What is the essence of Poincaré's theorem?
- E was proved by RED-haired Sophia, but she is also RED-haired....
- The bottom line is that the Universe is not shaped like a sphere, but like a donut.
- The meaning of the Poincaré conjecture in its original formulation is that for any three-dimensional body without holes there is a transformation that will allow it to be turned into a ball without cutting and gluing. If this seems obvious, then what if space is not three-dimensional, but contains ten or eleven dimensions (that is, we are talking about a generalized formulation of the Poincaré conjecture, which Perelman proved)
- you can't tell it in 2 words
- In 1900, Poincaré suggested that a three-dimensional manifold with all the homology groups of a sphere is homeomorphic to a sphere. In 1904, he also found a counterexample, now called the Poincaré sphere, and formulated the final version of his hypothesis. Attempts to prove the Poincaré conjecture have led to numerous advances in the topology of manifolds.
Proofs of the generalized Poincaré conjecture for n #10878; 5 were obtained in the early 1960s and 1970s almost simultaneously by Smale, independently and by other methods by Stallings (English) (for n #10878; 7, his proof was extended to the cases n = 5 and 6 by Zeeman (English)). A proof of the much more difficult case n = 4 was obtained only in 1982 by Friedman. From Novikov's theorem on the topological invariance of Pontryagin's characteristic classes it follows that there exist homotopy equivalent, but not homeomorphic, manifolds in high dimensions.
The proof of the original Poincaré conjecture (and the more general Trston conjecture) was found only in 2002 by Grigory Perelman. Subsequently, Perelman's proof was verified and presented in expanded form by at least three groups of scientists. 1 The proof uses Ricci flow with surgery and largely follows the plan outlined by Hamilton, who was also the first to use Ricci flow.
- who is this
- Poincare's theorem:
Poincaré's theorem on vector fields
Bendixson's Poincaré theorem
Poincaré's theorem on the classification of circle homeomorphisms
Poincaré's conjecture on the homotopy sphere
Poincaré's return theoremWhich one are you asking about?
- In the theory of dynamical systems, Poincaré's theorem on the classification of homeomorphisms of the circle describes possible types of invertible dynamics on the circle, depending on the rotation number p(f) of the iterated mapping f. Roughly speaking, it turns out that the dynamics of mapping iterations are to a certain extent similar to the dynamics of rotation by the corresponding angle.
Namely, let a circle homeomorphism f be given. Then:
1) The rotation number is rational if and only if f has periodic points. In this case, the denominator of the rotation number is the period of any periodic point, and the cyclic order on the circle of the points of any periodic orbit is the same as that of the points of the rotation orbit on p(f). Further, any trajectory tends to some periodicity both in forward and in reverse time (the a- and -w limit trajectories may be different).
2) If the rotation number f is irrational, then two options are possible:
i) either f has a dense orbit, in which case the homeomorphism of f is conjugate to a rotation by p(f). In this case, all orbits of f are dense (since this is true for irrational rotation);
ii) either f has a Cantor invariant set C, which is the only minimal set of the system. In this case, all trajectories tend to C both in forward and backward time. In addition, the mapping f is semiconjugate to the rotation by p(f): for some mapping h of degree 1, p o f =R p (f) o hMoreover, the set C is exactly the set of growth points of h; in other words, from a topological point of view, h collapses the complement intervals of C.
- the crux of the matter is $1 million
- The fact that no one understands her except 1 person
- In French foreign policy...
- Here Lka answered best of all http://otvet.mail.ru/question/24963208/
- A brilliant mathematician, 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, as 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 Trsten 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 an idea of a new generation, the solution of which opens up new horizons for mathematical science.
It is interesting that for some reason Perelman himself did not bother to bring his decision to final brilliance. Having described the solution as a whole in the preprint The entropy formula for the Ricci flow and its geometric applications in November 2002, in March 2003 he supplemented the proof and presented it in the preprint Ricci flow with surgery on three-manifolds, and also reported on the method in the series lectures that he 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 Clay Mathematical Institute 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 gave the final proof of the Poincaré conjecture.
- The generalized Poincaré conjecture states that:
For any n, any manifold of dimension n is homotopy equivalent to a sphere of dimension n if and only if it is homeomorphic to it.
The original Poincaré conjecture is a special case of the generalized conjecture for n = 3.
For clarification, go to the forest to pick mushrooms, Grigory Perelman goes there) - Poincaré's return theorem is one of the basic theorems of ergodic theory. Its essence is that with a measure-preserving mapping of space onto itself, almost every point will return to its initial neighborhood. The full formulation of the theorem is as follows: 1:
Let be a measure-preserving transformation of a space with finite measure, and let be a measurable set. Then for any natural
.
