Poincaré conjecture: formulation and proof. Forms of space Theorem that Perelman proved in simple words

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

There are several terms in this phrase that I will try to explain so that their general meaning is clear to non-mathematicians (I assume that the reader has graduated from high school and still remembers some of his school mathematics).

Let's start with the concept of homeomorphism, which is central to 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 breaks and gluing, or more precisely, if it is possible to establish a one-to-one and mutually continuous correspondence between two objects .

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

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

1. Three-dimensional manifold without edge. This is a geometric object in which each point has a neighborhood in the form of a three-dimensional ball. Examples of 3-manifolds include, firstly, the entire three-dimensional space, denoted 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 full torus, i.e., add its boundary points (the surface of the torus), then we obtain a manifold with an edge - the edge points do not have neighborhoods in the form of a ball, but only in the form of a half-ball.

2. Connected. The concept of connectivity here is the simplest. 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 extend beyond its boundaries.

3. Simply connected. The concept of simply connectedness is more complex. 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 (a rubber band, placed in any way on the surface of an apple, can be smoothly pulled to one point by smooth deformation without tearing the rubber band off the apple). On the other hand, the circle and the torus are not simply connected.

4. Compact. A variety 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 non-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 of a manifold is the number of degrees of freedom of 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 a one-dimensional case, a circle on a plane in two dimensions, a ball in three dimensions, etc. From the point of view of topology, there are only two one-dimensional connected manifolds without an edge: a 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 simply an interval (and all other points have such neighborhoods ). In such cases, mathematicians say that we are dealing with a special variety that has one special point.

Two-dimensional compact manifolds are well known. If we consider only orientable 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 space without self-intersections.

The figure shows surfaces of genus 0, 1, 2 and 3. What makes the 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, but 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 be classified in a sense, that is, 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 just like the 2D sphere in the list above. The fact that any curve on S 3 contracts to a point is proved as simply as in the two-dimensional case. But the opposite statement, namely that this property is unique specifically for the sphere, i.e., that on any other three-dimensional manifold there are non-contractible curves, is very difficult and exactly constitutes the content of the Poincaré conjecture that we are talking about.

It is important to understand that diversity can live on its own; it can be thought of as an independent object, not nested anywhere. (Imagine living as two-dimensional creatures on the surface of an ordinary sphere, unaware of the existence of a third dimension.) Fortunately, all of the two-dimensional surfaces in the list above can be nested in ordinary R3 space, making them easier to visualize. For the three-dimensional sphere S 3 (and in general for any compact three-dimensional manifold without boundary) this is no longer the case, so some effort is required to understand its structure.

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

Let us first explain this construction using simple examples. Let’s take an ordinary infinite straight line (a one-dimensional analogue of space) and add to it one “infinitely distant” point, assuming that when we move 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 line segment (without end points). Such a segment can be continuously bent in the form of an arc, bring the ends closer and glue the missing point at the junction. We will obviously get a circle - a one-dimensional analogue of a sphere.

In the same way, if I take an infinite plane and add one point at infinity, to which all straight 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 N, a certain point on 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 in a drawing. Therefore, I will limit myself to a verbal description of the 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) one point “at infinity” is added in such a way that when moving in a straight line in any direction you you get there (i.e., each spatial line closes into a circle). Then we get a compact three-dimensional manifold, which by definition is the sphere S 3 .

It is easy to understand that the sphere S 3 is simply connected. In fact, 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 ordinary space R 3, which easily contracts to a point through homotheties, that is, continuous compression in all three directions.

To understand how the variety S 3 is structured, it is very instructive to consider its partition into two solid tori. If we remove the solid torus from the space R 3, then something not very clear will remain. And if 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 that have a common boundary - a torus.

Here's how you can understand it. 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. Let us draw an arbitrary plane through the axis; it will intersect our solid torus along two circles shown in the figure green, and the additional part of the plane is divided into a continuous family of red circles. These include the central axis, highlighted more boldly, because in the sphere S 3 the straight line closes into a circle. A three-dimensional picture is obtained from this two-dimensional picture by rotation around an axis. A complete set of rotated circles will 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 an ordinary solid torus.

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 the starting material:

It has three pairs of edges: 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 transfer along the edge of the cube. That is, we will assume (purely abstractly, without the use of 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 it as usual. The cube itself is a manifold with an edge, but after the gluing is done, the edge closes on itself and disappears. In fact, the neighborhoods of points A and A" in the cube (they lie on the left and right shaded faces) are halves of 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.

To feel the structure of a 3-torus based on everyday ideas about physical space, you need to select 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 fairly 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 the one-point one used earlier to construct the sphere, but more complex.

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

So, 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 this evolution would turn our diversity into a sphere. It turned out that this is not true - in the three-dimensional case, the Ricci flow is capable of spoiling a manifold, i.e., making it a non-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 introduce corrections into the Ricci flow near singular points in such a way that during evolution the topology of the manifold does not change, no singular points arise, and in the end it turns into a round sphere . But we must finally explain what this Ricci flow is. The flows used by Hamilton and Perelman refer to changes in the intrinsic metric on an abstract manifold, and this is quite difficult to explain, so I will limit myself to describing the “external” Ricci flow on one-dimensional manifolds embedded in the plane.

Let's imagine a smooth closed curve on the Euclidean plane, choose a direction on it and consider a tangent vector of unit length at each point. Then, when going around the curve in the chosen direction, this vector will rotate with some angular velocity, which is called curvature. In those places where the curve is curved more steeply, the curvature (in absolute value) will be greater, and where it is smoother, the curvature will be less.

We will consider the curvature to be 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 agreement 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 (if measured in radians).

