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Grigory Perelman proved that there is no God. Mathematician Perelman Yakov: contribution to science

« Millennium Challenge”, solved by a Russian mathematical genius, is related to the origin of the Universe. Not every mathematician is given to understand the essence of the riddle ...

MIND GAME

Until recently, mathematics did not promise either glory or wealth to its "priests". They even Nobel Prize didn't give. There is no such nomination. Indeed, according to a very popular legend, Nobel's wife once cheated on him with a mathematician. And in retaliation, the rich man deprived all their chicane brethren of his respect and prize money.

The situation changed in 2000. The private Clay Mathematics Institute has selected seven of the most difficult tasks and promised to pay a million dollars for each decision.

Mathematicians were treated with respect. In 2001, the screens even released the film "A Beautiful Mind", the main character of which was a mathematician.

Now only people far from civilization are not aware: one of the promised millions - the very first one - has already been awarded. The prize was awarded to a Russian citizen, a resident of St. Petersburg Grigory Perelman. He proved the Poincaré conjecture, a puzzle that defied anyone for over 100 years and which, through his efforts, became a theorem.

Our cute 44-year-old bearded man wiped his nose around the world. And now continues to keep it - the world - in suspense. Since it is not known whether the mathematician will honestly deserve a million dollars or refuse. The progressive public in many countries is naturally agitated. At least the newspapers of all continents chronicle financial and mathematical intrigue.

And against the background of these fascinating activities - fortune-telling and sharing other people's money - the meaning of Perelman's achievement was somehow lost. The president of the Clay Institute, Jim Carlson, of course, stated at one time, they say, the goal prize fund- not so much a search for answers as an attempt to raise the prestige of mathematical science and to interest young people in it. But still, what is the point?

Grisha in his youth - even then he was a genius.

POINCARE HYPOTHESIS - WHAT IS IT?

The riddle, solved by the Russian genius, affects the foundations of the section of mathematics called topology. It - topology - is often called "geometry on a rubber sheet." It deals with properties geometric shapes, which are preserved if the form is stretched, twisted, bent. In other words, it is deformed without breaks, cuts and glues.

Topology is important for mathematical physics because it allows us to understand the properties of space. Or evaluate it without being able to look at the shape of this space from the outside. For example, our universe.

When explaining the Poincare conjecture, they start like this: imagine a two-dimensional sphere - take a rubber disk and pull it over the ball. So that the circumference of the disk is collected at one point. Similarly, for example, you can pull off a sports backpack with a cord. The result is a sphere: for us - three-dimensional, but from the point of view of mathematics - only two-dimensional.

Then they offer to pull the same disk on 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.

As he wrote in his popular book another Russian mathematician, Vladimir Uspensky, "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 is for Vasily Ivanovich from the well-known anecdote square trinomial."

So, according to the Poincaré hypothesis, a three-dimensional sphere is the only three-dimensional thing whose surface can be pulled into one point by some kind of hypothetical "hypercord".

Grigory Perelman: - Just think, Newton's binomial ...

Jules Henri Poincare 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 it allows us to quite reasonably assume that it is the same three-dimensional sphere.

But if the Universe is the only "figure" that can be contracted to a point, then, probably, it can also be stretched from a point. Which serves as an indirect confirmation of the Big Bang theory, which claims that the Universe originated just from the point.

It turns out that Perelman, together with Poincare, upset the so-called creationists - supporters divine beginning universe. And they spilled water on the mill of materialist physicists.

The ingenious mathematician from St. Petersburg, Grigory Perelman, who became famous throughout the world for proving the Poincaré conjecture, finally explained his refusal of the million dollar prize awarded for this. As states " TVNZ", the reclusive scientist revealed himself in a conversation with a journalist and producer of the film company "President-Film", which, with the consent of Perelman, will shoot the feature film "Formula of the Universe" about him.

Alexander Zabrovsky was lucky to talk to the great mathematician - he left Moscow for Israel a few years ago and guessed first to contact Grigory Yakovlevich's mother through the Jewish community of St. Petersburg, having helped her. She talked to her son, and after her good characterization, he agreed to a meeting. This can truly be called an achievement - the journalists could not "catch" the scientist, although they spent days sitting at his entrance.

As Zabrovsky told the newspaper, Perelman gave the impression of "an absolutely sane, healthy, adequate and normal person": "Realistic, pragmatic and sensible, but not devoid of sentimentality and excitement ... Everything that was attributed to him in the press, as if he was "out of his mind", - complete nonsense! He knows exactly what he wants, and knows how to achieve the goal. "

The film, for which the mathematician made contact and agreed to help, will not be about himself, but about the cooperation and confrontation of the three main world mathematical schools: Russian, Chinese and American, which are the most advanced in the path of studying and managing the Universe.

When asked why Perelman refused a million, he replied:

"I know how to manage the Universe. And tell me - why should I run after a million?"

The scientist is offended, as he is called in the Russian press

Perelman explained that he does not communicate with journalists, because they are not concerned with science, but with personal and domestic issues - from the reasons for refusing a million to the question of cutting hair and nails.

Specifically, he does not want to contact the Russian media because of the disrespectful attitude towards him. For example, in the press they call him Grisha, and such familiarity offends.