This theorem has an unexpected consequence: it turns out that if in a vessel divided by a partition into two compartments, one of which is filled with gas and the other is empty, the partition is removed, then after some time all the gas molecules will again gather in the original part of the vessel. The solution to this paradox is that some time is on the order of billions of years. - he has theorems like slaughtered dogs in Korea...
the universe is spherical... http://ru.wikipedia.org/wiki/Poincaré, _Henri
Yesterday scientists announced that the universe is a frozen substance... and asked for a lot of money to prove this... again the Merikos will turn on the printing press... for the amusement of eggheads...
- Try to prove where is up and down in zero gravity.
- Yesterday there was a wonderful film on CULTURE, in which this problem was explained in detail. Maybe they still have it?
http://video.yandex.ru/#search?text=РРР SR R РРРРР ССРРРwhere=allfilmId=36766495-03-12
Log in to Yandex and write Film about Perelman and go to the film
Grigory Perelman. refusenik
Vasily Maksimov
In August 2006, the names of the best mathematicians on the planet were announced who received the prestigious Fields Medal - a kind of analogue of the Nobel Prize, which mathematicians, at the whim of Alfred Nobel, were deprived of. The Fields Medal - in addition to a badge of honor, the winners are awarded a check for fifteen thousand Canadian dollars - is awarded by the International Congress of Mathematicians every four years. It was established by Canadian scientist John Charles Fields and was first awarded in 1936. Since 1950, the Fields Medal has been awarded regularly personally by the King of Spain for his contribution to the development of mathematical science. Prize winners can be from one to four scientists under the age of forty. Forty-four mathematicians, including eight Russians, have already received the prize.
Grigory Perelman. Henri Poincaré.
In 2006, the laureates were the Frenchman Wendelin Werner, the Australian Terence Tao and two Russians - Andrey Okunkov working in the USA and Grigory Perelman, a scientist from St. Petersburg. However, at the last moment it became known that Perelman refused this prestigious award - as the organizers announced, “for reasons of principle.”
Such an extravagant act by the Russian mathematician did not come as a surprise to people who knew him. This is not the first time he has refused mathematical awards, explaining his decision by saying that he does not like ceremonial events and unnecessary hype around his name. Ten years ago, in 1996, Perelman refused the European Mathematical Congress prize, citing the fact that he had not completed the work on the scientific problem nominated for the award, and this was not the last case. The Russian mathematician seemed to make it his life’s goal to surprise people, going against public opinion and the scientific community.
Grigory Yakovlevich Perelman was born on June 13, 1966 in Leningrad. From a young age, he was fond of exact sciences, brilliantly graduated from the famous 239th secondary school with in-depth study of mathematics, won numerous mathematical Olympiads: for example, in 1982, as part of a team of Soviet schoolchildren, he participated in the International Mathematical Olympiad, held in Budapest. Without exams, Perelman was enrolled in the Faculty of Mechanics and Mathematics at Leningrad University, where he studied with excellent marks, continuing to win mathematical competitions at all levels. After graduating from the university with honors, he entered graduate school at the St. Petersburg branch of the Steklov Mathematical Institute. His scientific supervisor was the famous mathematician Academician Aleksandrov. Having defended his Ph.D. thesis, Grigory Perelman remained at the institute, in the laboratory of geometry and topology. His work on the theory of Alexandrov spaces is known; he was able to find evidence for a number of important conjectures. Despite numerous offers from leading Western universities, Perelman prefers to work in Russia.
His most notable success was the solution in 2002 of the famous Poincaré conjecture, published in 1904 and since then remained unproven. Perelman worked on it for eight years. The Poincaré conjecture was considered one of the greatest mathematical mysteries, and its solution was considered the most important achievement in mathematical science: it would immediately advance research into the problems of the physical and mathematical foundations of the universe. The most prominent minds on the planet predicted its solution only in a few decades, and the Clay Institute of Mathematics in Cambridge, Massachusetts, included the Poincaré problem among the seven most interesting unsolved mathematical problems of the millennium, for the solution of each of which a million dollar prize was promised (Millennium Prize Problems). .
The conjecture (sometimes called the problem) of the French mathematician Henri Poincaré (1854–1912) is formulated as follows: any closed simply connected three-dimensional space is homeomorphic to a three-dimensional sphere. To clarify, use a clear example: if you wrap an apple with a rubber band, then, in principle, by tightening the tape, you can compress the apple into a point. If you wrap a donut with the same tape, you cannot compress it to a point without tearing either the donut or the rubber. In this context, an apple is called a “simply connected” figure, but a donut is not simply connected. Almost a hundred years ago, Poincaré established that a two-dimensional sphere is simply connected, and suggested that a three-dimensional sphere is also simply connected. The best mathematicians in the world could not prove this hypothesis.