Now let's forget about tangent vectors and, on the contrary, attach to each point of the curve 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 make each point move in the direction of the corresponding vector with speed proportional to its length. Here's an example:

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

Let us note in conclusion that Perelman’s reasoning 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 generally compact three-dimensional manifolds. 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

A brilliant mathematician and Parisian professor, Henri Poincaré, worked in a variety of areas of this science. Independently and independently of Einstein's work in 1905, he put forward the main principles of the Special Theory of Relativity. And he formulated his famous hypothesis back in 1904, so it took about a century to solve it.

Poincaré was one of the founders of topology - the science of the properties of geometric figures that do not change under deformations that occur without breaks. For example, a balloon can be easily deformed into a variety of shapes, just like they do for children in the park. But you will need to cut the ball in order to twist it into a donut (or, in geometric language, a torus) - there is no other way. And vice versa: take a rubber donut and try to “turn” it into a sphere. However, it still won't work. According to their topological properties, the surfaces of a sphere and a torus are incompatible, or non-homeomorphic. But any surfaces without “holes” (closed surfaces), on the contrary, are homeomorphic and are capable of becoming deformed and transforming into a sphere.

If everything was decided about the two-dimensional surfaces of the sphere and torus in the 19th century, it took much longer for more multidimensional cases. This, in fact, is the essence of the Poincaré conjecture, which extends the pattern to multidimensional cases. Simplifying a bit, the Poincaré conjecture states: “Every simply connected closed n-dimensional manifold is homeomorphic to an n-dimensional sphere.” It's funny that the option with three-dimensional surfaces turned out to be the most difficult. In 1960, the hypothesis was proven for dimensions 5 and higher, in 1981 - for n=4. The stumbling block was precisely three-dimensionality.

Developing the ideas of William Thurstan and Richard Hamilton, proposed by them in the 1980s, Grigory Perelman applied a special equation of “smooth evolution” to three-dimensional surfaces. And he was able to show that the original three-dimensional surface (if there are no discontinuities in it) will necessarily evolve into a three-dimensional sphere (this is the surface of a four-dimensional ball, and it exists in 4-dimensional space). According to a number of experts, this was a “new generation” idea, the solution of which opens up new horizons for mathematical science.

It is interesting that for some reason Perelman himself did not bother to bring his decision to final brilliance. Having described the solution “in general” in the preprint The entropy formula for the Ricci flow and its geometric applications in November 2002, in March 2003 he supplemented the proof and presented it in the preprint Ricci flow with surgery on three-manifolds, and also reported on the method in a series of lectures that he gave in 2003 at the invitation of a number of universities. None of the reviewers could find errors in the version he proposed, but Perelman did not publish a publication in a peer-reviewed scientific publication (which, in particular, was a necessary condition for receiving the prize). But in 2006, based on his method, a whole set of proofs was released, in which American and Chinese mathematicians examined the problem in detail and completely, supplemented the points omitted by Perelman, and produced the “final proof” of the Poincaré conjecture.

In 1904, Henri Poincaré proposed that any three-dimensional object that has certain properties of a 3-sphere can be converted into a 3-sphere. It took 99 years to prove this hypothesis. (Warning! The three-dimensional sphere is not what you think it is.) The Russian mathematician proved the Poincaré conjecture stated a century ago and completed the creation of a catalog of shapes of three-dimensional spaces. Perhaps he will receive a $1 million bonus.

Take a look around. The objects around you, like you yourself, are a collection of particles moving in three-dimensional space (3-manifold), which extends in all directions for many billions of light years.

Manifolds are mathematical constructions. Since the times of Galileo and Kepler, scientists have successfully described reality in terms of one branch of mathematics or another. Physicists believe that everything in the world happens in three-dimensional space and the position of any particle can be specified by three numbers, for example, latitude, longitude and altitude (let us leave aside for now the assumption made in string theory that in addition to the three dimensions we observe, there are several additional ).

According to classical and traditional quantum physics, space is fixed and unchanging. At the same time, the general theory of relativity considers it as an active participant in events: the distance between two points depends on passing gravitational waves and on how much matter and energy is located nearby. But in both Newtonian and Einsteinian physics, space - infinite or finite - in any case is a 3-manifold. Therefore, to fully understand the fundamentals on which almost all modern science, it is necessary to understand the properties of 3-manifolds (4-manifolds are of no less interest, since space and time together form one of them).

The branch of mathematics in which manifolds are studied is called topology. Topologists first asked fundamental questions: what is the simplest (i.e., least complex) type of 3-manifold? Does it have equally simple brothers or is it unique? What kind of 3-manifolds are there?

The answer to the first question has been known for a long time: the simplest compact 3-manifold is a space called a 3-sphere (Non-compact manifolds are infinite or have edges. Below, only compact manifolds are considered). Two other questions remained open for a century. Only in 2002 were they answered by the Russian mathematician Grigory Perelman, who, apparently, was able to prove the Poincaré conjecture.

Exactly one hundred years ago, the French mathematician Henri Poincaré proposed that the 3-sphere is unique and that no other compact 3-manifold has the properties that make it so simple. More complex 3-manifolds have boundaries that stand up like a brick wall, or multiple connections between certain areas, like a forest path that branches and then joins again. Any three-dimensional object with the properties of a 3-sphere can be transformed into it itself, so to topologists it appears to be simply a copy of it. Perelman's proof also allows us to answer the third question and classify all existing 3-manifolds.