Grigory Perelman said that since school years used to what is called "brain training". Remembering how, being a "delegate" from the USSR, he received gold medal at the Mathematical Olympiad in Budapest, he said: “We tried to solve problems where the ability to think abstractly was an indispensable condition.

It was in this abstraction from mathematical logic that main point daily workouts. To find the right solution, it was necessary to imagine a "piece of the world".

As an example of such a "difficult" task, he cited the following: "Remember biblical legend about how Jesus Christ walked on water, like on dry land. So I had to calculate how fast he had to move through the waters so as not to fall through.

Since then, Perelman has devoted all his activities to studying the problem of studying the properties of the three-dimensional space of the Universe: “This is very interesting. I am trying to embrace the immensity.

The scientist wrote his dissertation under the guidance of Academician Aleksandrov. "The topic was simple: 'Saddle surfaces in Euclidean geometry'. Can you imagine surfaces that are equal in size and unevenly spaced from each other at infinity? We need to measure the 'hollows' between them," the mathematician explained.

What does Perelman's discovery mean, frightening the intelligence services of the world

The "Formula of the Universe" Poincare's statement is called because of its importance in the study of complex physical processes in the theory of the universe and because it gives an answer to the question about the shape of the Universe. This evidence will play a big role in the development of nanotechnology."

“I learned how to calculate voids, together with my colleagues we will learn the mechanisms for filling social and economic “voids,” he said. “Voids are everywhere. They can be calculated, and this provides great opportunities ...

According to the publication, the scale of what Grigory Yakovlevich discovered, which actually steps ahead of today's world science, has made him the object of constant interest of special services, not only Russian, but also foreign.

He comprehended some super-knowledge that helps to understand the universe. And here questions of this kind arise: "What will happen if his knowledge finds practical implementation?"

In fact, the secret services need to know - is Perelman, or rather, his knowledge, a threat to humanity? After all, if with the help of his knowledge it is possible to turn the Universe into a point, and then unfold it, then we can die or be reborn in a different capacity? And then will we be? And do we need to manage the universe at all?

AND AT THIS TIME

Genius mom: "Don't ask us questions about money!"

When it became known that the mathematician was awarded the Millennium Prize, a crowd of journalists gathered in front of his door. Everyone wanted to personally congratulate Perelman and find out if he would take his legitimate million.

We knocked on the flimsy door for a long time (if only we could replace it with premium money), but the mathematician did not open it. But his mother quite intelligibly dotted the "i" right from the hallway.

We do not want to talk to anyone and are not going to give any interviews, - Lyubov Leibovna shouted. - And do not ask us questions about this award and money.

People living in the same entrance were very surprised to see a sudden interest in Perelman.

Is our Grisha married? one of the neighbors chuckled. - Oh, I got an award. Again. No, he won't take it. He doesn’t need anything at all, lives on a penny, but is happy in his own way.

They say that on the eve of the mathematician was seen with full packages of products from the store. He was preparing to "keep the siege" with his mother. The last time, when the hype about the award began in the press, Perelman did not leave the apartment for three weeks.

BY THE WAY

For what else will they give a million dollars ...

In 1998, with the funds of billionaire Landon T. Clay, the Clay Mathematics Institute was founded in Cambridge (USA) to popularize mathematics. On May 24, 2000, the institute's experts chose the seven most puzzling problems, in their opinion. And they appointed one million dollars for each.

The list is named .

1. Cook's problem

It is necessary to determine whether the verification of the correctness of the solution of a problem can be longer than obtaining the solution itself. This logical task is important for specialists in cryptography - data encryption.

2. Riemann hypothesis

There are so-called prime numbers, such as 2, 3, 5, 7, etc., which are only divisible by themselves. How many there are is not known. Riemann believed that this could be determined and the regularity of their distribution could be found. Whoever finds it will also provide cryptography services.

3. Birch and Swinnerton-Dyer hypothesis

The problem is related to solving equations with three unknowns raised to a power. We need to figure out how to solve them, no matter how difficult.

4. Hodge hypothesis

In the twentieth century, mathematicians discovered a method for studying the form complex objects. The idea is to use simple “bricks” instead of the object itself, which are glued together and form its likeness. We need to prove that this is always admissible.

5. Navier - Stokes equations

It is worth remembering them on the plane. The equations describe the air currents that keep it in the air. Now the equations are solved approximately, according to approximate formulas. It is necessary to find exact ones and prove that in three-dimensional space there is a solution of the equations, which is always true.

6. Yang-Mills equations

There is a hypothesis in the world of physics: if an elementary particle has a mass, then its lower limit also exists. But which one is not clear. You need to get to him. This is perhaps the most difficult task. To solve it, it is necessary to create a "theory of everything" - equations that combine all the forces and interactions in nature. Whoever succeeds will certainly receive the Nobel Prize.

The last great achievement of pure mathematics is the proof of the Poincaré conjecture, expressed in 1904 and stating: “every connected, simply connected, compact three-dimensional manifold without boundary, is homeomorphic to the sphere S 3 ” by Grigory Perelman from St. Petersburg in 2002–2003.