To qualify for the Clay Institute Prize, Perelman only had to publish his solution in one of the scientific journals, and if within two years no one could find an error in his calculations, then the solution would be considered correct. However, Perelman deviated from the rules from the very beginning, publishing his decision on the preprint website of the Los Alamos Scientific Laboratory. Perhaps he was afraid that an error had crept into his calculations - a similar story had already happened in mathematics. In 1994, the English mathematician Andrew Wiles proposed a solution to Fermat’s famous theorem, and a few months later it turned out that an error had crept into his calculations (although it was later corrected, and the sensation still took place). There is still no official publication of the proof of the Poincaré conjecture, but there is an authoritative opinion of the best mathematicians on the planet confirming the correctness of Perelman’s calculations.
The Fields Medal was awarded to Grigory Perelman precisely for solving the Poincaré problem. But the Russian scientist refused the prize, which he undoubtedly deserves. “Gregory told me that he feels isolated from the international mathematical community, outside this community, and therefore does not want to receive the award,” Englishman John Ball, president of the World Union of Mathematicians (WUM), said at a press conference in Madrid.
There are rumors that Grigory Perelman is going to leave science altogether: six months ago he resigned from his native Steklov Mathematical Institute, and they say that he will no longer study mathematics. Perhaps the Russian scientist believes that by proving the famous hypothesis, he has done everything he could for science. But who will undertake to discuss the train of thought of such a bright scientist and extraordinary person?.. Perelman refuses any comments, and he told The Daily Telegraph newspaper: “None of what I can say is of the slightest public interest.” However, leading scientific publications were unanimous in their assessments when they reported that “Grigory Perelman, having resolved the Poincaré theorem, stood on a par with the greatest geniuses of the past and present.”
Monthly literary and journalistic magazine and publishing house.
Scientists believe that 38-year-old Russian mathematician Grigory Perelman proposed the correct solution to the Poincaré problem. Keith Devlin, a professor of mathematics at Stanford University, said this at the science festival in Exeter (UK).
Poincaré's problem (also called a problem or hypothesis) is one of the seven most important mathematical problems, for the solution of each of which he awarded a prize of one million dollars. This is what attracted such widespread attention to the results obtained by Grigory Perelman, an employee of the laboratory of mathematical physics.
Scientists around the world learned about Perelman’s achievements from two preprints (articles preceding a full-fledged scientific publication), posted by the author in November 2002 and March 2003 on the website of the archive of preliminary works of the Los Alamos Scientific Laboratory.
According to the rules adopted by the Clay Institute's Scientific Advisory Board, a new hypothesis must be published in a specialized journal of "international reputation." In addition, according to the Institute's rules, the decision to pay the prize is ultimately made by the "mathematical community": the proof must not be refuted within two years after publication. Every proof is checked by mathematicians in different countries of the world.
Poincaré problem
Born on June 13, 1966 in Leningrad, into a family of employees. He graduated from the famous secondary school No. 239 with in-depth study of mathematics. In 1982, as part of a team of Soviet schoolchildren, he participated in the International Mathematical Olympiad, held in Budapest. He was enrolled in mathematics and mechanics at Leningrad State University without exams. He won faculty, city and all-Union student mathematical Olympiads. Received a Lenin scholarship. After graduating from the university, Perelman entered graduate school at the St. Petersburg branch of the Steklov Mathematical Institute. Candidate of Physical and Mathematical Sciences. Works in the laboratory of mathematical physics.
Poincaré's problem relates to the area of the so-called topology of manifolds - spaces arranged in a special way that have different dimensions. Two-dimensional manifolds can be visualized, for example, using the example of the surface of three-dimensional bodies - a sphere (the surface of a ball) or a torus (the surface of a donut).
It is easy to imagine what will happen to a balloon if it is deformed (bent, twisted, pulled, compressed, pinched, deflated or inflated). It is clear that with all the above deformations, the ball will change its shape over a wide range. However, we will never be able to turn a ball into a donut (or vice versa) without breaking the continuity of its surface, that is, without tearing it apart. In this case, topologists say that the sphere (ball) is non-homeomorphic to the torus (donut). This means that these surfaces cannot be mapped to one another. In simple terms, a sphere and a torus are different in their topological properties. And the surface of a balloon, under all its possible deformations, is homeomorphic to a sphere, just as the surface of a lifebuoy is to a torus. In other words, any closed two-dimensional surface that does not have through holes has the same topological properties as a two-dimensional sphere.
TOPOLOGY, a branch of mathematics that deals with the study of the properties of figures (or spaces) that are preserved under continuous deformations, such as stretching, compression or bending. Continuous deformation is a deformation of a figure in which there are no breaks (i.e., violation of the integrity of the figure) or gluing (i.e., identification of its points).
TOPOLOGICAL TRANSFORMATION of one geometric figure to another is a mapping of an arbitrary point P of the first figure to point P' of another figure, which satisfies the following conditions: 1) each point P of the first figure must correspond to one and only one point P' of the second figure, and vice versa; 2) The mapping must be mutually continuous. For example, there are two points P and N belonging to the same figure. If, when point P moves to point N, the distance between them tends to zero, then the distance between points P' and N' of another figure should also tend to zero, and vice versa.