You will need a fair amount of imagination to imagine a 3-sphere (see MULTI-DIMENSIONAL MUSIC OF THE SPHERES). Fortunately, it has a lot in common with the 2-sphere, a typical example of which is the rubber of a round balloon: it is two-dimensional, since any point on it is defined by only two coordinates - latitude and longitude. If you examine a fairly small area of it under a powerful magnifying glass, it will seem like a piece of a flat sheet. To a tiny insect crawling on a balloon, it will appear to be a flat surface. But if the booger moves in a straight line long enough, it will eventually return to its point of departure. In the same way, we would perceive a 3-sphere the size of our Universe as “ordinary” three-dimensional space. If we flew far enough in any direction, we would eventually "circumnavigate" it and end up back at our starting point.

As you may have guessed, an n-dimensional sphere is called an n-sphere. For example, the 1-sphere is familiar to everyone: it is just a circle.

Grigory Perelman presents his proof of the Poincaré conjecture and the completion of Thurston's geometrization program at a seminar at Princeton University in April 2003.

Hypothesis testing

Half a century passed before the matter of the Poincaré conjecture got off the ground. In the 60s XX century Mathematicians have proven similar statements to her for spheres of five or more dimensions. In each case, the n-sphere is indeed the only and simplest n-manifold. Oddly enough, it turned out to be easier to obtain results for multidimensional spheres than for 3- and 4-spheres. The proof for four dimensions appeared in 1982. And only the original Poincaré conjecture about the 3-sphere remained unconfirmed.

The decisive step was taken in November 2002, when Grigory Perelman, a mathematician from the St. Petersburg branch of the Mathematical Institute. Steklov, sent the article to the website www.arxiv.org, where physicists and mathematicians from all over the world discuss the results of their scientific activity. Topologists immediately grasped the connection between the Russian scientist’s work and the Poincaré conjecture, although the author did not directly mention it. In March 2003, Perelman published a second article and in the spring of that year he visited the United States and gave several seminars at the Massachusetts Institute of Technology and the State University of New York at Stony Brook. Several groups of mathematicians at leading institutes immediately began a detailed study of the submitted works and search for errors.

REVIEW: PROOF OF POINCARES HYPOTHESIS

For a whole century, mathematicians tried to prove Henri Poincaré's assumption about the exceptional simplicity and uniqueness of the 3-sphere among all three-dimensional objects.
The rationale for the Poincaré conjecture finally appeared in the work of the young Russian mathematician Grigory Perelman. He also completed an extensive program of classification of three-dimensional manifolds.
Perhaps our Universe has the shape of a 3-sphere. There are other intriguing connections between mathematics and particle physics and general relativity.

In Stony Brook, Perelman gave several lectures over two weeks, speaking from three to six hours a day. He presented the material very clearly and answered all the questions that arose. There is still one small step left before the final result is obtained, but there is no doubt that it is about to be done. The first article introduces the reader to the underlying ideas and is considered fully verified. The second article covers applied issues and technical nuances; it does not yet inspire the same complete confidence as its predecessor.

In 2000, the Institute of Mathematics named after. Clay in Cambridge, Massachusetts, has established a $1 million prize for proving each of the seven Millennium Problems, one of which is considered to be the Poincaré conjecture. Before a scientist can claim a prize, his proof must be published and carefully reviewed for two years.

Perelman's work expands and completes the research program conducted in the 90s. last century by Richard S. Hamilton from Columbia University. At the end of 2003, the works of the American mathematician were awarded the Clay Institute Prize. Perelman managed to brilliantly overcome a number of obstacles that Hamilton could not cope with.

In fact, Perelman's proof, the correctness of which no one has yet been able to question, solves a much wider range of issues than the Poincaré conjecture itself. The geometrization procedure proposed by William P. Thurston of Cornell University allows for a complete classification of 3-manifolds based on the 3-sphere, unique in its sublime simplicity. If the Poincaré conjecture were false, i.e. If there were many spaces as simple as a sphere, then the classification of 3-manifolds would turn into something infinitely more complex. Thanks to Perelman and Thurston, we have a complete catalog of all the mathematically possible forms of three-dimensional space that our Universe could take (if we consider only space without time).

Rubber bagels

To better understand the Poincaré conjecture and Perelman's proof, you should take a closer look at topology. In this branch of mathematics, the shape of an object does not matter, as if it were made of dough that can be stretched, compressed and bent in any way. Why should we think about things or spaces made from imaginary dough? The fact is that the exact shape of an object - the distance between all its points - refers to a structural level called geometry. By examining an object from a dough, topologists identify its fundamental properties that do not depend on the geometric structure. Studying topology is like searching for the most common traits that people have by looking at a “plasticine man” who can be turned into any specific individual.

In popular literature, there is often a hackneyed statement that, from a topological point of view, a cup is no different from a donut. The fact is that a cup of dough can be turned into a donut by simply crushing the material, i.e. without blinding anything or making holes (see SURFACE TOPOLOGY). On the other hand, to make a donut from a ball, you definitely need to make a hole in it or roll it into a cylinder and mold the ends, so a ball is not a donut at all.

Topologists are most interested in the sphere and donut surfaces. Therefore, instead of solid bodies, you should imagine balloons. Their topology is still different because a spherical balloon cannot be converted into a ring one, which is called a torus. First, scientists decided to figure out how many objects with different topologies exist and how they can be characterized. For 2-manifolds, which we are accustomed to calling surfaces, the answer is elegant and simple: everything is determined by the number of “holes” or, what is the same, the number of handles (see TOPOLOGY OF SURFACES). By the end of the 19th century. mathematicians figured out how to classify surfaces and determined that the simplest of them was the sphere. Naturally, topologists began to think about 3-manifolds: is the 3-sphere unique in its simplicity? The century-long history of searching for an answer is full of missteps and flawed evidence.

