the surface of a four-dimensional ball, and it exists in a four-dimensional space) . According to a number of experts, this was an idea of ​​a new generation, the solution of which opens up new horizons for mathematical science.

    It is interesting that for some reason Perelman himself did not bother to bring his decision to its final brilliance. Having described the solution as a whole in the preprint The entropy formula for the Ricci flow and its geometric applications in November 2002, in March 2003 he completed the proof and presented it in the preprint Ricci flow with surgery on three-manifolds, and also reported on the method in the series lectures that he read in 2003 at the invitation of a number of universities. None of the reviewers could find errors in the version he proposed, but Perelman did not issue publications in the refereed scientific publication (namely, such, in particular, was necessary condition receiving the Clay Mathematical Institute Prize). But in 2006, based on his method, a whole set of proofs came out in which American and Chinese mathematicians consider the problem in detail and completely, supplement the points omitted by Perelman, and give the final proof of the Poincaré conjecture.

14. The generalized Poincare conjecture states that:
    For any n, any manifold of dimension n is homotopy equivalent to a sphere of dimension n if and only if it is homeomorphic to it.
    The original Poincare conjecture is a special case of the generalized conjecture for n = 3.
    For explanations - go to the forest for mushrooms, Grigory Perelman goes there)
15. The Poincare recurrence theorem is one of the basic theorems of ergodic theory. Its essence is that under a measure-preserving mapping of space onto itself, almost every point will return to its initial neighborhood. The full statement of the theorem is as follows1:
    Let be a measure-preserving transformation of a space with a finite measure and let be a measurable set. Then for any natural
    .
    This theorem has an unexpected consequence: it turns out that if in a vessel divided by a partition into two compartments, one of which is filled with gas and the other is empty, the partition is removed, then after a while all the gas molecules will again gather in the original part of the vessel. The key to this paradox is that some time is on the order of billions of years.
16. he has theorems like cut dogs in Korea ...

    the universe is spherical... http://ru.wikipedia.org/wiki/Poincare, _Henri

    yesterday, scientists announced that the universe is a frozen substance ... and asked for a lot of money to prove this ... again, the Merikos will turn on the printing press ... for the joy of eggheads ...

17. Try to prove where the top and bottom are in weightlessness.
18. Yesterday was great movie on CULTURE, in which this problem was explained on the fingers. Maybe they still have it?

    http://video.yandex.ru/#search?text=РРР СР Р РРРР СРРРwhere=allfilmId=36766495-03-12
    Enter Yandex and write a film about Perelman and go to the film

By school course Everyone is familiar with the concepts of theorems and hypotheses. As a rule, in life the most simple and primitive laws are affected, while mathematicians make very complex assumptions and pose interesting problems. Far from always, they themselves manage to find solutions and evidence, and in some cases, their followers and just colleagues have been struggling with this for many years.

The Clay Institute in 2000 compiled a list of 7 so-called Millennium Problems, similar to the list of hypotheses compiled in 1900. Almost all of those tasks have been solved by now, only one of them has migrated to the updated version. Now the list of problems looks like this:

• the Hodge hypothesis;
• equality of classes P and NP;
• the Poincaré hypothesis;
• Yang-Mills theory;
• the Riemann hypothesis;
• existence and smoothness of the solution of the Navier-Stokes equations;
• the Birch-Swinnerton-Dyer hypothesis.

All of them belong to various disciplines within mathematics and are essential. For example, the Navier-Stokes equations are related to hydrodynamics, but in practice they can describe the behavior of matter in terrestrial magma or come in handy in weather prediction. But all these problems are still looking for their proof or refutation. Except one.

Poincaré's theorem

Explain in simple words, what this problem is, is quite difficult, but you can try. Imagine a sphere, for example, a soap bubble. All points of its surface are equidistant from its center, which does not belong to it. But this is a two-dimensional body, and the hypothesis speaks of a three-dimensional one. It is already impossible to imagine, but we have theoretical mathematics for that. In this case, of course, all points of this body will also be removed from the center.