HOMEOMORPHISM. Geometric figures that transform into one another during topological transformations are called homeomorphic. The circle and the boundary of a square are homeomorphic, since they can be converted into each other by a topological transformation (i.e., bending and stretching without breaking or gluing, for example, stretching the boundary of a square to the circle circumscribed around it). A region in which any closed simple (i.e., homeomorphic to a circle) curve can be contracted to a point while remaining in this region all the time is called simply connected, and the corresponding property of the region is simply connected. If some closed simple curve of this region cannot be contracted to a point, remaining all the time in this region, then the region is called multiply connected, and the corresponding property of the region is called multiply connected.
Poincaré's problem states the same thing for three-dimensional manifolds (for two-dimensional manifolds, such as the sphere, this point was proven back in the 19th century). As the French mathematician noted, one of the most important properties of a two-dimensional sphere is that any closed loop (for example, a lasso) lying on it can be pulled to one point without leaving the surface. For a torus, this is not always true: a loop passing through its hole will be pulled to a point either when the torus is broken, or when the loop itself is broken. In 1904, Poincaré proposed that if a loop can contract to a point on a closed three-dimensional surface, then such a surface is homeomorphic to a three-dimensional sphere. Proving this hypothesis turned out to be an extremely difficult task.
Let us immediately clarify: the formulation of the Poincaré problem we mentioned does not speak at all about a three-dimensional ball, which we can imagine without much difficulty, but about a three-dimensional sphere, that is, about the surface of a four-dimensional ball, which is much more difficult to imagine. But in the late 1950s, it suddenly became clear that high-dimensional manifolds were much easier to work with than three- and four-dimensional ones. Obviously, the lack of clarity is far from the main difficulty that mathematicians face in their research.
A problem similar to Poincaré's for dimensions 5 and higher was solved in 1960 by Stephen Smale, John Stallings, and Andrew Wallace. The approaches used by these scientists, however, turned out to be inapplicable to four-dimensional manifolds. For them, the Poincaré problem was proven only in 1981 by Michael Freedman. The three-dimensional case turned out to be the most difficult; Grigory Perelman proposes his solution.
It should be noted that Perelman has a rival. In April 2002, Martin Dunwoody, a professor of mathematics at the British University of Southampton, proposed his method for solving the Poincaré problem and is now awaiting a verdict from the Clay Institute.
Experts believe that solving the Poincaré problem will make it possible to take a serious step in the mathematical description of physical processes in complex three-dimensional objects and will give new impetus to the development of computer topology. The method proposed by Grigory Perelman will lead to the opening of a new direction in geometry and topology. The St. Petersburg mathematician may well qualify for the Fields Prize (analogous to the Nobel Prize, which is not awarded in mathematics).
Meanwhile, some find Grigory Perelman's behavior strange. Here is what the British newspaper The Guardian writes: “Most likely, Perelman’s approach to solving the Poincaré problem is correct. But not everything is so simple. Perelman does not provide evidence that the work was published as a full-fledged scientific publication (preprints are not considered such). And this necessary if a person wants to receive an award from the Clay Institute. Besides, he shows no interest in money at all."
Apparently, for Grigory Perelman, as for a real scientist, money is not the main thing. For solving any of the so-called “millennium problems”, a true mathematician will sell his soul to the devil.
Millennium List
On August 8, 1900, at the International Congress of Mathematics in Paris, mathematician David Hilbert outlined a list of problems that he believed would have to be solved in the twentieth century. There were 23 items on the list. Twenty-one of them have been resolved so far. The last problem on Hilbert's list to be solved was Fermat's famous theorem, which scientists had been unable to solve for 358 years. In 1994, Briton Andrew Wiles proposed his solution. It turned out to be true.
Following the example of Gilbert, at the end of the last century, many mathematicians tried to formulate similar strategic tasks for the 21st century. One of these lists became widely known thanks to Boston billionaire Landon T. Clay. In 1998, with his funds, prizes were founded and established in Cambridge (Massachusetts, USA) for solving a number of the most important problems of modern mathematics. On May 24, 2000, the institute's experts selected seven problems - according to the number of millions of dollars allocated for the prize. The list is called Millennium Prize Problems:
1. Cook's problem (formulated in 1971)
Let's say that you, being in a large company, want to make sure that your friend is there too. If they tell you that he is sitting in the corner, then a split second will be enough for you to take a glance and be convinced of the truth of the information. Without this information, you will be forced to walk around the entire room, looking at the guests. This suggests that solving a problem often takes longer than checking the correctness of the solution.
Stephen Cook formulated the problem: can checking the correctness of a solution to a problem take longer than obtaining the solution itself, regardless of the verification algorithm. This problem is also one of the unsolved problems in the field of logic and computer science. Its solution could revolutionize the fundamentals of cryptography used in data transmission and storage.