There are several terms in this phrase which I will try to explain in such a way that their general meaning becomes clear to non-mathematicians (I assume that the reader has finished high school and still remembers something from school mathematics).

Let's start with the concept of homeomorphism, which is central in topology. In general, topology is often defined as "rubber geometry", i.e., as the science of the properties of geometric images that do not change during smooth deformations without gaps and gluing, or rather, if it is possible to establish a one-to-one and one-to-one correspondence between two objects .

The main idea is easiest to explain using the classic example of a mug and a bagel. The first can be turned into the second by continuous deformation.

These figures clearly show that the mug is homeomorphic to the donut, and this fact is true both for their surfaces (two-dimensional manifolds, called a torus) and for filled bodies (three-dimensional manifolds with boundary).

Let us give an interpretation of the rest of the terms appearing in the formulation of the hypothesis.

A three-dimensional 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 R 3 , as well as any open sets points in R 3 , 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 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.
Connected. The concept of connectivity is the simplest here. A manifold is connected if it consists of one piece, or, what is the same, any two of its points can be connected by a continuous line that does not go beyond its limits.
Simply connected. The notion of single-connectedness is more complicated. 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 R 3 is simply connected (an elastic band, arbitrarily attached 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 a boundary: for any continuous deformation, the ends go to some specific points, and the entire segment must go into a bounded curve connecting these points.

Dimension manifolds is the number of degrees of freedom at the point that "lives" on it. Each point has a neighborhood in the form of a disk of the corresponding dimension, i.e., an interval of a line in the one-dimensional case, a circle on the plane in the two-dimensional case, a ball in the three-dimensional case, etc. From the point of view of topology, there are only two one-dimensional connected manifolds without boundary: this is the line and circle. Of these, only the circle is compact.

An example of a space that is not a manifold is, for example, a pair of intersecting lines - after all, at the point of intersection of two lines, any neighborhood has the shape of a cross, it does not have a neighborhood that would itself be just an interval (and all other points have such neighborhoods ). Mathematicians in such cases say that we are dealing with a singular manifold, which has one singular point.

Two-dimensional compact manifolds are well known. If we consider only oriented manifolds without boundary, then from a topological point of view they form a simple, albeit infinite, list: and so on. Each such manifold is obtained from a sphere by gluing several handles, the number of which is called the genus of the surface.

The figure shows surfaces of genus 0, 1, 2, and 3. How does a sphere stand out from all the surfaces in this list? It turns out that it is simply connected: on a sphere, any closed curve can be contracted to a point, and on any other surface, it is always possible to indicate a curve that cannot be contracted to a point along the surface.

It is curious that three-dimensional compact manifolds without boundary can also be classified in a certain sense, i.e., arranged in a certain list, although not as straightforward as in the two-dimensional case, but having a rather complex structure. However, the 3D sphere S 3 stands out in this list in exactly the same way as the 2D sphere in the list above. The fact that any curve on S 3 contracts to a point is just as easy to prove as in the two-dimensional case. But the converse assertion, namely, that this property is unique precisely for the sphere, i.e., that there are non-contractible curves on any other three-dimensional manifold, is very difficult and exactly constitutes the content of the Poincare conjecture we are talking about.

It is important to understand that the manifold can live on its own, it can be thought of as an independent object, not nested anywhere. (Imagine living two-dimensional beings on the surface of an ordinary sphere, unaware of the existence of a third dimension.) Fortunately, all of the two-dimensional surfaces from the above list can be embedded in the usual R 3 space, which makes them easier to visualize. For the 3-sphere S 3 (and in general for any compact 3-manifold without boundary) this is no longer the case, so some effort is needed to understand its structure.

Apparently simplest way to explain the topological structure of the three-dimensional sphere S 3 is with the help of one-point compactification. Namely, the three-dimensional sphere S 3 is a one-point compactification of the usual three-dimensional (unbounded) space R 3 .

Let us explain this construction first on simple examples. Let's take an ordinary infinite straight line (a one-dimensional analogue of space) and add one “infinitely distant” point to it, assuming that when moving along a straight line to the right or left, we eventually get to this point. From a topological point of view, there is no difference between an infinite line and a bounded open segment (without endpoints). Such a segment can be continuously bent in the form of an arc, bring the ends closer together and glue the missing point into the junction. We get, obviously, a circle - a one-dimensional analogue of a sphere.

Similarly, if I take an infinite plane and add one point at infinity, to which all lines of the original plane, passing in any direction, tend, then we get a two-dimensional (ordinary) sphere S 2 . This procedure can be observed by means of a stereographic projection, which assigns to each point P of the sphere, with the exception of the north pole of N, a certain point of the plane P.

Thus, a sphere without one point is topologically the same as a plane, and adding a point turns the plane into a sphere.

In principle, exactly the same construction is applicable to a three-dimensional sphere and three-dimensional space, only for its implementation it is necessary to enter the fourth dimension, and this is not so easy to depict on the drawing. So I will limit myself verbal description one-point compactification of the space R 3 .

Imagine that to our physical space (which we, following Newton, consider to be an unlimited Euclidean space with three coordinates x, y, z) has one point “at infinity” added in such a way that when moving along a straight line in any direction, you you fall (i.e., each spatial line closes into a circle). Then we get a compact three-dimensional manifold, which is, by definition, the sphere S 3 .