Henri Poincaré took up this issue closely. He was one of the two most powerful mathematicians of the early 20th century. (the other was David Gilbert). He was called the last universalist - he successfully worked in all areas of both pure and applied mathematics. In addition, Poincaré made enormous contributions to the development of celestial mechanics, the theory of electromagnetism, as well as to the philosophy of science, about which he wrote several popular books.

Poincaré became the founder of algebraic topology and, using its methods, in 1900 he formulated a topological characteristic of an object, called homotopy. To determine the homotopy of a manifold, you need to mentally immerse a closed loop in it (see TOPOLOGY OF SURFACES). Then you should find out whether it is always possible to contract the loop to a point by moving it inside the manifold. For a torus, the answer will be negative: if you place a loop around the circumference of the torus, you will not be able to tighten it to a point, because the donut “hole” will get in the way. Homotopy is the number of different paths that can prevent a loop from contracting.

MULTI-DIMENSIONAL MUSIC OF THE SPHERES

It is not so easy to imagine a 3-sphere. Mathematicians who prove theorems about higher-dimensional spaces do not have to imagine the object of study: they deal with abstract properties, guided by intuitions based on analogies with fewer dimensions (such analogies must be treated with caution and not taken literally). We will also consider the 3-sphere, based on the properties of objects with fewer dimensions.

1. Let's start by looking at a circle and its enclosing circle. For mathematicians, a circle is a two-dimensional ball, and a circle is a one-dimensional sphere. Further, a ball of any size is a filled object, reminiscent of a watermelon, and a sphere is its surface, more like a balloon. A circle is one-dimensional because the position of a point on it can be specified by a single number.

2. From two circles we can construct a two-dimensional sphere, turning one of them into the Northern Hemisphere and the other into the Southern Hemisphere. All that remains is to glue them together, and the 2-sphere is ready.

3. Let's imagine an ant crawling from the North Pole along a large circle formed by the prime and 180th meridians (left). If we map its path onto the two original circles (on the right), we see that the insect moves in a straight line (1) to the edge of the northern circle (a), then crosses the border, hits the corresponding point on the southern circle and continues to follow the straight line (2 and 3). Then the ant again reaches the edge (b), crosses it and again finds itself on the northern circle, rushing towards the starting point - the North Pole (4). Note that when traveling around the world on a 2-sphere, the direction of movement is reversed when moving from one circle to another.

4. Now consider our 2-sphere and the volume it contains (a three-dimensional ball) and do with them the same thing as with a circle and a circle: take two copies of the ball and glue their boundaries together. It is impossible and not necessary to clearly show how balls are distorted in four dimensions and turn into an analogue of hemispheres. It is enough to know that the corresponding points on the surfaces, i.e. 2-spheres are connected to each other in the same way as in the case of circles. The result of connecting two balls is a 3-sphere - the surface of a four-dimensional ball. (In four dimensions, where a 3-sphere and a 4-ball exist, the surface of an object is three-dimensional.) Let's call one ball the northern hemisphere and the other the southern hemisphere. By analogy with circles, the poles are now located in the centers of the balls.

5. Imagine that the balls in question are large empty regions of space. Let's say an astronaut sets off from the North Pole on a rocket. Over time, it reaches the equator (1), which is now a sphere surrounding the northern ball. Crossing it, the rocket enters the southern hemisphere and moves in a straight line through its center - the South Pole - to the opposite side of the equator (2 and 3). There the transition to the northern hemisphere occurs again, and the traveler returns to the North Pole, i.e. to the starting point (4). This is the scenario for traveling around the world on the surface of a 4-dimensional ball! The three-dimensional sphere considered is the space referred to in the Poincaré conjecture. Perhaps our Universe is precisely a 3-sphere.
The reasoning can be extended to five dimensions and construct a 4-sphere, but this is extremely difficult to imagine. If you glue two n-balls along the (n–1)-spheres surrounding them, you get an n-sphere bounding the (n+1)-ball.

On the n-sphere, any loop, even an intricately twisted one, can always be unraveled and pulled together to a point. (The loop is allowed to pass through itself.) Poincaré assumed that the 3-sphere is the only 3-manifold on which any loop can be contracted to a point. Unfortunately, he was never able to prove his conjecture, which later became known as the Poincaré conjecture. Over the past hundred years, many have offered their own version of the proof, but only to be convinced of its fallacy. (For ease of exposition, I am neglecting two special cases: the so-called non-orientable manifolds and manifolds with edges. For example, a sphere with a segment cut out of it has an edge, and a Möbius loop not only has edges, but is also non-orientable.)

Geometrization

Perelman's analysis of 3-manifolds is closely related to the geometrization procedure. Geometry deals with the actual shape of objects and manifolds, no longer made of dough, but of ceramics. For example, a cup and a donut are geometrically different because their surfaces are curved differently. It is said that a cup and a donut are two examples of a topological torus that is given different geometric shapes.

To understand why Perelman used geometrization, consider the classification of 2-manifolds. Each topological surface is assigned a unique geometry whose curvature is distributed evenly across the manifold. For example, for a sphere, this is a perfectly spherical surface. Another possible geometry for a topological sphere is an egg, but its curvature is not evenly distributed everywhere: the sharp end is more curved than the blunt end.

2-manifolds form three geometric types (see GEOMETRIZATION). The sphere is characterized by positive curvature. A geometrized torus is flat and has zero curvature. All other 2-manifolds with two or more "holes" have negative curvature. They correspond to a surface similar to a saddle, which curves upward in front and behind, and downward on the left and right. Poincaré developed this geometric classification (geometrization) of 2-manifolds together with Paul Koebe and Felix Klein, after whom the Klein bottle is named.

There is a natural desire to apply a similar method to 3-manifolds. Is it possible to find for each of them a unique configuration in which the curvature would be distributed evenly throughout the entire variety?