2. Riemann hypothesis (formulated in 1859)
Some integers cannot be expressed as the product of two smaller integers, such as 2, 3, 5, 7, and so on. Such numbers are called prime numbers and play an important role in pure mathematics and its applications. The distribution of prime numbers among the series of all natural numbers does not follow any pattern. However, the German mathematician Riemann made a conjecture concerning the properties of a sequence of prime numbers. If the Riemann Hypothesis is proven, it will lead to a revolutionary change in our knowledge of encryption and an unprecedented breakthrough in Internet security.
3. Birch and Swinnerton-Dyer hypothesis (formulated in 1960)
Associated with the description of the set of solutions to some algebraic equations in several variables with integer coefficients. An example of such an equation is the expression x 2 + y 2 = z 2. Euclid gave a complete description of the solutions to this equation, but for more complex equations, finding solutions becomes extremely difficult.
4. Hodge's hypothesis (formulated in 1941)
In the 20th century, mathematicians discovered a powerful method for studying the shape of complex objects. The main idea is to use simple “bricks” instead of the object itself, which are glued together and form its likeness. Hodge's hypothesis is associated with some assumptions regarding the properties of such “building blocks” and objects.
5. Navier - Stokes equations (formulated in 1822)
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If you sail in a boat on a lake, waves will arise, and if you fly in an airplane, turbulent currents will arise in the air. It is assumed that these and other phenomena are described by equations known as the Navier-Stokes equations. The solutions to these equations are unknown, and it is not even known how to solve them. It is necessary to show that a solution exists and is a sufficiently smooth function. Solving this problem will significantly change the methods of carrying out hydro- and aerodynamic calculations.
6. Poincaré problem (formulated in 1904)
If you pull a rubber band over an apple, you can, by slowly moving the band without lifting it from the surface, compress it to a point. On the other hand, if the same rubber band is suitably stretched around a donut, there is no way to compress the band to a point without tearing the tape or breaking the donut. They say that the surface of an apple is simply connected, but the surface of a donut is not. It turned out to be so difficult to prove that only the sphere is simply connected that mathematicians are still looking for the correct answer.
7. Yang-Mills equations (formulated in 1954)
The equations of quantum physics describe the world of elementary particles. Physicists Young and Mills, having discovered the connection between geometry and particle physics, wrote their equations. Thus, they found a way to unify the theories of electromagnetic, weak and strong interactions. The Yang-Mills equations implied the existence of particles that were actually observed in laboratories all over the world, so the Yang-Mills theory is accepted by most physicists despite the fact that within the framework of this theory it is still not possible to predict the masses of elementary particles.
Mikhail Vitebsky
"The problem that was solved Perelman, is the requirement to prove a hypothesis put forward in 1904 by the great French mathematician Henri Poincaré(1854-1912) and bearing his name. It is difficult to say better about the role of Poincaré in mathematics than is done in the encyclopedia: “Poincaré’s works in the field of mathematics, on the one hand, complete the classical direction, and on the other, open the way to the development of new mathematics, where, along with quantitative relationships, facts are established that have qualitative character" (TSB, 3rd ed., vol. 2). The Poincaré conjecture is precisely of a qualitative nature - like the entire area of mathematics (namely topology) to which it relates and in the creation of which Poincaré took a decisive part.
In modern language, the Poincaré conjecture sounds like this: every simply connected compact three-dimensional manifold without boundary is homeomorphic to a three-dimensional sphere.
In the following paragraphs we will try to at least partially and very roughly explain the meaning of this terrifying verbal formula. To begin with, we note that an ordinary sphere, which is the surface of an ordinary ball, is two-dimensional (and the ball itself is three-dimensional). A two-dimensional sphere consists of all points of three-dimensional space that are equidistant from some selected point, called the center, which does not belong to the sphere. A three-dimensional sphere consists of all points of four-dimensional space that are equidistant from its center (which does not belong to the sphere). Unlike two-dimensional spheres, three-dimensional spheres not available our direct observation, and it is as difficult for us to imagine them as it was for Vasily Ivanovich to imagine the square trinomial from the famous joke. It is possible, however, that we are all in the three-dimensional sphere, that is, that our Universe is a three-dimensional sphere.
This is the meaning of the result Perelman for physics and astronomy. The term “simply connected compact three-dimensional manifold without edge” contains indications of the supposed properties of our Universe. The term “homeomorphic” means a certain high degree of similarity, in a certain sense, indistinguishability. The formulation as a whole means, therefore, that if our Universe has all the properties of a simply connected compact three-dimensional manifold without an edge, then it - in the same “known sense” - is a three-dimensional sphere.
