Mathematics / A Russian-Jewish mathematician apparently succeeded in proving the Poincaré hypothesis, one of the seven "millennium problems". Mathematicians are now working on testing it
Uriel Brizon
Poincare. Despite the relative simplicity with which his hypothesis was formulated, the best minds failed to prove it * Perlman presents his proof in a lecture at Princeton. Locked himself in his house for about a decade and worked on the problem
Direct link to this page: https://www.hayadan.org.il/poincare_perlman.html
Has Poincaré's hypothesis been solved?
The world of mathematics as a concoction: a Russian researcher apparently succeeded in solving one of the most complicated conjectures in history. Researchers from all over the world are now checking the researcher's work and will soon determine whether another mathematical legend has fallen
IP
Did a Russian researcher manage to solve the Poincaré conjecture? This is one of the oldest and most difficult to decipher conjectures in modern mathematical history. The claimant to the crown is Gregory Perlman, a Jewish researcher from Russia, who apparently solved the complicated problem that aims to explain the geometric space of three-dimensional space.
The evidence presented by Perlman has been examined for three months and now the probability is rising that he did manage to solve the problem. Scholars from around the world are now searching for errors in Perlman's complicated work who apparently succeeded where many scholars have failed over the past hundred years.
If he is indeed successful, Perlman will win a grant in the amount of one million dollars that will be awarded to him by the Clay Foundation of Cambridge, which has set itself the goal of solving the seven most complex mathematical problems in the world.
"This is undoubtedly one of the most famous problems in the history of mathematics," said a professor from the University of Michigan who examines Perlman's work.
The excitement is at its peak
In the mathematical world, excitement is at its peak for what could be the biggest event since Fermat's theorem was deciphered about ten years ago. "There was never such a case in the past that concentrated the attention of so many mathematicians," said one of the researchers.
Perlman himself was defined as a brilliant and shy scientist who hates exposure. He is employed at the Staklov Institute in St. Petersburg, and before that he studied for several years in the USA. His friends, who call him "Grisha", say that for eight years he did not publish any document or research work. The question of whether he would like to accept the amount of money that would be offered to him also remained unknown after he temporarily refused to answer For press inquiries on the subject.
Poincaré: predates chaos theory
You can learn about Gilles-Henri Poincaré from the entry bearing his name in the ynet encyclopedia: Poincaré dealt with the n-body problem in astronomy (the question of the stability of a gravitational system with a number of bodies exceeding 2), and provided a solution for the case of 3 bodies. But after he submitted the solution, and won the prize offered by Oscar the 2nd King of Sweden to whoever solves the problem of the stability of the orbits in the solar system (in 1889), he discovered that there was an error in his solution: in practice, the orbit of the third body will depend to a large extent on slight changes in the initial conditions.
In doing so, Poincaré preceded the chaos theory developed in the second half of the 20th century.
For news on Yahoo's news site
Fermat took 358 years to prove; 100 years was probably enough for Poincaré
12/5/03
by Uri Brizon
There is a good chance that the Poincaré conjecture, one of the most well-known unsolved problems in mathematics, was solved by the Russian-Jewish mathematician Gregory Perlman. Mathematicians close to Perlman say that for nearly a decade he sat cooped up in his apartment in St. Petersburg and worked on the proof. If the proof turns out to be correct, a hundred years of failed attempts to deal with the Poincaré conjecture will come to an end, and Professor Perlman will win a place of honor in the history of mathematics and a million dollar prize. The prize will be given by the "Clay" Institute in the USA, which has pledged to award a prize of one million dollars to whoever solves each of the seven problems defined as the "millennium problems" - the most important unsolved problems of mathematics.
Perlman, who is considered a brilliant mathematician and has published many articles in the past, decided to devote himself to trying to prove the Poincaré hypothesis. Due to the desire to devote all his time to proving the law, he refused to accept various jobs offered to him at universities in the USA and also rejected the application of Tel Aviv University, which offered him the position of professor of mathematics in the Faculty of Exact Sciences (some of Perlman's family members immigrated to Israel a few years ago).
In his publications, Perlman thanks the American universities that invited him to stay for short periods. With the money given to him by these universities, he financed his living in Petersburg and thus could devote all his time to his work. Although Perlman is listed as a faculty member of the Staklov Mathematics Institute in St. Petersburg, he is apparently not active at the institute and at least until recently spent his time at home, working on the work of the proof.
After the first articles on the proof were published, last November, Perlman was invited to present his ideas to his colleagues in the USA. About two weeks ago, about a hundred of the world's leading mathematicians gathered in the lecture hall at Princeton University in New Jersey to hear him speak. Among those present at the lecture were Nobel Prize winner C Van Nash, known from the film "The Wonders of Reason" which reviewed his life, and Andrew Wales, the British mathematician who proved Fermat's theorem - the mathematical hypothesis that survived without proof for 358 years.
Peter Sarnak, a professor of mathematics who was present at the lecture, told the Princeton University newspaper that "it is clear that a breakthrough has been made here", but noted that "even though Sperlman has not previously made false claims, every detail must be checked carefully". Many mathematicians take a position similar to this; They express a deep impression of the work along with a cautious reservation until the rigorous tests are completed. Perlman himself refused to be interviewed on the grounds that it was not yet time for public announcements. Over the years, there have been many cases of proofs that failed at the testing stage, but it is evident that this time the expectations are very high.
