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What did Grigory Perelman prove? Mathematician Perelman Yakov: contribution to science. Famous Russian mathematician Grigory Perelman

« 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 didn't even get a Nobel Prize. 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. Clay Mathematics Institute, a private mathematical institute, chose seven of the most difficult problems and promised to pay a million dollars for each solution.

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 pool- 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 the properties of geometric shapes that are preserved if the shape 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 bagel. 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 another wrote in his popular book 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. According to Komsomolskaya Pravda, the reclusive scientist revealed himself in a conversation with a journalist and producer of the President-Film film company, 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.

This distraction from mathematical logic was the main point of daily training. 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 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 device 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 count as in fantasy stories that when moving in any of these directions for a sufficiently 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 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 mathematician Perelman is a very famous person, despite the fact that he leads a solitary life and avoids the press in every possible way. His proof of the Poincare conjecture placed him on a par with the greatest scientists in world history. The mathematician Perelman refused many awards provided by the scientific community. This man lives very modestly and is completely devoted to science. Of course, it is worth talking about him and his discovery in detail.

Father Grigory Perelman

On June 13, 1966, Grigory Yakovlevich Perelman, a mathematician, was born. Photo of him in free access a little, but the most famous are presented in this article. He was born in Leningrad - cultural capital our country. His father was an electrical engineer. He had nothing to do with science, as many believe.

Yakov Perelman

It is widely believed that Grigory is the son of Yakov Perelman, a well-known popularizer of science. However, this is a misconception, because he died in besieged Leningrad in March 1942, so there was no way he could be a father. This man was born in Bialystok, a city that previously belonged to Russian Empire and is now part of Poland. Yakov Isidorovich was born in 1882.

Yakov Perelman, which is very interesting, was also attracted to mathematics. In addition, he was fond of astronomy and physics. This man is considered the founder of entertaining science, as well as one of the first who wrote works in the genre of popular science literature. He is the creator of the book "Live Mathematics". Perelman wrote many other books. In addition, his bibliography includes more than a thousand articles. As for such a book as "Live Mathematics", Perelman presents in it various puzzles related to this science. Many of them are designed in the form of short stories. This book is intended primarily for teenagers.

In one respect, another book is especially interesting, the author of which is Yakov Perelman (" Entertaining mathematics"). Trillion - do you know what this number is? It's 10 21. In the USSR for a long time there were two scales in parallel - "short" and "long". According to Perelman, "short" was used in financial calculations and everyday life, and "long" - in scientific papers dedicated to physics and astronomy. So, a trillion on a "short" scale does not exist. 10 21 is called a sextillion in it. These scales generally differ significantly.

However, we will not dwell on this in detail and move on to a story about the contribution to science that was made by Grigory Yakovlevich, and not by Yakov Isidorovich, whose achievements were less modest. By the way, it was not his well-known namesake who instilled in Gregory a love for science.

Perelman's mother and her influence on Grigory Yakovlevich

The mother of the future scientist taught mathematics at a vocational school. In addition, she was a talented violinist. Probably a love of mathematics, as well as classical music Grigory Yakovlevich adopted it from her. Both equally attracted Perelman. When he faced the choice of where to enter - to the conservatory or to a technical university, he could not decide for a long time. Who knows who Grigory Perelman could have become if he had decided to get a musical education.

The childhood of the future scientist

Already from a young age, Gregory was distinguished competent speech both written and oral. He often amazed teachers at school with this. By the way, before the 9th grade, Perelman studied at a secondary school, apparently typical, of which there are so many on the outskirts. And then teachers from the Palace of Pioneers noticed a talented young man. He was taken to courses for gifted children. This contributed to the development of Perelman's unique talents.

Victory at the Olympics, graduation from school

Since then, the milestone of victories for Gregory begins. In 1982, he received at the International Mathematical Olympiad held in Budapest. Perelman participated in it together with a team of Soviet schoolchildren. He received a full score, solving all the problems flawlessly. Gregory graduated from the eleventh grade of the school in the same year. The very fact of participation in this prestigious Olympiad opened the doors of the best educational institutions of our country for him. But Grigory Perelman not only participated in it, but also received a gold medal.

It is not surprising that he was enrolled without exams in the Leningrad State University, at the Faculty of Mechanics and Mathematics. By the way, Gregory, oddly enough, did not receive a gold medal at school. This was prevented by the assessment in physical education. Passing sports standards at that time was mandatory for everyone, including those who could hardly imagine themselves at the pole for jumping or at the bar. In other subjects, he studied for five.