The concept of simply connectedness is a fairly simple concept. Let's imagine a rubber band (that is, a rubber thread with glued ends) so elastic that if you don't hold it, it will shrink to a point. We will also require from our elastic band that when pulled to a point, it does not extend beyond the surface on which we placed it. If we stretch such an elastic band on a plane and release it, it will immediately shrink to a point. The same thing will happen if we place an elastic band on the surface of a globe, that is, on a sphere. For the surface of a lifebuoy, the situation will be completely different: the kind reader will easily find such arrangements of the elastic on this surface in which it is impossible to pull the elastic to a point without going beyond the surface in question. A geometric figure is called simply connected if any closed contour located within the limits of this figure can be contracted to a point without going beyond the named limits. We have just seen that the plane and the sphere are simply connected, but the surface of the lifebuoy is not simply connected. A plane with a hole cut in it is not simply connected either. The concept of simply connectedness also applies to three-dimensional figures. Thus, a cube and a ball are simply connected: any closed contour located in their thickness can be contracted to a point, and during the contraction process the contour will always remain in this thickness. But the bagel is not simply connected: in it you can find a contour that cannot be contracted to a point so that during the process of contraction the contour is always in the dough of the bagel. The pretzel is not monoconnected either. It can be proven that the three-dimensional sphere is simply connected.
We hope that the reader has not forgotten the difference between a segment and an interval, which is taught at school. A segment has two ends; it consists of these ends and all the points located between them. An interval consists only of all the points located between its ends; the ends themselves are not included in the interval: we can say that an interval is a segment with the ends removed from it, and a segment is an interval with the ends added to it. An interval and a segment are the simplest examples of one-dimensional manifolds, where an interval is a manifold without an edge, and a segment is a manifold with an edge; an edge in the case of a segment consists of two ends. The main property of manifolds, which underlies their definition, is that in the manifold the neighborhoods of all points, with the exception of points on the edge (which may not exist), are arranged in exactly the same way.
In this case, the neighborhood of a point A is the collection of all points located close to this point A. A microscopic creature living in a manifold without an edge and capable of seeing only the points of this manifold closest to itself is not able to determine at which point it is, being, is: around itself it always sees the same thing. More examples of one-dimensional manifolds without edge: the entire straight line, a circle. An example of a one-dimensional figure that is not a manifold is a line in the shape of the letter T: there is a special point, the neighborhood of which is not similar to the neighborhood of other points - this is the point where three segments meet. Another example of a one-dimensional manifold is a figure-eight line; Four lines converge at a special point here. A plane, a sphere, and the surface of a lifebuoy are examples of two-dimensional manifolds without an edge. A plane with a hole cut out in it will also be a manifold - but with or without an edge, it depends on where we place the contour of the hole. If we refer it to a hole, we get a manifold without an edge; if we leave the contour on the plane, we get a manifold with an edge, which is what this contour will serve as. Of course, we had in mind here an ideal mathematical cutting, and in real physical cutting with scissors, the question of where the contour belongs does not make any sense.
A few words about three-dimensional manifolds. The sphere, together with the sphere that serves as its surface, is a manifold with an edge; the indicated sphere is precisely this edge. If we remove this ball from the surrounding space, we get a manifold without an edge. If we peel off the surface of a ball, we get what is called a “sanded ball” in mathematical jargon, and an open ball in more scientific language. If we remove an open ball from the surrounding space, we get a manifold with an edge, and the edge will be the very sphere that we tore off from the ball. The bagel, together with its crust, is a three-dimensional manifold with an edge, and if you tear off the crust (which we treat as infinitely thin, that is, as a surface), we get a manifold without an edge in the form of a “sanded bagel.” All space as a whole, if we understand it as it is understood in high school, is a three-dimensional manifold without an edge.
The mathematical concept of compactness partly reflects the meaning that the word “compact” has in everyday Russian: “close”, “compressed”. A geometric figure is called compact if, for any arrangement of an infinite number of its points, they accumulate to one of the points or to many points of the same figure. A segment is compact: for any infinite set of its points in the segment there is at least one so-called limit point, any neighborhood of which contains infinitely many elements of the set under consideration. An interval is not compact: you can specify a set of its points that accumulates towards its end, and only towards it - but the end does not belong to the interval!
Due to lack of space, we will limit ourselves to this commentary. Let's just say that of the examples we have considered, the compact ones are a segment, a circle, a sphere, the surfaces of a bagel and a pretzel, a ball (together with its sphere), a bagel and a pretzel (together with its crusts). In contrast, interval, plane, sanded ball, bagel and pretzel are not compact. Among three-dimensional compact geometric figures without an edge, the simplest is the three-dimensional sphere, but such figures do not fit in our usual “school” space. Perhaps the most profound of those concepts that are connected by the hypothesis Poincare, is the concept of homeomorphy. Homeomorphy is the highest level of geometric sameness . Now we will try to give an approximate explanation of this concept by gradually approaching it.