It is easy to see that the sphere S 3 is simply connected. Indeed, any closed curve on this sphere can be shifted slightly so that it does not pass through the added point. Then we get a curve in the usual space R 3 , which is easily contracted to a point by means of homotheties, i.e. continuous contraction in all three directions.

To understand how the manifold S 3 is structured, it is very instructive to consider its partition into two solid tori. If the solid torus is omitted from the space R 3, then something not very clear remains. And if the space is compactified into a sphere, then this complement also turns into a solid torus. That is, the sphere S 3 is divided into two solid tori having a common boundary - a torus.

Here is how it can be understood. Let's embed the torus in R 3 as usual, in the form of a round donut, and draw a vertical line - the axis of rotation of this donut. We draw an arbitrary plane through the axis, it will intersect our solid torus in two circles shown in the figure in green, and the additional part of the plane is divided into a continuous family of red circles. Among them is the central axis, highlighted in bolder, because in the sphere S 3 the line closes into a circle. A three-dimensional picture is obtained from this two-dimensional one by rotating around an axis. A complete set of rotated circles will then fill a three-dimensional body, homeomorphic to a solid torus, only looking unusual.

In fact, the central axis will be an axial circle in it, and the rest will play the role of parallels - circles that make up the usual solid torus.

In order to have something to compare the 3-sphere with, I will give another example of a compact 3-manifold, namely a three-dimensional torus. A three-dimensional torus can be constructed as follows. Let's take an ordinary three-dimensional cube as a source material:

It has three pairs of faces: left and right, top and bottom, front and back. In each pair of parallel faces, we identify in pairs the points obtained from each other by transferring along the edge of the cube. That is, we will assume (purely abstractly, without applying physical deformations) that, for example, A and A "are the same point, and B and B" are also one point, but different from point A. All internal points of the cube we will consider as usual. The cube itself is a manifold with a boundary, but after the gluing done, the boundary closes on itself and disappears. Indeed, the neighborhoods of the points A and A" in the cube (they lie on the left and right shaded faces) are the halves of the balls, which, after gluing the faces together, merge into a whole ball, which serves as a neighborhood of the corresponding point of the three-dimensional torus.

In order to feel the structure of a 3-torus based on ordinary ideas about physical space, you need to choose three mutually perpendicular directions: forward, left and up - and mentally consider, as in science fiction stories, that when moving in any of these directions, a rather long, but finite time , we will return to the starting point, but from the opposite direction. This is also a “compactification of space”, but not a one-point one, used earlier to construct a sphere, but more complex.

There are non-contractible paths on the 3-torus; for example, this is the segment AA" in the figure (on the torus it depicts a closed path). It cannot be contracted, because for any continuous deformation, the points A and A" must move along their faces, remaining strictly opposite each other (otherwise the curve will open).

Thus, we see that there are simply connected and non-simply connected compact 3-manifolds. Perelman proved that a simply connected manifold is exactly one.

The starting point of the proof is the use of the so-called "Ricci flow": we take a simply connected compact 3-manifold, endow it with an arbitrary geometry (i.e., introduce some metric with distances and angles), and then consider its evolution along the Ricci flow. Richard Hamilton, who proposed this idea in 1981, hoped that with this evolution our manifold would turn into a sphere. It turned out that this is not true - in the three-dimensional case, the Ricci flow is capable of spoiling the manifold, i.e., making it a little manifold (something with singular points, as in the above example of intersecting lines). Perelman, by overcoming incredible technical difficulties, using the heavy apparatus of partial differential equations, managed to amend the Ricci flow near singular points in such a way that during evolution the topology of the manifold does not change, there are no singular points, and in the end, it turns into a round sphere. But it is necessary to explain, finally, what this flow of Ricci is. The flows used by Hamilton and Perelman refer to a change in the intrinsic metric on an abstract manifold, and this is rather difficult to explain, so I will limit myself to describing the "external" Ricci flow on one-dimensional manifolds embedded in a plane.

Imagine a smooth closed curve on the Euclidean plane, choose a direction on it, and consider at each point a tangent vector of unit length. Then, when going around the curve in the chosen direction, this vector will rotate with some angular velocity, which is called curvature. Where the curve is steeper, the curvature (in absolute value) will be greater, and where it is smoother, the curvature will be less.

The curvature will be considered positive if the velocity vector turns towards the inner part of the plane divided by our curve into two parts, and negative if it turns outward. This convention is independent of the direction in which the curve is traversed. At inflection points where the rotation changes direction, the curvature will be 0. For example, a circle of radius 1 has a constant positive curvature of 1 (measured in radians).

Now let's forget about tangent vectors and attach to each point of the curve, on the contrary, a vector perpendicular to it, equal in length to the curvature at a given point and directed inward if the curvature is positive, and outward if it is negative, and then we will force each point to move in the direction of the corresponding vector with speed proportional to its length. Here is an example:

It turns out that any closed curve on the plane behaves in a similar way during such an evolution, i.e., it eventually turns into a circle. This is the proof of the one-dimensional analogue of the Poincare conjecture using the Ricci flow (however, the statement itself in this case is already obvious, just the method of proof illustrates what happens in dimension 3).