It turned out that 3-manifolds are much more complex than their two-dimensional counterparts and most of them cannot be assigned a homogeneous geometry. They should be divided into parts that correspond to one of the eight canonical geometries. This procedure is reminiscent of decomposing a number into prime factors.

SURFACE TOPOLOGY

IN TOPOLOGY the exact form, i.e. geometry is irrelevant: objects are treated as if they were made of dough and can be stretched, compressed and twisted. However, nothing can be cut or glued. Thus, any object with one hole, such as a coffee cup (left), is equivalent to a donut or torus (right).

ANY TWO-DIMENSIONAL manifold or surface (limited to compact orientable objects) can be made by adding handles to sphere (a). Let's stick one and make a surface of the 1st kind, i.e. a torus or a donut (top right), add a second one - we get a surface of the 2nd kind (b), etc.

The uniqueness of the 2-sphere among surfaces is that any closed loop embedded in it can be contracted to point (a). On a torus, this can be prevented by the middle hole (b). Any surface except the 2-sphere has handles that prevent the loop from tightening. Poincaré suggested that the 3-sphere is unique among three-dimensional manifolds: only on it can any loop be contracted to a point.

This classification procedure was first proposed by Thurston in the late 70s. last century. Together with his colleagues, he substantiated most of it, but they were unable to prove some key points (including the Poincaré conjecture). Is the 3-sphere unique? A reliable answer to this question first appeared in Perelman's articles.

How can a manifold be geometrized and given uniform curvature everywhere? You need to take some arbitrary geometry with various protrusions and recesses, and then smooth out all the irregularities. In the early 90s. XX century Hamilton began analyzing 3-manifolds using the Ricci flow equation, named after the mathematician Gregorio Ricci-Curbastro. It is somewhat similar to the heat conduction equation, which describes heat flows flowing in an unevenly heated body until its temperature becomes the same everywhere. In the same way, the Ricci flow equation specifies a change in the curvature of the manifold that leads to the alignment of all protrusions and recesses. For example, if you start with an egg, it will gradually become spherical.

GEOMETRIZATION

TO CLASSIFY 2-manifolds, you can use uniformization or geometrization: assign them a certain geometry, a rigid form. In particular, every manifold can be transformed so that its curvature is distributed uniformly. Sphere (a) is a unique shape with constant positive curvature: it is curved everywhere like the top of a hill. The torus (b) can be made flat, i.e. everywhere having zero curvature. To do this, you need to cut it and straighten it. The resulting cylinder should be cut lengthwise and unfolded to form a rectangular plane. In other words, a torus can be mapped onto a plane. Surfaces of type 2 and higher (c) can be given a constant negative curvature, and their geometry will depend on the number of handles. Below is a saddle-shaped surface with constant negative curvature.

CLASSIFYING 3-VARIETIES is much more difficult. The 3-manifold has to be divided into parts, each of which can be transformed into one of the eight canonical 3-dimensional geometries. The example below (illustrated as a 2-manifold for simplicity) blue) is composed of 3-geometries with constant positive (a), zero (b) and constant negative (c) curvature, as well as the “products” of a 2-sphere and a circle (d) and a surface with negative curvature and a circle (e).

However, Hamilton encountered certain difficulties: in some cases, the Ricci flow leads to compression of the manifold and the formation of an infinitely thin neck. (This is different from heat flow: at the pinch points the temperature would be infinitely high.) One example is a dumbbell-shaped manifold. The spheres grow by drawing material from the bridge, which tapers into a point in the middle (see COMBATING FEATURES). In another case, when a thin rod protrudes from the manifold, the Ricci flow causes the appearance of a so-called cigar-shaped singularity. In a regular 3-manifold, the neighborhood of any point is a piece of ordinary three-dimensional space, which cannot be said about singular pinch points. The work of a Russian mathematician helped overcome this difficulty.

In 1992, after defending his Ph.D. thesis, Perelman arrived in the United States and spent several semesters at the State University of New York at Stony Brook, and then two years at the University of California at Berkeley. He quickly earned a reputation as a rising star, obtaining several important and profound results in one of the branches of geometry. Perelman was awarded a prize from the European Mathematical Society (which he declined) and received a prestigious invitation to speak at the International Congress of Mathematicians (which he accepted).

In the spring of 1995, he was offered positions at several prominent mathematical institutions, but he chose to return to his native St. Petersburg and essentially disappeared from view. For many years, the only sign of his activity were letters to former colleagues indicating errors made in the articles they published. Inquiries about the status of his own works went unanswered. And then at the end of 2002, several people received an email from Perelman informing them of an article that he had sent to a mathematical server. Thus began his attack on the Poincaré conjecture.

FIGHTING THE FEATURES

TRYING TO USE Ricci flow equation to prove the Poincaré conjecture and the geometrization of 3-manifolds, scientists encountered difficulties that Grigory Perelman managed to overcome. Using Ricci flow to gradually change the shape of a 3-manifold sometimes results in singularities. For example, when part of an object has the shape of a dumbbell (a), the tube between the spheres may become pinched to a point section, violating the properties of the manifold (b). It is also possible that the so-called cigar-shaped feature will appear.

PERELMAN SHOWED, that “surgeries” can be performed on features. When the manifold begins to pinch, cut out small sections on either side of the constriction point (c), cover the cut points with small spheres, and then use Ricci flow again (d). If the pinch occurs again, the procedure must be repeated. Perelman also proved that the cigar-shaped feature never appears.