The hypothesis known as the Poincaré hypothesis was formulated by the French mathematician Henri Poincare at the end of the nineteenth century (he published the final formulation at the beginning of the twentieth century). The hypothesis deals with a mathematical field called topology, a field that split from geometry during the 19th century. Topologists study those properties of objects that do not change when they are deformed - expanded, contracted or twisted - without severing their shell and without connecting parts of it that were not connected before so. For example: a triangle is equivalent, topologically, to a circle but not to a segment of a straight line - the transition from a triangle to a circle only requires a curve, but the straight line must be "broken" to produce a triangle. Similarly, the shape known as a sphere (the outer shell of a sphere) is equivalent to the shell of a pyramid or a cube, but not to the shell of a ring- or tube-like shape (known by mathematicians as a "torus") and this is because a cube or a pyramid can be curved into a sphere (and vice versa) But the creation of the torus cannot be done without breaking the shell of the sphere and creating the hole in the center.
A topological test for the equivalence of different shapes is the loop test: any loop drawn on the envelope of the sphere (or cube) can be reduced in size to one point without exceeding the envelope. You can imagine a rubber band wrapped around an apple: you can move it around so that it shrinks. Even on a shape like a ring, similar to a sphere, a loop can be stretched in such a way that it can be reduced to a point without exceeding the envelope of the ring. But unlike a sphere, on the surface of a ring, for example, loops can also be stretched around the hole in its center, and these cannot be reduced to a point without exceeding the envelope. Therefore, a sphere and a torus are topologically inequivalent forms.
Topology, as is the way of mathematical research fields, aims to determine general theorems as much as possible and is therefore not satisfied with the study of two-dimensional or three-dimensional bodies. The topological research includes, therefore, all possible bodies in all dimensions. Although humans are not able to imagine a body with more than three dimensions directly, it is certainly possible to represent it mathematically. Poincaré knew that every three-dimensional object, which through contraction, expansion or curvature can be turned into a sphere, has the property that any loop we draw on its envelope can be reduced to a point without exceeding the envelope. He hypothesized that the same rule holds in the four-dimensional space as well (for the sake of accuracy, we will clarify that a three-dimensional ball envelope, as it is known to us, is called by topologists an envelope in two-dimensional space that "curves" into a third dimension, similar to, for example, a sheet of paper that wraps a tennis ball .for simplicity, we call it a 3D shell here).
A four-dimensional count, which cannot, as mentioned, be perceived directly, should be thought of using its mathematical definition. A count is the set of points that are equidistant from the starting point in an axis system. In 3D space there are three axes. If we measure one unit in every possible direction from the origin of the axes, we will get a shell of a sphere, that is, a count. Now you can try to think of a space where there are four axes instead of three, that is, four perpendicular lines come from the beginning of the axes. The shape that will result if we measure one unit in every possible direction from the origin of the axes in this space is the envelope of a four-dimensional "sphere".
Poincaré hypothesized, as mentioned, that even in the four-dimensional space every count, and every object that can become a count through contraction, expansion or curvature, has the property that every loop marked on its shell can be reduced to a point. This is Poincaré's hypothesis, and despite the relative simplicity with which it is formulated, the best minds have failed for a century in attempts to prove it. During the twentieth century it became clear that it is easier to prove the equivalent of Poincaré's theorem for larger dimensions and indeed they succeeded in proving it for all dimensions larger than four, but the original hypothesis remained unproven.
Perlman's work is, in fact, aimed at a more ambitious goal than merely proving the Poincaré hypothesis. In the 1970s, the mathematician William Thurston formulated the "Geometry Hypothesis", a basic hypothesis that enables a change of perception in topology and shows that under certain conditions it is possible to divide any three-dimensional shell of a body in a space with a larger number of dimensions into parts, each of which belongs to one of eight possible geometric categories that he defined By using this process it is possible to perform many topological operations, including a new approach to the proof of Poincaré's theorem. Perlman claims that he was able to complete Thurston's work, to prove the geometry conjecture and, in addition, using the new abilities at his disposal, to also provide a solution, almost casually, to the Poincaré conjecture.
The process of examining Perlman's work is accompanied by great excitement, although mixed feelings are evident from mathematicians who specialize in the topology relevant to the Poincaré hypothesis. Whether Perlman turns out to be right or whether his proof is found to be flawed, his work as a whole is expected to be a basis for development in various mathematical fields and perhaps even in physics; If it turns out that it is indeed true - it will probably mark, ironically, the end of this field of topological research.
Mathematician Michael Friedman, who spent many years working on the Poincaré hypothesis and topology, revealed his feelings on the subject in an interview with the "New York Times": "Suppose you are an advanced student of mathematics facing the choice of your academic field of study. Will you choose the field in which the central problem has just been solved?"
For news on this topic at the BBC
Books on sale: Secrets of Encryption + Ferma's Last Theorem - on the Mythos site
The math genius
https://www.hayadan.org.il/BuildaGate4/general2/data_card.php?Cat=~~~525785579~~~133&SiteName=hayadan