In conclusion, we note that Perelman's argument proves not only the Poincaré conjecture, but also the much more general Thurston geometrization conjecture, which in in a certain sense describes the structure of all compact 3-manifolds in general. But this subject lies beyond the scope of this elementary article.

For lack of space, I will not talk about non-orientable manifolds, an example of which is the famous Klein bottle - a surface that cannot be embedded in a space without self-intersections.

The Clay Institute of Mathematics awarded Grigory Perelman the Millennium Prize, thus officially recognizing the proof of the Poincaré conjecture, performed by a Russian mathematician, as correct. It is noteworthy that in doing so, the institute had to break its own rules - according to them, only an author who has published his work in peer-reviewed journals can claim to receive about a million dollars, this is exactly the size of the prize. Grigory Perelman's work never formally saw the light of day - it remained as a set of several preprints on the arXiv.org website (one, two and three). However, it is not so important what caused the institute's decision - the award of the Millennium Prize puts an end to the history of more than 100 years.

Mug, donut and some topology

Before finding out what the Poincaré conjecture is, it is necessary to understand what kind of branch of mathematics - topology - to which this very hypothesis belongs. The topology of manifolds deals with the properties of surfaces that do not change under certain deformations. Let's explain with a classic example. Suppose the reader has a donut and an empty cup in front of him. From the point of view of geometry and common sense, these are different objects, if only because you won’t be able to drink coffee from a donut with all your desire.

However, the topologist will say that the cup and donut are the same thing. And he will explain it this way: imagine that a cup and a donut are surfaces that are hollow inside, made of a very elastic material (a mathematician would say that there is a pair of compact two-dimensional manifolds). Let's conduct a speculative experiment: first we inflate the bottom of the cup, and then its handle, after which it will turn into a torus (this is how the donut shape is mathematically called). You can see how this process looks like.

Of course, an inquisitive reader has a question: since surfaces can be wrinkled, how can they be distinguished? After all, for example, it is intuitively clear - no matter how you imagine a torus, you cannot get a sphere from it without gaps and gluings. Here the so-called invariants come into play - surface characteristics that do not change under deformation - a concept necessary for the formulation of the Poincaré hypothesis.

Common sense tells us that a hole distinguishes a torus from a sphere. However, a hole is far from a mathematical concept, so it needs to be formalized. This is done as follows - imagine that we have a very thin elastic thread on the surface that forms a loop (in this speculative experiment, unlike the previous one, we consider the surface itself to be solid). We will move the loop without tearing it off the surface and without breaking it. If the thread can be contracted to a very small circle (almost a point), then the loop is said to be contractible. Otherwise, the loop is called non-retractable.

The fundamental group of a torus is denoted by n 1 (T 2). Because it is non-trivial, the mouse's arms form a non-retractable loop. The sadness on the face of the animal is the result of the realization of this fact.

So, it’s easy to see that any loop on a sphere is contractible (you can see how it looks approximately), but for a torus this is no longer the case: there are as many as two loops on a donut - one is threaded into a hole, and the other bypasses the hole "along the perimeter ", - which cannot be pulled. In this picture, examples of non-contractible loops are shown in red and purple respectively. When there are loops on the surface, mathematicians say that "the fundamental group of a variety is non-trivial", and if there are no such loops, then it is trivial.

Now, in order to honestly formulate the Poincare conjecture, the inquisitive reader has to be patient a little more: we need to figure out what a three-dimensional manifold in general and a three-dimensional sphere in particular are.

Let's go back for a moment to the surfaces we discussed above. Each of them can be cut into such small pieces that each will almost resemble a piece of the plane. Since the plane has only two dimensions, the manifold is also said to be two-dimensional. A three-dimensional manifold is a surface that can be cut into small pieces, each of which is very similar to a piece of ordinary three-dimensional space.

chief" actor"Hypothesis is a three-dimensional sphere. It is probably impossible to imagine a three-dimensional sphere as an analogue of an ordinary sphere in four-dimensional space without losing your mind. However, it is quite easy to describe this object, so to speak, "in parts" quite easily. Everyone who seen a globe, they know that an ordinary sphere can be glued together from the northern and southern hemisphere along the equator. So, a three-dimensional sphere is glued together from two balls (northern and southern) along a sphere, which is an analogue of the equator.

On three-dimensional manifolds, one can consider the same loops that we took on ordinary surfaces. So, the Poincaré conjecture states: "If the fundamental group of a three-dimensional manifold is trivial, then it is homeomorphic to a sphere." The incomprehensible phrase "homeomorphic to a sphere" translated into informal language means that the surface can be deformed into a sphere.

A bit of history

Generally speaking, in mathematics it is possible to formulate a large number of complex statements. However, what makes this or that hypothesis great, distinguishes it from the rest? Oddly enough, but the great hypothesis is distinguished by a large number of incorrect proofs, each of which contains a great error - inaccuracy, which often leads to the emergence of a whole new section of mathematics.