This problem belongs to topology - the science of the properties of geometric shapes. And one of basic terms it is homeomorphic, that is, a high degree of similarity. To give an example, one can imagine a ball and a torus. One figure cannot be obtained from another in any way, avoiding gaps, but a cone, a cube or a cylinder from the first one will turn out quite easily. Here Poincare's hypothesis is devoted to these metamorphoses with only one difference - we are talking about multidimensional space and bodies.

Story

The French mathematician Henri Poincaré worked in various fields of science. His achievements can be said, for example, by the fact that, quite independently of Albert Einstein, he put forward the main provisions of the special theory of relativity. In 1904, he raised the problem of proving that any three-dimensional body that has some of the properties of a sphere is it, up to a deformation. It was later extended and generalized to become a special case of Thurston's conjecture formulated in 1982.

Wording

Poincaré originally left the following assertion: every simply connected compact three-dimensional manifold without boundary is homeomorphic to a three-dimensional sphere. Later it was expanded and generalized. And yet, for a long time, it was the original problem that caused the most problems, and was solved only 100 years after its appearance.

Interpretation and meaning

We have already talked about what homeomorphism is. Now it is worth talking about compactness and simply connectedness. The first means only that the manifold has limited dimensions, cannot be continuously and infinitely extended.

With regard to single-linkedness, we can try to give a simple example. A two-dimensional sphere - an apple - has one interesting property. If you take an ordinary closed rubber band and attach it to the surface, then by smooth deformation it can be reduced to one point. This is the property of singly connectedness, but it is rather difficult to present it in relation to three-dimensional space.

To put it quite simply, the problem was to prove that being simply connected is a property unique to a sphere. And if, relatively speaking, the experiment with the rubber band ended with such a result, then the body is homeomorphic to it. As for the application of this theory to life, Poincaré believed that the universe is in some sense a three-dimensional sphere.

Proof

It should not be thought that of the dozens of mathematicians who have worked all over the world, no one has moved one iota on this problem. On the contrary, there was progress, and in the end it led to a result. Poincare himself did not have time to finish the work, but his research seriously advanced the entire topology.

In the 1930s, interest in the hypothesis returned. First of all, the wording has been expanded to " n-dimensional space", and then the American Whitehead reported a successful proof, later abandoning it. In the 60-70s, two mathematicians at once - Smale and Stallings - almost simultaneously, but different ways developed a solution for all n greater than 4.

In 1982, a proof was found for 4 as well, leaving only 3. In the same year, Thurston formulated the geometrization conjecture, with the Poincaré theory becoming its particular case.

For 20 years, the Poincaré hypothesis seemed to have been forgotten. In 2002, Russian mathematician Grigory Perelman presented a solution in in general terms, after six months making some additions. Later, this proof was checked and brought "to a shine" by American and Chinese scientists. And Perelman himself seemed to have lost all interest in the problem, although he decided more common task about geometrization, for which the Poincare conjecture is only a particular case.

Recognition and ratings

Of course, this immediately became a sensation, because the solution to one of the Millennium Problems simply could not go unnoticed. Even more surprising was the fact that Grigory Perelman refused all awards and prizes, saying that he already had a great life. In the minds of the townsfolk, he immediately became an example of that very half-crazy genius who is only interested in science.

All this caused a lot of discussion in the press and the media that the popularity of the mathematician began to weigh him down. In the summer of 2014, there was information that Perelman had left to work in Sweden, but this turned out to be just a rumor, he still lives modestly in St. Petersburg and hardly communicates with anyone. Among the awards given to him were not only the Clay Institute Prize, but also the prestigious Fields Medal, but he refused everything. However, Hamilton, who, according to Perelman, made no less contribution to the proof, was also not forgotten. In 2009 and 2011, he also received some prestigious awards and prizes.

Reflection in culture

Despite the fact that for ordinary people both the statement and the solution of this problem make little sense, the proof became known rather quickly. In 2008, on this occasion, Japanese director Masahito Kasuga filmed the documentary film "The Enchantment of the Poincaré Hypothesis", dedicated to a century of attempts to solve this problem.

Many mathematicians involved in this problem took part in the filming, but the main character, Grigory Perelman, did not want to do this. More or less close acquaintances of his were also involved in the filming. Documentary, having appeared on the screens in the wake of public outcry about the scientist's refusal to accept the award, he gained fame in certain circles and also received several awards. As for popular culture, simple people people are still wondering what arguments the Petersburg mathematician was guided by when he refused to take money when he could give it, for example, to charity.