Already in school geometry we encounter two types of sameness - the congruence of figures and their similarity. Recall that figures are called congruent if they coincide with each other when superimposed. At school, congruent figures do not seem to be distinguished, and therefore congruence is called equality. Congruent figures have the same dimensions in all their details. Similarity, without requiring the same size, means the same proportions of these sizes; therefore, similarity reflects a more essential similarity of figures than congruence. Geometry in general is a higher level of abstraction than physics, and physics is higher than materials science.
Take for example the ball bearing, billiard ball, croquet ball and ball. Physics does not delve into such details as the material from which they are made, but is only interested in such properties as volume, weight, electrical conductivity, etc. For mathematics, they are all balls, differing only in size. If the balls have different sizes, then they are different for metric geometry, but they are all the same for similarity geometry. From the point of view of geometry, all balls and all cubes are similar, but a ball and a cube are not the same.
Now let's look at the torus. Top is the geometric figure whose shape is shaped like a steering wheel and a lifebuoy. The Encyclopedia defines a torus as a figure obtained by rotating a circle around an axis located outside the circle. We urge the kind reader to realize that the ball and the cube are “more alike” with each other than each of them with the torus. The following thought experiment allows us to fill this intuitive awareness with precise meaning. Let's imagine a ball made of a material so pliable that it can be bent, stretched, compressed and, in general, deformed in any way - it just cannot be torn or glued together. Obviously, the ball can then be turned into a cube, but it is impossible to turn into a torus. Ushakov's explanatory dictionary defines a pretzel as a pastry (literally: like a buttery twisted bun) in the shape of the letter B. With all due respect to this wonderful dictionary, the words “in the shape of the number 8” seem to me more accurate; However, from the point of view expressed in the concept of homeomorphy, baking in the shape of the number 8, baking in the shape of the letter B, and baking in the shape of fita have the same shape. Even if we assume that bakers were able to obtain dough that has the above-mentioned pliability properties, a bun is impossible - without tears and gluing! - turn into neither a bagel nor a pretzel, just like the last two baked goods into each other. But you can turn a spherical bun into a cube or a pyramid. The kind reader will undoubtedly be able to find a possible form of baking into which neither a bun, nor a pretzel, nor a bagel can be turned.
Without naming this concept, we have already become acquainted with homeomorphy. Two figures are called homeomorphic if one can be transformed into the other by continuous (i.e., without breaking or gluing) deformation; such deformations themselves are called homeomorphisms. We just found out that the ball is homeomorphic to the cube and the pyramid, but not homeomorphic to either the torus or the pretzel, and the last two bodies are not homeomorphic to each other. We ask the reader to understand that we have given only an approximate description of the concept of homeomorphy, given in terms of mechanical transformation.
Let us touch upon the philosophical aspect of the concept of homeomorphy. Let us imagine a thinking being living inside some geometric figure and Not having the opportunity to look at this figure from the outside, “from the outside.” For him, the figure in which it lives forms the Universe. Let us also imagine that when the enclosing figure is subjected to continuous deformation, the being is deformed along with it. If the figure in question is a ball, then the creature cannot in any way distinguish whether it is in a ball, a cube, or a pyramid. However, it is possible for him to be convinced that his Universe is not shaped like a torus or a pretzel. In general, a creature can establish the shape of the space surrounding it only up to homeomorphy, that is, it is not able to distinguish one form from another, as long as these forms are homeomorphic.
For mathematics, the meaning of a hypothesis Poincare, which has now turned from a hypothesis into the Poincaré-Perelman theorem, is enormous (it’s not for nothing that a million dollars were offered for solving the problem), just as the significance of the method found by Perelman to prove it is enormous, but explaining this significance here is beyond our ability. As for the cosmological side of the matter, perhaps the significance of this aspect was somewhat exaggerated by journalists.
However, some authoritative experts say that Perelman’s scientific breakthrough can help in the study of the processes of formation of black holes. Black holes, by the way, serve as a direct refutation of the thesis about the knowability of the world - one of the central provisions of that most advanced, only true and omnipotent teaching, which for 70 years was forcibly drummed into our poor heads. After all, as physics teaches, no signals from these holes can reach us in principle, so it is impossible to find out what is happening there. We generally know very little about how our Universe as a whole works, and it is doubtful that we will ever find out. And the very meaning of the question about its structure is not entirely clear. It is possible that this question is one of those that, according to the teaching Buddha, Not there is an answer. Physics offers only models of devices that more or less agree with known facts. In this case, physics, as a rule, uses already developed preparations provided to it by mathematics.
Mathematics does not, of course, pretend to establish any geometric properties of the Universe. But it allows us to comprehend those properties that have been discovered by other sciences. Moreover. It allows us to make more understandable some properties that are difficult to imagine; it explains how this can be. Such possible (we emphasize: just possible!) properties include the finiteness of the Universe and its non-orientability.
