Grigory Perelman: multidimensional figure

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Grigory Perelman: multidimensional figure
Grigory Perelman: multidimensional figure
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About one of the most outstanding mathematicians of our time. Grigory Perelman.

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The USSR's course on exact sciences, which paved the way for the achievements of nuclear physics, astronautics and sports chess, was based on a strong mathematical tradition. Having formed in the 1930s, she gave the world such scientists as Andrei Kolmogorov, Alexander Gelfond, Pavel Alexandrov and many others who excelled in the traditional (algebra, number theory) and new areas of mathematics (topology, probability theory, mathematical statistics). In terms of the scale of interests and intellectual resources, only the American and Chinese schools could be compared with the Soviet one. But they were not limited to comparison: at the macro level, the queen of sciences developed in a contradictory atmosphere of friendly suspicion. Such mutual influences also played an important role in the professional life of Grigory Perelman, a recognized mathematical genius who finally proved Poincaré's hypothesis and thus solved one of the seven "Millennium Problems".

Curriculum vitæ. First pages

Grigory Yakovlevich Perelman was born on June 13, 1966 in Leningrad in the family of an electrical engineer and a mathematics teacher, and ten years later he had a sister - in the future, too, a candidate (more precisely, PhD) of mathematical sciences. In addition to the love of classical music instilled by his mother, Grigory showed interest in the exact sciences from childhood: in the fifth grade, he began to attend the mathematics center at the Palace of Pioneers, and after the eighth grade he moved to school number 239 with in-depth study of mathematics, which he graduated without a gold medal only from - for lack of points according to the TRP standards. In 1982, as part of a school team, he received a gold medal at the 23rd International Mathematical Olympiad in Budapest and was soon enrolled in the Faculty of Mathematics and Mechanics of Leningrad State University without passing exams.

At the university, for exemplary studies, Perelman received a Lenin scholarship. After graduating from the university with honors, he entered graduate school at the Leningrad Branch of the Steklov Mathematical Institute of the Russian Academy of Sciences. In 1990, under the scientific guidance of Academician Alexander Danilovich Aleksandrov (the founder of the so-called geometry of Aleksandrov - a section of metric geometry), Perelman defended his Ph.D. thesis on the topic “Saddle surfaces in Euclidean spaces”. Then, as a senior researcher, he continued to work in the laboratory of mathematical physics at the Steklov Institute, successfully developing the theory of Aleksandrov spaces.

In the early 1990s, Perelman worked for several respected US research institutions: the State University of New York at Stony Brook, the Courant Institute of Mathematical Sciences, and the University of California at Berkeley.

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The turning point for the young mathematician was a meeting with Richard Hamilton, whose area of ​​scientific interests extended in the plane of differential geometry - a new direction widely used in general relativity. In his works on the topology of manifolds, the American scientist was the first to use a system of differential equations called the Ricci flow - a nonlinear analogue of the heat equation, which describes not the temperature distribution, but the deformation of a Hausdorff space, which is locally equivalent to a Euclidean one.

Thanks to this system of equations, Hamilton was able to outline a solution to one of the seven "Millennium Problems" - in fact, to develop an approach to proving Poincaré's conjecture.

The favor of his foreign colleague and such a fundamental problem made a great impression on Perelman.At that time, he continued to smooth out the corners of Aleksandrov's spaces - the technical difficulties seemed insurmountable, and the scientist returned again and again to the idea of ​​the Ricci flow. According to the Soviet mathematician Mikhail Gromov, focusing on these problems, Perelman became even more ascetic, which caused anxiety among his loved ones.

In 1994, he received an invitation to lecture at the International Congress of Mathematicians in Zurich, and several scientific organizations, including Princeton and Tel Aviv universities, offered him a place in the state. When asked by Stanford University for a resume and recommendation, the scientist remarked, “If they know my work, they don't need my CV. If they need my CV, they don't know my work. " Despite such an abundance of tempting offers, in 1995 he decided to return to his "native" Steklov Institute.

In 1996, the European Mathematical Society awarded Perelman his first international prize, which for some reason he refused to receive.

In addition to unpretentiousness in everyday life, an addiction to music (Perelman plays the violin) and a strict adherence to scientific ethics, the scientist was already distinguished by his interest in the parallel solution of complex problems. In 1994, he proved the soul hypothesis. In differential geometry, by “soul” (S) we mean a compact, totally convex, totally geodesic submanifold of a Riemannian manifold (M, g). In the simplest case, that is, in the case of the Euclidean space Rn (n reflects the dimension), any point of this space will be the soul.

Perelman proved that the soul of a complete connected Riemannian manifold with sectional curvature K ≥ 0, the sectional curvature of one of the points at which is strictly positive in all directions, is a point, and the manifold itself is diffeomorphic to Rn. Mathematicians were shocked by the rare elegance of Perelman's proof: the calculations took only two pages, while the "pre-Perelman" attempts at a solution were presented in long articles and remained incomplete.