So, initially Henri Poincaré, who, among other things, was distinguished by the ability to make brilliant mistakes, formulated the hypothesis in a slightly different form than we wrote above. Some time later, he gave a counterexample to his assertion, which became known as the homological Poincaré 3-sphere, and in 1904 formulated a conjecture already in modern form. By the way, quite recently, scientists adapted the sphere in astrophysics - it turned out that the Universe may well turn out to be a homologous Poincaré 3-sphere.

It must be said that the hypothesis did not cause much excitement among fellow geometers. So it was until 1934, when the British mathematician John Henry Whitehead presented his version of the proof of the hypothesis. Very soon, however, he himself found an error in the reasoning, which later led to the emergence of the whole theory of Whitehead manifolds.

After that, the glory of an extremely difficult task was gradually entrenched in the hypothesis. Many great mathematicians tried to take it by storm. For example, the American R.H.Bing, a mathematician who (absolutely officially) had initials written instead of a name in documents. He made several unsuccessful attempts to prove the hypothesis, formulating his own statement during this process - the so-called "property P conjecture" (Property P conjecture). It is noteworthy that this statement, which was considered by Bing as an intermediate one, turned out to be almost more complicated than the proof of the Poincaré conjecture itself.

There were among the scientists and people who laid down their lives to prove this mathematical fact. For example, the famous mathematician of Greek origin Christos Papakiriakopoulos. For more than ten years, while working at Princeton, he unsuccessfully tried to prove the conjecture. He died of cancer in 1976.

It is noteworthy that the generalization of the Poincaré conjecture to manifolds of dimensions above three turned out to be noticeably simpler than the original - extra dimensions made it easier to manipulate manifolds. Thus, for n-dimensional manifolds (when n is at least 5), the conjecture was proved by Stephen Smale in 1961. For n = 4, the conjecture was proved by a completely different method from Smale's in 1982 by Michael Friedman. For his proof, the latter received the Fields Medal - the highest award for mathematicians.

The works described are far from full list attempts to solve more than a century of hypotheses. And although each of the works led to the emergence of a whole direction in mathematics and can be considered successful and significant in this sense, only the Russian Grigory Perelman managed to finally prove the Poincaré conjecture.

Perelman and proof

In 1992, Grigory Perelman, then an employee of the Mathematical Institute. Steklov, got to the lecture of Richard Hamilton. The American mathematician talked about Ricci flows - a new tool for studying Thurston's geometrization conjecture - a fact from which the Poincaré conjecture was obtained as a simple consequence. These flows, constructed in a sense by analogy with the heat transfer equations, caused the surfaces to deform over time in much the same way as we deformed two-dimensional surfaces at the beginning of this article. It turned out that in some cases the result of such a deformation was an object whose structure is easy to understand. The main difficulty was that during the deformation, singularities with infinite curvature arose, analogous in some sense to black holes in astrophysics.

After the lecture, Perelman approached Hamilton. He later said that Richard pleasantly surprised him: “He smiled and was very patient. He even told me some facts that were published only a few years later. He did this without hesitation. His openness and kindness amazed me. I can’t say that most modern mathematicians behave like this."

After a trip to the United States, Perelman returned to Russia, where he began to work on solving the problem of singularities of Ricci flows and proving the geometrization hypothesis (and not at all on the Poincaré hypothesis) in secret. It is not surprising that the appearance of Perelman's first preprint on November 11, 2002 shocked the mathematical community. After some time, a couple more works appeared.

After that, Perelman withdrew from the discussion of evidence and even, they say, stopped doing mathematics. He did not interrupt his solitary lifestyle even in 2006, when he was awarded the Fields Medal, the most prestigious award for mathematicians. It makes no sense to discuss the reasons for this behavior of the author - a genius has the right to behave strangely (for example, being in America, Perelman did not cut his nails, allowing them to grow freely).

Be that as it may, Perelman's proof took on a life of its own: three preprints haunted modern mathematicians. The first results of testing the ideas of the Russian mathematician appeared in 2006 - major geometers Bruce Kleiner and John Lott from the University of Michigan published a preprint own work, more like a book in size - 213 pages. In this work, scientists carefully checked all the calculations of Perelman, explaining in detail the various statements that were only briefly indicated in the work of the Russian mathematician. The verdict of the researchers was unequivocal: the evidence is absolutely correct.

An unexpected turn in this story came in July of the same year. In the journal Asian Journal of Mathematics An article by the Chinese mathematicians Xiping Zhu and Huaidong Cao entitled "A Complete Proof of the Thurston Geometrization Conjecture and the Poincaré Conjecture" appeared. Within the framework of this work, Perelman's results were considered important, useful, but only intermediate. this work caused surprise among specialists in the West, but received very favorable reviews in the East. In particular, the results were supported by Shintan Yau - one of the founders of the Calabi-Yau theory, which laid the foundation for string theory - as well as the teacher of Cao and Ju. By a happy coincidence, it was Yau who was the editor-in-chief of the magazine. Asian Journal of Mathematics in which the work was published.

After that, the mathematician began to travel around the world with popular lectures, talking about the achievements of Chinese mathematicians. As a result, there was a danger that very soon the results of Perelman and even Hamilton would be relegated to the background. This has happened more than once in the history of mathematics - many theorems bearing the names of specific mathematicians were invented by completely different people.