Studying at LSU

Over the next few years, the future scientist continued his education at Leningrad State University. He participated, and with great success, in various mathematical competitions. Perelman even managed to get the prestigious Lenin Scholarship. So he became the owner of 120 rubles - a lot of money at that time. He must have been doing well at the time.

It must be said that the Faculty of Mathematics and Mechanics of this university, which is now called St. Petersburg, was in Soviet years one of the best in Russia. In 1924, for example, V. Leontiev graduated from it. Almost immediately after completing his studies, he received the Nobel Prize in Economics. This scientist is even called the father of the American economy. Leonid Kantorovich, the only domestic laureate of this award, who received it for his contribution to this science, was a professor of mathematics.

Continuing education, life in the USA

After graduating from Leningrad State University, Grigory Perelman entered the Steklov Mathematical Institute to continue his postgraduate studies. Soon he flew to the USA in order to present it. educational institution. This country has always been considered a state of unlimited freedom, especially in Soviet time among the inhabitants of our country. Many dreamed of seeing her, but the mathematician Perelman was not one of them. It seems that the temptations of the West have gone unnoticed for him. The scientist still led a modest lifestyle, even somewhat ascetic. He ate sandwiches with cheese, which he washed down with kefir or milk. And of course, the mathematician Perelman worked hard. In particular, he was a teacher. The scientist met with his fellow mathematicians. America bored him after 6 years.

Return to Russia

Grigory returned to Russia, to his native institute. Here he worked for 9 years. It was at this time that he must have begun to understand that the road to " pure art"lies through isolation, isolation from society. Grigory decided to break off all his relations with his colleagues. The scientist decided to lock himself in his Leningrad apartment and start a grandiose work ...

Topology

It is not easy to explain what Perelman proved in mathematics. Only great lovers of this science can fully understand the significance of his discovery. We will try in plain language talk about the hypothesis that Perelman brought out. Grigory Yakovlevich was attracted by topology. This is a branch of mathematics, often also called geometry on a rubber sheet. Topology studies geometric shapes that persist when the shape is bent, twisted, or stretched. In other words, if it is absolutely elastically deformed - without gluing, cutting and tearing. Topology is very important for a discipline like mathematical physics. It gives an idea of the properties of space. In our case, we are talking about an infinite space that is continuously expanding, that is, about the Universe.

Poincare conjecture

The great French physicist, mathematician and philosopher J. A. Poincaré was the first to hypothesize this. This happened at the beginning of the 20th century. But it should be noted that he made an assumption, and did not give a proof. Perelman made it his task to prove this hypothesis, to derive a mathematical solution, logically verified, after a whole century.

When talking about its essence, they usually begin as follows. Take the rubber disk. It should be pulled over the ball. Thus, you have a two-dimensional sphere. It is necessary that the circumference of the disk be collected at one point. For example, you can do this with a backpack by pulling it off and tying it with a cord. It turns out a sphere. Of course, for us it is three-dimensional, but from the point of view of mathematics it will be two-dimensional.

Then figurative projections and reasoning begin, which are difficult to understand for an unprepared person. One should now imagine a three-dimensional sphere, that is, a ball stretched over something that goes into another dimension. A three-dimensional sphere, according to the hypothesis, is the only existing three-dimensional object that can be pulled together by a hypothetical "hypercord" at one point. The proof of this theorem helps us understand what shape the Universe has. In addition, thanks to it, one can reasonably assume that the Universe is such a three-dimensional sphere.

The Poincaré Hypothesis and the Big Bang Theory

It should be noted that this hypothesis is a confirmation of the Big Bang theory. If the Universe is the only "figure" whose distinguishing feature is the ability to contract it into one point, this means that it can be stretched in the same way. The question arises: if it is a sphere, what is outside the universe? Is man, who is a by-product belonging to the planet Earth alone and not even to the cosmos as a whole, capable of cognizing this mystery? Those who are interested can be invited to read the works of another world-famous mathematician - Stephen Hawking. However, he cannot yet say anything concrete on this score. Let's hope that in the future another Perelman will appear and he will be able to solve this riddle, which torments the imagination of many. Who knows, maybe Grigory Yakovlevich himself will still be able to do it.

Nobel Prize in Mathematics

Perelman did not receive this prestigious award for his great achievement. Strange, isn't it? In fact, this is explained very simply, given that such an award simply does not exist. A whole legend has been created about the reasons why Nobel deprived representatives of such an important science. To this day, the Nobel Prize in mathematics has not been awarded. Perelman would probably get it if it existed. There is a legend that the reason for Nobel's rejection of mathematicians is the following: it was to the representative of this science that his bride left him. Like it or not, it was only with the advent of the 21st century that justice finally prevailed. It was then that another prize for mathematicians appeared. Let's briefly talk about its history.