For a long time, the only conceivable model of the geometric structure of the Universe was three-dimensional Euclidean space, that is, the space that is known to everyone from high school. This space is infinite; it seemed that no other ideas were possible; It seemed crazy to think about the finitude of the Universe. However, now the idea of the finitude of the Universe is no less legitimate than the idea of its infinity. In particular, the three-dimensional sphere is finite. From communicating with physicists, I was left with the impression that some answered “most likely. The Universe is infinite,” while others said, “most likely, the Universe is finite.”
Uspensky V.A. , Apology of mathematics, or about mathematics as part of spiritual culture, magazine “New World”, 2007, N 12, p. 141-145.
Almost every person, even those who have nothing to do with mathematics, have heard the words “Poincaré conjecture,” but not everyone can explain what its essence is. For many, higher mathematics seems to be something very complex and inaccessible to understanding. Therefore, let's try to figure out what the Poincaré hypothesis means in simple words.
Content:
What is Poincaré's conjecture?
The original formulation of the hypothesis sounds like this: “ Every compact simply connected three-dimensional manifold without boundary is homeomorphic to a three-dimensional sphere».
A ball is a geometric three-dimensional body, its surface is called a sphere, it is two-dimensional and consists of points of three-dimensional space that are equidistant from one point that does not belong to this sphere - the center of the ball. In addition to two-dimensional spheres, there are also three-dimensional spheres, consisting of many points of four-dimensional space, which are also equidistant from one point that does not belong to the sphere - its center. If we can see two-dimensional spheres with our own eyes, then three-dimensional ones are not subject to our visual perception.
Since we do not have the opportunity to see the Universe, we can assume that it is the three-dimensional sphere in which all humanity lives. This is the essence of the Poincaré conjecture. Namely, that the Universe has the following properties: three-dimensionality, boundlessness, simply connectedness, compactness. The concept of “homeomorphy” in the hypothesis means the highest degree of similarity, similarity, in the case of the Universe - indistinguishability.
Who is Poincare?
Jules Henri Poincaré- the greatest mathematician who was born in 1854 in France. His interests were not limited only to mathematical science, he studied physics, mechanics, astronomy, and philosophy. He was a member of more than 30 scientific academies around the world, including the St. Petersburg Academy of Sciences. Historians of all times and peoples rank David Hilbert and Henri Poincaré among the world's greatest mathematicians. In 1904, the scientist published a famous paper that contained an assumption known today as the “Poincaré conjecture.” It was three-dimensional space that turned out to be very difficult for mathematicians to study; finding evidence for other cases was not difficult. Over the course of about one century, the truth of this theorem was proven.
At the beginning of the 21st century, a prize of one million US dollars was established in Cambridge for solving this scientific problem, which was included in the list of problems of the millennium. Only a Russian mathematician from St. Petersburg, Grigory Perelman, was able to do this for a three-dimensional sphere. In 2006, he was awarded the Fields Medal for this achievement, but he declined to receive it.
To the merits of Poincaré's scientific activities The following achievements can be attributed:
- foundation of topology (development of theoretical foundations of various phenomena and processes);
- creation of a qualitative theory of differential equations;
- development of the theory of amorphous functions, which became the basis of the special theory of relativity;
- putting forward the return theorem;
- development of the latest, most effective methods of celestial mechanics.
Proof of the hypothesis
A simply connected three-dimensional space is assigned geometric properties and is divided into metric elements that have distances between them to form angles. To simplify, we take as a sample a one-dimensional manifold, in which on the Euclidean plane, tangent vectors equal to 1 are drawn at each point to a smooth closed curve. When traversing the curve, the vector rotates with a certain angular velocity equal to the curvature. The more the line bends, the greater the curvature. The curvature has a positive slope if the velocity vector is rotated toward the inside of the plane that the line divides, and a negative slope if it is rotated outward. In places of inflection, the curvature is equal to 0. Now, each point of the curve is assigned a vector perpendicular to the angular velocity vector, and with a length equal to the value of the curvature. It is turned inward when the curvature is positive, and outward when it is negative. The corresponding vector determines the direction and speed at which each point on the plane moves. If you draw a closed curve anywhere, then with such evolution it will turn into a circle. This is true for three-dimensional space, which was what needed to be proven.
Example: When deformed without breaking, a balloon can be made into different shapes. But you can’t make a bagel; to do this you just need to cut it. And vice versa, having a bagel, you can’t make a solid ball. Although from any other surface without discontinuities during deformation it is possible to obtain a sphere. This indicates that this surface is homeomorphic to a ball. Any ball can be tied with a thread with one knot, but this is impossible to do with a donut.
A ball is the simplest three-dimensional plane that can be deformed and folded into a point and vice versa.
Important! The Poincaré conjecture states that a closed n-dimensional manifold is equivalent to an n-dimensional sphere if it is homeomorphic to it. It became the starting point in the development of the theory of multidimensional planes.