However, this did not happen and probably will not happen now. The presentation of the Clay Award to Perelman (even if he refuses) forever cemented in public consciousness fact: Russian mathematician Grigory Perelman proved the Poincaré conjecture. It does not matter that in fact he proved a more general fact, developing along the way a completely new theory of singularities of Ricci flows. Even so. The award has found a hero.

Photo by N. Chetverikova The last great achievement of pure mathematics is the proof of the Poincaré conjecture, expressed in 1904 and stating: “every connected, simply connected, compact three-dimensional manifold without boundary, is homeomorphic to the sphere S 3 ” by Grigory Perelman from St. Petersburg in 2002-2003.

There are several terms in this phrase, which I will try to explain in such a way that their general meaning becomes clear to non-mathematicians (I assume that the reader has graduated from high school and still remembers something from school mathematics).

The main idea is easiest to explain using the classic example of a mug and a bagel. The first can be turned into the second by a continuous deformation: These figures clearly show that the mug is homeomorphic to the donut, and this fact is true both for their surfaces (two-dimensional manifolds, called a torus) and for filled bodies (three-dimensional manifolds with boundary).

Let us give an interpretation of the rest of the terms appearing in the formulation of the hypothesis.

1. Three-dimensional 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 R 3 , as well as any open sets of points in R 3 , 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 the torus), then we already get a manifold with 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.

2. Connected. The concept of connectivity is the simplest here. A manifold is connected if it consists of one piece, or, something the same, any two of its points can be connected by a continuous line that does not go beyond its limits.

3. Simply connected. The notion of single-connectedness is more complicated. 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 R 3 is simply connected (an elastic band, arbitrarily attached 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.

4. 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 a boundary: for any continuous deformation, the ends go to some specific points, and the entire segment must go into a bounded curve connecting these points.

Dimension manifold is the number of degrees of freedom at the point that "lives" on it. Each point has a neighborhood in the form of a disk of the corresponding dimension, i.e., an interval of a line in the one-dimensional case, a circle on the plane in the two-dimensional case, a ball in the three-dimensional case, etc. From the point of view of topology, there are only two one-dimensional connected manifolds without boundary: this is the line and circle. Of these, only the circle is compact.

Two-dimensional compact manifolds are well known. If we consider only oriented 1 manifolds without boundary, then from a topological point of view they form a simple, albeit infinite, list: and so on. Each such manifold is obtained from a sphere by gluing several handles, the number of which is called the genus of the surface.

1 For lack of space, I will not talk about non-orientable manifolds, an example of which is the famous Klein bottle - a surface that cannot be embedded in a space without self-intersections.

Apparently, the simplest way to explain the topological structure of the three-dimensional sphere S 3 is with the help of one-point compactification. Namely, the three-dimensional sphere S 3 is a one-point compactification of the usual three-dimensional (unbounded) space R 3 .

Let us explain this construction first with simple examples. Let's take an ordinary infinite straight line (a one-dimensional analogue of space) and add one “infinitely distant” point to it, assuming that when moving along a straight line to the right or left, we eventually get to this point. From a topological point of view, there is no difference between an infinite line and a bounded open segment (without endpoints). Such a segment can be continuously bent in the form of an arc, bring the ends closer together and glue the missing point into the junction. We obtain, obviously, a circle - a one-dimensional analogue of a sphere.

Similarly, if I take an infinite plane and add one point at infinity, to which all lines of the original plane, passing in any direction, tend, then we get a two-dimensional (ordinary) sphere S 2 . This procedure can be observed using a stereographic projection, which assigns to each point P of the sphere, with the exception of the north pole of N, a certain point of the plane P ":

Thus, a sphere without one point is topologically the same as a plane, and adding a point turns the plane into a sphere.

In principle, exactly the same construction is applicable to a three-dimensional sphere and three-dimensional space, only for its implementation it is necessary to enter the fourth dimension, and this is not so easy to depict on the drawing. Therefore, I confine myself to a verbal description of the one-point compactification of the space R 3 .

Here is how it can be understood. Let's embed the torus in R 3 as usual, in the form of a round donut, and draw a vertical line - the axis of rotation of this donut. Draw an arbitrary plane through the axis, it will intersect our solid torus along two circles shown in green in the figure, and the additional part of the plane is divided into a continuous family of red circles. Among them is the central axis, highlighted in bolder, because in the sphere S 3 the line closes into a circle. A three-dimensional picture is obtained from this two-dimensional one by rotating around an axis. A complete set of rotated circles will then fill a three-dimensional body, homeomorphic to a solid torus, only looking unusual.

In fact, the central axis will be an axial circle in it, and the rest will play the role of parallels - circles that make up the usual solid torus.

It has three pairs of faces: left and right, top and bottom, front and back. In each pair of parallel faces, we identify in pairs the points obtained from each other by transferring along the edge of the cube. That is, we will assume (purely abstractly, without applying physical deformations) that, for example, A and A "are the same point, and B and B" are also one point, but different from point A. All internal points of the cube we will consider as usual. The cube itself is a manifold with an edge, but after the gluing done, the edge closes on itself and disappears. Indeed, the neighborhoods of the points A and A" in the cube (they lie on the left and right shaded faces) are the halves of the balls, which, after gluing the faces together, merge into a whole ball, which serves as a neighborhood of the corresponding point of the three-dimensional torus.