How did the Clay Institute Award come about?

At a mathematical congress held in Paris in 1900, he proposed a list of 23 problems that needed to be solved in the new, 20th century. To date, 21 of them are already allowed. By the way, in 1970 Yu. V. Matiyasevich, a graduate of mathematics and mechanics at Leningrad State University, completed the solution of the 10th of these problems. At the beginning of the 21st century, the American Clay Institute compiled a list similar to it, consisting of seven problems in mathematics. They should have been solved already in the 21st century. A million dollar reward was announced for solving each of them. As early as 1904, Poincaré formulated one of these problems. He put forward the conjecture that all three-dimensional surfaces that are homotypically equivalent to a sphere are homeomorphic to it. talking in simple terms, if a three-dimensional surface is somewhat similar to a sphere, then it is possible to flatten it into a sphere. This statement of the scientist is sometimes called the formula of the universe because of its great importance in understanding complex physical processes, and also because the answer to it means solving the question of the shape of the universe. It should also be said that this discovery plays an important role in the development of nanotechnologies.

So, the Clay Mathematics Institute decided to choose the 7 most difficult problems. For the solution of each of them was promised a million dollars. And now Grigory Perelman appears with his discovery. The prize in mathematics, of course, goes to him. He was noticed quite quickly, since since 2002 he has been publishing his work on foreign Internet resources.

How Perelman was awarded the Clay Award

So, in March 2010, Perelman was awarded the well-deserved award. The prize in mathematics meant receiving an impressive fortune, the size of which was $ 1 million. Grigory Yakovlevich was supposed to receive it for the proof. However, in June 2010, the scientist ignored the mathematical conference held in Paris, at which this award was to be presented. And on July 1, 2010, Perelman announced his refusal publicly. Moreover, he never took the money allotted to him, despite all the requests.

Why did the mathematician Perelman refuse the prize?

Grigory Yakovlevich explained this by the fact that his conscience did not allow him to receive a million, which was due to several other mathematicians. The scientist noted that he had many reasons both to take the money and not to take it. It took him a long time to decide. Grigory Perelman, a mathematician, cited disagreement with the scientific community as the main reason for refusing the award. He noted that he considers his decisions unfair. Grigory Yakovlevich stated that he believed that the contribution of Hamilton, a German mathematician, to the solution of this problem was no less than his.

By the way, a little later there was even an anecdote on this topic: mathematicians need to allocate millions more often, perhaps someone will still decide to take them. A year after Perelman's refusal, Demetrios Christodoul and Richard Hamilton were awarded the Shaw Prize. The amount of this award in mathematics is one million dollars. This award is sometimes also referred to as Nobel Prize East. Hamilton received it for the creation of a mathematical theory. It was it that the Russian mathematician Perelman then developed in his works devoted to the proof of the Poincaré conjecture. Richard accepted the award.

Other awards refused by Grigory Perelman

By the way, in 1996 Grigory Yakovlevich was awarded a prestigious prize for young mathematicians from the European Mathematical Society. However, he refused to receive it.

Ten years later, in 2006, the scientist was awarded the Fields Medal for solving the Poincare conjecture. Grigory Yakovlevich also refused her.

The journal Science in 2006 called the proof of the hypothesis created by Poincaré the scientific breakthrough of the year. It should be noted that this is the first work in the field of mathematics that has earned such a title.

David Gruber and Sylvia Nazar published an article in 2006 called Manifold Destiny. It talks about Perelman, about his solution to the Poincaré problem. In addition, the article talks about the mathematical community and the ethical principles that exist in science. It also features a rare interview with Perelman. Much is also said about the criticism of Yau Xingtang, the Chinese mathematician. Together with his students, he tried to challenge the completeness of the evidence presented by Grigory Yakovlevich. In an interview, Perelman noted: "Those who violate ethical standards in science are not considered outsiders. People like me are who are isolated."

In September 2011, he also refused membership in Russian Academy mathematician Perelman. His biography is presented in a book published in the same year. From it you can learn more about the fate of this mathematician, although the information collected is based on the testimony of third parties. Its author - The book was compiled on the basis of interviews with classmates, teachers, colleagues and colleagues of Perelman. Sergei Rukshin, Grigory Yakovlevich's teacher, spoke critically of her.

Grigory Perelman today

And today he leads a solitary life. The mathematician Perelman ignores the press in every possible way. Where does he live? Until recently, Grigory Yakovlevich lived with his mother in Kupchino. And since 2014, the famous Russian mathematician Grigory Perelman has been in Sweden.

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).

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 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 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.

Full torus (full torus) - geometric body, homeomorphic to the product of a two-dimensional disk and the 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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