Henri Poincaré (1854-1912), one of the greatest mathematicians, in 1904 formulated the famous idea of ​​a deformed three-dimensional sphere and, in the form of a little marginal note placed at the end of a 65 page article on a completely different issue, scrawled a few lines of a rather strange conjecture with the words: "Well, this question can take us too far" ...

Marcus Du Sotoy of Oxford University believes that Poincaré's theorem- "This the central problem of mathematics and physics , trying to understand what form may be Universe It's very hard to get close to her."

Once a week, Grigory Perelman traveled to Princeton to take part in a seminar at the Institute for Advanced Study. At the seminar, one of the mathematicians Harvard University answers Perelman’s question: “The theory of William Thurston (1946-2012, mathematician, works in the field of “Three-dimensional geometry and topology”), called the geometrization hypothesis, describes all possible three-dimensional surfaces and is a step forward compared to the Poincaré hypothesis. If you prove the assumption of William Thurston, then the Poincare conjecture will open all its doors to you and more its solution will change the entire topological landscape of modern science ».

Six leading American universities in March 2003 invite Perelman to read a series of lectures explaining his work. In April 2003, Perelman makes a scientific tour. His lectures become an outstanding scientific event. John Ball (chairman of the International Mathematical Union), Andrew Wiles (mathematician, works in the field of arithmetic of elliptic curves, proved Fermat's theorem in 1994), John Nash (mathematician working in the field of game theory and differential geometry) come to Princeton to listen to him.

Grigory Perelman managed to solve one of the seven tasks of the millennium and describe mathematically the so-called the formula of the universe , to prove the Poincaré conjecture. The brightest minds fought over this hypothesis for more than 100 years, and for the proof of which the world mathematical community (the Clay Mathematical Institute) promised $ 1 million. It was presented on June 8, 2010. Grigory Perelman did not appear on it, and the world mathematical community " jaws dropped."

In 2006, for solving the Poincare conjecture, the mathematician was awarded the highest mathematical award - the Fields Prize (Fields Medal). John Ball personally visited St. Petersburg in order to persuade him to accept the award. He refused to accept it with the words: Society is unlikely to seriously appreciate my work».

“The Fields Prize (and medal) is awarded once every 4 years at each international mathematical congress to young scientists (under 40 years old) who have made a significant contribution to the development of mathematics. In addition to the medal, the awardees are awarded 15,000 Canadian dollars ($13,000).”

In its original formulation, the Poincaré conjecture reads as follows: "Every simply connected compact three-dimensional manifold without boundary is homeomorphic to a three-dimensional sphere." AT translation into common language, this means that any three-dimensional object, for example, a glass, can be transformed into a ball by deformation alone, that is, it will not need to be cut or glued. In other words, Poincaré suggested that space is not three-dimensional, but contains significantly more measurements , and Perelman 100 years later proved it mathematically .

Grigory Perelman's expression of Poincaré's theorem on the transformation of matter into another state, form is similar to the knowledge set forth in Anastasia Novykh's book "Sensei IV": needles". As well as the ability to control the material Universe through transformations introduced by the Observer from controlling dimensions above the sixth (from 7 to 72 inclusive) (report "" topic "Ezoosmic Grid").

Grigory Perelman was distinguished by the austerity of life, the severity of ethical requirements both for himself and for others. Looking at him, one gets the feeling that he is only bodily resides in common with all other contemporaries space , a Spiritually in some other , where even for $1 million don't go for the most "innocent" compromises with conscience . And what kind of space is this, and is it possible to even look at it from the corner of your eye? ..

Exceptional the importance of the hypothesis, put forward about a century ago by a mathematician Poincaré, concerns three-dimensional structures and is a key element contemporary research foundations of the universe . This riddle, according to experts from the Clay Institute, is one of the seven fundamentally important for the development of mathematics of the future.