Perelman added a new term to Ricci's flow equation. This change did not eliminate the peculiarity problem, but it did allow for much more in-depth analysis. The Russian scientist showed that a “surgical” operation can be performed on a dumbbell-shaped manifold: cut off a thin tube on both sides of the emerging constriction and seal the open tubes protruding from the balls with spherical caps. Then one should continue changing the “operated” manifold in accordance with the Ricci flow equation, and apply the above procedure to all emerging constrictions. Perelman also showed that cigar-shaped features cannot appear. Thus, any 3-manifold can be reduced to a set of parts with homogeneous geometry.

When Ricci flow and "surgery" are applied to all possible 3-manifolds, any one of them, if it is as simple as a 3-sphere (in other words, characterized by the same homotopy), necessarily reduces to the same homogeneous geometry as and 3-sphere. This means, from a topological point of view, the manifold in question is a 3-sphere. Thus, the 3-sphere is unique.

The value of Perelman's articles lies not only in the proof of the Poincaré conjecture, but also in new methods of analysis. Scientists around the world are already using the results obtained by the Russian mathematician in their work and applying the methods he developed in other areas. It turned out that the Ricci flow is associated with the so-called renormalization group, which determines how the strength of interactions changes depending on the particle collision energy. For example, at low energies the strength of the electromagnetic interaction is characterized by the number 0.0073 (approximately 1/137). However, when two electrons collide head-on at nearly the speed of light, the force approaches 0.0078. The mathematics that describes the change in physical forces is very similar to the mathematics that describes the geometrization of manifolds.

Increasing the collision energy is equivalent to studying the force at smaller distances. Therefore, the renormalization group is similar to a microscope with a variable magnification factor, which allows you to study the process at different levels of detail. Similarly, Ricci flow is a microscope for viewing manifolds. Protrusions and depressions visible at one magnification disappear at another. It is likely that on the Planck length scale (about $10^(–35)$ m) the space in which we live looks like foam with a complex topological structure (see the article “Atoms of Space and Time”, “In the World of Science”, No. 4, 2004). In addition, the equations of general relativity, which describe the characteristics of gravity and the large-scale structure of the Universe, are closely related to the Ricci flow equation. Paradoxically, the term Perelman added to the expression used by Hamilton originates in string theory, which purports to be a quantum theory of gravity. It is possible that in the articles of the Russian mathematician, scientists will find a lot more useful information not only about abstract 3-manifolds, but also about the space in which we live.

Graham P. Collins, Ph.D., is an editor at Scientific American. More information about Poincaré's theorem is available at www.sciam.com/ontheweb.

ADDITIONAL READING:

The Poincare Conjecture 99 Years Later: A Progress Report. John W. Milnor. February 2003. Available at www.math.sunysb.edu/~jack/PREPRINTS/poiproof.pdf
Jules Henri Poincare' (biography). October 2003. Available atwww-groups.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html
Millennium Problems. The Clay Mathematics Institute: www.claymath.org/millennium/
Notes and commentary on Perelman’s Ricci flow papers. Compiled by Bruce Kleiner and John Lott. Available at www.math.lsa.umich.edu/research/ricciflow/perelman.html
Topology. Eric W. Weisstein in Mathworld-A Wolfram Web Resource. Available at

Poincare's hypothesis and features of the Russian mentality.

In short: An unemployed professor, who is only 40 years old, has solved one of the 7 most difficult problems of humanity, lives in a panel house on the outskirts of the city with his mother, and instead of receiving the prize that all mathematicians in the world dream of, and a million dollars to boot, he left collect mushrooms and asked him not to disturb him.

And now in more detail:

http://lenta.ru/news/2006/08/16/perelman/

Grigory Perelman, who proved the Poincaré conjecture, refuses numerous awards and cash prizes awarded to him for this achievement, the Guardian newspaper reports. After extensive review of the evidence, which lasted almost four years, the scientific community concluded that Perelman's solution was correct.

The Poincaré conjecture is one of the seven most important mathematical “problems of the millennium,” for the solution of each of which the Clay Mathematics Institute awarded a prize of one million dollars. Thus, Perelman should receive a reward. The scientist does not communicate with the press, but the newspaper became it is known that Perelman does not want to take this money. According to the mathematician, the committee that awarded the award is not qualified enough to evaluate his work.

“It’s not safe to own a million dollars in St. Petersburg,” the professional community jokingly suggests another reason for Perelman’s unusual behavior. Nigel Hitchin, professor of mathematics at Oxford University, told the newspaper about this.

Next week, according to rumors, it will be announced that Perelman has been awarded the most prestigious international Fields Medal in this field, consisting of a precious medal and a monetary award. The Fields Medal is considered the mathematical equivalent of the Nobel Prize. It is awarded every four years at the International Mathematical Congress, and the prize winners should not be older than 40 years. Perelman, who will turn forty in 2006 and lose the chance to ever receive this prize, does not want to accept this award either.

It has long been known about Perelman that he avoids formal events and does not like to be admired. But in the current situation, the behavior of the scientist goes beyond the eccentricity of an armchair theorist. Perelman has already left his academic work and refuses to perform professorial functions. Now he wants to hide from recognition of his services to mathematics - his life’s work.

Grigory Perelman worked on the proof of Poincaré's theorem for eight years. In 2002, he posted a solution to the problem on the Los Alamos Scientific Laboratory preprint website. Until now, he has never published his work in a peer-reviewed journal, which is a prerequisite for most awards.

Perelman can be considered a reference product Soviet education. He was born in 1966 in Leningrad. He still lives in this city. Perelman studied at specialized school No. 239 with in-depth study of mathematics. He won countless Olympics. I was enrolled in mathematics and mechanics at Leningrad State University without exams. Received a Lenin scholarship. After university, he entered graduate school at the Leningrad branch of the V.A. Steklov Mathematical Institute, where he remained to work. In the late eighties, Perelman moved to the USA, taught at several universities, and then returned to his old place.