In order to feel the structure of the 3-torus based on ordinary ideas about physical space, you need to choose three mutually perpendicular directions: forward, left and up - and mentally consider, as in science fiction stories, that when moving in any of these directions, a rather long, but finite time , we will return to the starting point, but from the opposite direction. This is also a “compactification of space”, but not a one-point one, used earlier to construct a sphere, but more complex.

Thus, we see that there are simply connected and non-simply connected compact 3-manifolds. Perelman proved that a simply connected manifold is exactly one.

The initial idea of the proof is to use the so-called "Ricci flow": we take a simply connected compact 3-manifold, endow it with an arbitrary geometry (i.e., introduce some metric with distances and angles), and then consider its evolution along the Ricci flow. Richard Hamilton, who proposed this idea in 1981, hoped that with this evolution our manifold would turn into a sphere. It turned out that this is not true - in the three-dimensional case, the Ricci flow is capable of spoiling the manifold, i.e., making it a little manifold (something with singular points, as in the above example of intersecting lines). Perelman, by overcoming incredible technical difficulties, using the heavy apparatus of partial differential equations, managed to amend the Ricci flow near singular points in such a way that during evolution the topology of the manifold does not change, there are no singular points, and in the end it turns into a round sphere . But we must finally explain what this flow of Ricci is. The flows used by Hamilton and Perelman refer to a change in the intrinsic metric on an abstract manifold, and this is rather difficult to explain, so I will limit myself to describing the "external" Ricci flow on one-dimensional manifolds embedded in a plane.

The curvature will be considered positive if the velocity vector turns towards the inner part of the plane divided by our curve into two parts, and negative if it turns outward. This convention does not depend on the direction in which the curve is traversed. At inflection points where the rotation changes direction, the curvature will be 0. For example, a circle of radius 1 has a constant positive curvature of 1 (measured in radians).

It turns out that any closed curve in the plane behaves in a similar way during such an evolution, i.e., it eventually turns into a circle. This is the proof of the one-dimensional analogue of the Poincare conjecture using the Ricci flow (however, the statement itself in this case is already obvious, just the method of proof illustrates what happens in dimension 3).

In conclusion, we note that Perelman's argument proves not only the Poincaré conjecture, but also the much more general Thurston geometrization conjecture, which in a certain sense describes the structure of all compact 3-manifolds in general. But this subject lies beyond the scope of this elementary article.

Sergey Duzhin,
Doctor of Physics and Mathematics Sciences,
senior Researcher
St. Petersburg branch
Mathematical Institute of the Russian Academy of Sciences

Poincaré's theorem is the mathematical formula of the "Universe". Grigory Perelman. Part 1 (from the series " Real man in science")

Henri Poincare (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 the University of Oxford believes that Poincaré's theorem is "this central problem 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 Harvard University mathematicians 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 in compared with 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 Poincaré 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 hardly able to seriously evaluate 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." Translated into a 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 a much larger number of dimensions, 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 "PRIMARY ALLATRA PHYSICS" 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? ..

The exceptional importance of the hypothesis put forward about a century ago by the mathematician Poincaré concerns three-dimensional structures and is 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 "PRIMORDIAL ALLATRA PHYSICS" and in the book "AllatRa" the last 50 pages about the way to build a creative society). AND cash(energy), which businessmen are ready to donate to science, if it is necessary to use it, then it is 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 (about the dimensions of the global universe and about spiritual world see book"AllatRa" and report"PRIMORDIAL ALLATRA PHYSICS"), 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) is a one-to-one correspondence between two topological spaces, under which both mutually inverse mappings defined by this correspondence are continuous. These mappings are called homeomorphic or topological mappings, as well as homeomorphisms, and spaces are said to belong to the same topological type are called homeomorphic, or topologically equivalent.

A three-dimensional 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. If we add its boundary points (the surface of a torus), then we will 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.

A solid torus (solid torus) is a geometric body homeomorphic to the product of a two-dimensional disk and a circle D2 * S1. Informally, a solid torus is a donut, while a torus is only its surface (a hollow chamber of a wheel).

Simply 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 to one point by a smooth deformation without removing 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 a boundary: for any continuous deformation, the ends go to some specific points, and the entire segment must go into a bounded curve connecting these points.

To be continued...

Ilnaz Basharov

Literature:

– Report "PRIMARY ALLATRA PHYSICS" of the international group of scientists of the ALLATRA International Public Movement, ed. Anastasia Novykh, 2015 http://allatra-science.org/pub... ;

- New ones. A. "AllatRa", K.: AllatRa, 2013 http://schambala.com.ua/book/a... .

- New ones. A., "Sensei-IV", K.: LOTOS, 2013, 632 p. http://schambala.com.ua/book/s...

– Sergey Duzhin, Doctor of Physics and Mathematics Sci., Senior Researcher, St. Petersburg Branch of the Mathematical Institute of the Russian Academy of Sciences

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