Perelman, rejecting medals and prizes, asks: “Why do I need them? They are absolutely useless to me. Everyone understands that if the proof is correct, then no other recognition is required. Until I developed suspicion, I had the choice of either speaking out loud about the disintegration of the mathematical community as a whole, due to its low moral level, or saying nothing and allowing myself to be treated like cattle. Now, when I have become more than suspicious, I cannot remain a cattle and continue to be silent, so I can only leave.

In order to do modern mathematics, you need to have a totally pure mind, without the slightest admixture that disintegrates it, disorients it, replaces values, and accepting this award means demonstrating weakness. The ideal scientist is engaged only in science, does not care about anything else (power and capital), he must have a pure mind, and for Perelman there is no greater importance than living in accordance with this ideal. Is this whole idea with millions useful for mathematics, and does a real scientist need such an incentive? And this desire of capital to buy and subjugate everything in this world is not insulting? Or you can sell its purity for a million? Money, no matter how much there is, is equivalent the truth of the Soul ? After all, we are dealing with an a priori assessment of problems that money simply should not have to do with, right?! To make of all this something like a lotto-million, or a tote, means to indulge the disintegration of the scientific, and indeed the human community as a whole (see the report and the last 50 pages in the AllatRa book about the way to build a creative society). And the money (energy) that businessmen are ready to give to science, if it is necessary to use it, is it correct, or something, without humiliating The Spirit of True Service , whatever one may say, an invaluable monetary equivalent: “ What is a million, compared , with purity, or Majesty those spheres (for the dimensions of the global Universe and the Spiritual World, see the book "AllatRa" and report ) , in which unable to penetrate even human imagination (mind) ?! What is a million starry sky for time?

Let us give an interpretation of the remaining terms appearing in the formulation of the hypothesis:

- Topology- (from the Greek. topos - place and logos - teaching) - a branch of mathematics that studies the topological properties of figures, i.e. properties that do not change under any deformations produced without discontinuities and gluings (more precisely, under one-to-one and continuous mappings). Examples of topological properties of figures are the dimension, the number of curves that bound a given area, and so on. So, a circle, an ellipse, a square contour have the same topological properties, since these lines can be deformed one into the other in the manner described above; at the same time, the ring and the circle have different topological properties: the circle is bounded by one contour, and the ring by two.

- Homeomorphism(Greek ομοιο - similar, μορφη - shape) - a one-to-one correspondence between two topological spaces, in which both mutually inverse mappings defined by this correspondence are continuous. These mappings are called homeomorphic or topological mappings, as well as homeomorphisms, and the spaces are said to belong to the same topological type are called homeomorphic, or topologically equivalent.

- 3-manifold without boundary. This is such a geometric object, in which each point has a neighborhood in the form of a three-dimensional ball. Examples of 3-manifolds are, firstly, the entire three-dimensional space, denoted by R3 , as well as any open sets of points in R3 , for example, the interior of a solid torus (donut). If we consider a closed solid torus, i.e. add its boundary points (the surface of a torus), then we will already get a manifold with a boundary - the boundary points do not have neighborhoods in the form of a ball, but only in the form of a half of the ball.

- Full torus (full torus) — geometric body, homeomorphic to the product of a two-dimensional disk and a circle D 2 * S 1 . Informally, a solid torus is a donut, while a torus is only its surface (a hollow chamber of a wheel).

- singly connected. It means that any continuous closed curve located entirely within a given manifold can be smoothly contracted to a point without leaving this manifold. For example, an ordinary two-dimensional sphere in R3 is simply connected (an elastic band, arbitrarily applied to the surface of an apple, can be contracted by a smooth deformation to one point without tearing the elastic band from the apple). On the other hand, the circle and the torus are not simply connected.

- Compact. A manifold is compact if any of its homeomorphic images has bounded dimensions. For example, an open interval on a line (all points of a segment except its ends) is not compact, since it can be continuously extended to an infinite line. But a closed segment (with ends) is a compact manifold with boundary: for any continuous deformation, the ends go into some certain points, and the entire segment must pass into a bounded curve connecting these points.

Ilnaz Basharov

Literature:

Report "PRIMORDIAL ALLATRA PHYSICS" of the international group of scientists of the ALLATRA International Public Movement, ed. Anastasia Novykh, 2015;

New. A. "AllatRa", K.: AllatRa, 2013