The state of Count Muravyov's St. Petersburg mansion on the Fontanka, where the Mathematical Institute is located, makes Perelman's lack of silver especially inadequate. The building, as the Izvestia newspaper reports, can collapse at any moment and fall into the river. The purchase of computer equipment (the only equipment needed by mathematicians) can still be financed with the help of various grants, but charitable organizations are not ready to pay for the restoration of the historical building.

==========================

http://www.newsinfo.ru/news/2006/08/news1325575.php

A hermit mathematician who proved one of the most difficult scientific hypotheses, Poincaré's theorem, is no less mysterious than the problem itself.

Little is known about him. I entered the institute based on the results of school Olympiads and received a Lenin scholarship. In St. Petersburg special school No. 239, he is remembered - the son of Yakov Perelman, the author of the famous textbook " Entertaining physics". Photo of Grisha Perelman - on the board of the greats along with Lobachevsky and Leibniz.

“He was such an excellent student, only in physical education... Otherwise there would have been a medal,” recalls his teacher Tamara Efimova, director of Physics and Mathematics Lyceum 239 in an interview with Channel One.

He was always for pure science, against formalities - these are the words of his former school teacher, one of the few with whom Perelman kept in touch throughout the eight years of his search. As he says, the mathematician had to leave his job because he had to write articles and reports, and Poincare absorbed all his time. Mathematics comes first.

Perelman spent eight years of his life solving one of the seven unsolvable mathematical problems. He worked alone, somewhere in the attic, secretly. He lectured in America to support himself at home. He left a job that distracted him from the main goal, does not answer calls and does not communicate with the press.

A million dollars is awarded for solving one of the seven unsolvable mathematical problems; this is the Fields Medal, a Nobel prize for mathematicians. Grigory Perelman became the main candidate for receiving it.

The scientist knows this, but, apparently, he is clearly not interested in monetary recognition. According to colleagues, he didn’t even submit documents for the award.

“As I understand, Grigory Yakovlevich himself does not care about a million at all,” says Ildar Ibragimov, academician of the Russian Academy of Sciences. “In fact, the people who are able to solve these problems are mostly people who will not work because of this money. They Something completely different will worry you."

Perelman published work on the Poincaré conjecture for the only time three years ago on the Internet. More likely not even a work, but a sketch of 39 pages. He does not agree to write a more detailed report with detailed evidence. Even the vice-president of the World Mathematical Society, who specially came to St. Petersburg to find Perelman, failed to do this.

Over the past three years, no one has been able to find an error in Perelman’s calculations, as required by the Fields Prize regulations. Q.E.D.

==============================

http://elementy.ru/news/430288

The process of proving the Poincaré conjecture is now apparently entering its final stage. Three groups of mathematicians have finally figured out the ideas of Grigory Perelman and over the past couple of months have presented their versions of a complete proof of this hypothesis.

A conjecture formulated by Poincaré in 1904 states that all three-dimensional surfaces in four-dimensional space that are homotopically equivalent to a sphere are homeomorphic to it. Speaking in simple words, if a three-dimensional surface is somewhat similar to a sphere, then if it is straightened out, it can only become a sphere and nothing else. For details about this conjecture and the history of its proof, read the popular article Problems of 2000: Poincaré's conjecture in the journal Computerra.

For the proof of the Poincaré conjecture, Mathematical Institute. Clay was awarded a prize of a million dollars, which may seem surprising: after all, we are talking about a very private, uninteresting fact. In fact, what is important for mathematicians is not so much the properties of a three-dimensional surface as the fact that the proof itself is difficult. This problem formulates in a concentrated form what could not be proven using previously existing ideas and methods of geometry and topology. It allows you to look at a deeper level, into that layer of problems that can only be solved with the help of the ideas of the “new generation”.

As in the situation with Fermat's theorem, it turned out that the Poincaré conjecture is a special case of a much more general statement about the geometric properties of arbitrary three-dimensional surfaces - Thurston's Geometrization Conjecture. Therefore, the efforts of mathematicians were not aimed at solving this special case, but to build a new mathematical approach that can cope with such problems.

The breakthrough was made in 2002-2003 by Russian mathematician Grigory Perelman. In his three articles math.DG/0211159, math.DG/0303109, math.DG/0307245, proposing a number of new ideas, he developed and completed the method proposed in the 1980s by Richard Hamilton. In his works, Perelman claims that the theory he constructed makes it possible to prove not only the Poincaré conjecture, but also the geometrization hypothesis.

The essence of the method is that for geometric objects it is possible to define some equation of “smooth evolution”, similar to the equation of the renormalization group in theoretical physics. The initial surface will be deformed during this evolution and, as Perelman showed, will eventually smoothly transform into a sphere. The strength of this approach is that, bypassing all the intermediate moments, you can immediately look “into infinity,” at the very end of evolution, and discover a sphere there.

Perelman's works marked the beginning of the intrigue. In his articles, he developed a general theory and outlined the key points of the proof of not only the Poincaré conjecture, but also the geometrization hypothesis. Perelman did not provide complete proof in all details, although he claimed that he had proven both hypotheses. Also in 2003, Perelman toured the United States with a series of lectures, during which he answered any technical questions from listeners clearly and in detail.

Immediately after the publication of Perelman's preprints, experts began checking the key points of his theory, and not a single error has yet been found. Moreover, over the past years, several teams of mathematicians have been able to absorb the ideas proposed by Perelman to such an extent that they began to write down the complete proof “in full.”

In May 2006, a paper by B. Kleiner, J. Lott, math.DG/0605667, appeared, in which a detailed derivation of omitted points in Perelman’s proof was given. (By the way, these authors maintain a web page dedicated to Perelman's articles and related work.)

Then in June 2006, the Asian Journal of Mathematics published a 327-page paper by Chinese mathematicians Huai-Dong Cao and Xi-Ping Zhu entitled "A complete proof of the Poincaré and geometrization conjectures - an application of the Hamilton-Perelman theory of Ricci flows." The authors themselves do not claim to have a completely new proof, but only claim that Perelman’s approach really works.

Finally, the other day a 473-page article (or already a book?) appeared by J. W. Morgan, G. Tian, math.DG/0607607, in which the authors, following in the footsteps of Perelman, present their proof of the Poincaré conjecture (and not the more general geometrization hypothesis). John Morgan is considered one of the main experts on this problem, and after the publication of his work, it can apparently be considered that the Poincaré conjecture has been finally proven.

It is interesting, by the way, that at first the article by Chinese mathematicians was distributed only in a paper version at a price of $69, so not everyone had the opportunity to look at it. But the very next day after Morgan-Tian’s article appeared in the preprint archive, the Asian Journal of Mathematics website also appeared electronic version articles.

Time will tell whose refinement of Perelman’s evidence is more accurate and transparent. It is possible that in the coming years it will be simplified, as happened with Fermat’s theorem. So far, we can only see an increase in the volume of publications: from Perelman’s 30-page articles to a thick book by Morgan and Tian, but this is not due to the complication of the proof, but to a more detailed derivation of all intermediate steps.

In the meantime, the final proof of the conjecture and, perhaps, who will be awarded the Clay Institute Prize are expected to be “officially” announced at the International Congress of Mathematicians in Madrid this August. In addition, there are rumors that Grigory Perelman will become one of four Fields Medalists, the highest distinction for young mathematicians.

This news spread across the CIS media. 39-year-old St. Petersburg scientist GRIGORY PERELMAN is a real candidate for the Fields Medal ($1 million), the highest award in the mathematical world (as is known, the Nobel Prize is not awarded to mathematicians).

The French mathematician Poincaré tried to find out whether three-dimensional space is a sphere. He could not find evidence of this thesis or refute it. Among the strange consequences of the Poincaré hypothesis, which run counter to our everyday ideas, we highlight the following: with the help of some super-powerful telescope, peering into the cosmic distance from the Earth, you can clearly see your native... Earth, or, flying off on a long space journey, end up in departure point.

Every few years, attempts to prove the Poincaré conjecture are published in scientific journals, but none of the proposed solutions has yet passed the scientific test. In the end it turned out that the proof was incorrect. Grigory Perelman published his works on the Internet in 2002, and no one refuted them (control period - 2 years). Moreover, many prominent scientists believe that Perelman’s decision is correct. And they complain that his works are very concise, concise and take up only a few dozen pages (60).

The rules for receiving the award require publication in the pages of a regularly published scientific journal and compliance with some other formalities. Petersburg resident Perelman, who earns about $200 (6,000 rubles) at his home institute, ignores them. These are his rules of life. Firm adherence to them may have made it possible to achieve unique scientific results. St. Petersburg journalists tried to meet the original, so consistent with popular ideas about geniuses. All they managed to find out: Perelman is a regular at concerts classical music Petersburg Philharmonic, eats porridge, is indifferent to clothes, is considered strange even in his scientific community and cannot stand the press.

So, about an unexpected consequence of Poincaré's theorem. A million dollars is nothing to someone who knows what space is. We would like Mr. Perelman's iron confidence.

Comment from a specialist - corresponding member of the National Academy of Sciences of Ukraine, mathematician Vladimir Sharko:

Now, in addition to the works of the Russian mathematician, a proof has appeared by Chinese professors Zhu Xiping and Lehai Cao, and the second is presented by the Americans, headed by John Morgan. But the championship, of course, belongs to Perelman. Although in fact there is no proof of it. Precisely because it has not been published, but exists only briefly, in abstracts. Perelman’s work “hangs” on websites, just like any other unofficial work.

- Is Perelman really that eccentric?

He is a sweet, pleasant person to talk to. A typical St. Petersburg intellectual. We met at various scientific conferences. It can hardly be called strange. Perhaps he is somewhat annoyed by journalists and is playing a prank on them.

It only seems that the bonus is already in his pocket, so his behavior is considered strange. Awards of this rank require the support of colleagues and the scientific community. And the Russians, unfortunately, cannot provide adequate support. Therefore, it is too early to talk about the award. Although the St. Petersburg resident really refused other awards.

- Does Perelman’s discovery have any practical significance?

Not yet. But, as a rule, mathematical discoveries eventually find application. For example, the achievements of mathematics are actively used in modern weather forecasting. Biologists now work closely with mathematicians. After all, it was with the help of the former that the genome was deciphered. Computers also appeared thanks to the work of mathematicians. It is actually a very useful and practical science.

- Can the people of Kiev boast of any breakthrough?

The most pleasant news: young people are appearing at the Kiev Institute of Mathematics. It's no secret what happened hard time and people left, especially young people. But the director of the institute, academician Anatoly Samoilenko, managed to keep it at the proper level, which was very difficult. Now we can talk about normalizing the situation.

Recently, a Kiev guy from Polytechnic took first place at the European Student Olympiad. Which, in general, indicates a good level of teaching mathematics, scientific work in Kyiv. There are famous mathematical schools in Ukraine: in Donetsk, Kharkov; The famous pre-war Lviv school of mathematicians began to revive. Perhaps someday we will delight the scientific community with brilliant works.

My digression: The Poincaré conjecture states: Every simply connected compact three-dimensional manifold without boundary is homeomorphic to a three-dimensional sphere.