Proof of Poincaré's Hypothesis, or the Blessed Fusion of the Kitchen with the Operating Room

At the turn of the 19th and 20th centuries, the brilliant French mathematician Henri Poincaré enthusiastically laid the foundation of topology - the science of the properties of spaces that remain unchanged under continuous deformations. In 1900, the scientist suggested that a three-dimensional manifold, all of whose homology groups are like those of a sphere, is homeomorphic to a sphere (topologically equivalent to it). In the general case, for manifolds of any dimension, the conjecture sounds something like this: every simply connected closed n-manifold is homeomorphic to an n-dimensional sphere. Here it is necessary to decipher at least a little the terms with which Poincaré so freely operated.

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A two-dimensional manifold is a plane: for example, the surface of a sphere or a torus ("donut"). It is more difficult to imagine a three-dimensional manifold: one of its models is a dodecahedron, the opposite faces of which are "glued" to each other in a special way - they are identified. It was for the case of a three-dimensional manifold that Poincaré's conjecture remained a tough nut to crack for a century. As for homeomorphism, any closed, without holes, surfaces are homeomorphic, that is, they can continuously and uniquely transform (map) into each other and deform into a sphere, but with a torus, for example, this will not work without breaking the surface, therefore it is not homeomorphic to a sphere, but homeomorphic … to the mug - the same one from the kitchen cabinet. Homology - a concept that allows one to construct specific algebraic objects (groups, rings) for the study of topological spaces - it is believed that general algebraic structures are simpler than topological ones.Here are the simplest examples of homology: a closed line on a surface is homologous to zero if it serves as the boundary of some section of this surface; any closed line on the sphere is homologous to zero, while for a torus such a line may not be homologous to zero.

Groups - various sets that satisfy special conditions - have proven extremely useful for describing topological invariants - characteristics of a space that do not change under its deformations. In particular, homology and fundamental groups are in great demand. The homology group is associated with a topological space for the algebraic study of its properties. The fundamental group is a set of mappings of a segment into space (loops) fixed (starting and ending) at a marked point, measuring the number of “holes” in this space (“holes” arise due to the impossibility of continuously deforming a segment into a point). Such a group is one of the topological invariants: homeomorphic spaces have the same fundamental group.

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In the original version, Poincaré's conjecture for three-dimensional manifolds remained "decidable": it made it possible to weaken the condition on the fundamental group to a condition on the homology group. However, Poincaré soon ruled out this assumption, demonstrating an example of a non-standard three-dimensional homological sphere with a finite fundamental group - the "Poincaré sphere". Such an object could be obtained, for example, by gluing each face of the dodecahedron with the opposite one, rotated by an angle of π / 5 clockwise. The uniqueness of the Poincaré sphere lies in the fact that it is homologous to the three-dimensional sphere, but at the same time it differs from it in Euclidean space.

In the final formulation, Poincaré's conjecture sounded as follows: any simply connected compact three-dimensional manifold without boundary is homeomorphic to a three-dimensional sphere. The proof of this hypothesis promised new possibilities for modeling multidimensional spaces. In particular, the data obtained with the WMAP space probe made it possible to consider the Poincaré dodecahedral space as a possible mathematical model of the shape of the Universe.

And so, in 2002-2003 (by that time the thematic correspondence between Perelman and Hamilton had already disappeared) a user with the nickname Grisha Perelman, with an interval of several months, posted three articles on the arXiv.org preprint server (1, 2, 3) containing the solution of a problem even more general than Poincaré's conjecture - Thurston's geometrization conjecture. And the very first publication became an international scientific sensation, although due to the author's antipathy to bureaucracy, none of the articles ever made it to the pages of peer-reviewed journals. Perelman's calculations were so laconic and at the same time complex that mistrust simply could not creep into the general delight, therefore, from 2004 to 2006, three groups of scientists from the United States and China carried out verification of Perelman's works at once.

To deform the Riemannian metric on a simply connected 3-manifold to a smooth metric of the target manifold, Perelman introduced a new method for studying the Ricci flow, which is quite rightly called the Hamilton - Perelman theory. The highlight of the method was that when approaching the singularity arising from the deformation of the metric, stop the flow applied to the manifold and cut out the “neck” (an open region diffeomorphic to the direct product) or discard the small connected component by “gluing” the two resulting “holes” with balls … As this surgical operation is repeated, everything is thrown away, with each piece being diffeomorphic to a spherical spatial shape, and the resulting manifold is a sphere.

As a result, Perelman succeeded not only in proving Poincaré's conjecture, but also in completely classifying compact three-dimensional manifolds.This probably would never have happened if Perelman's long list of hallmarks had not been unshakable perseverance. Former teacher of mathematics, candidate of physical and mathematical sciences Sergei Rushkin recalled: “Grisha began to work very hard in the ninth grade, and he turned out to have a very valuable quality for doing mathematics: the ability to concentrate for a very long time without much success within a problem.

Still, a person needs psychological support, psychological success is needed in order to do something further. In fact, Poincaré's hypothesis is almost nine years without knowing whether the problem will be solved or not. You see, even partial results were impossible there. The theorem has not been proved in full - sometimes you can even publish a twenty-page article on what actually happened. And there - either pan, or disappeared. " Eternity in your pocket

In 2003, Grigory Perelman accepted an invitation to read a series of public lectures and talks about his work in the United States. But neither students nor colleagues understood him. For several months, the mathematician patiently explained, including in personal conversations, his methods and ideas. During the "American tour" Perelman also counted on a fruitful conversation with Hamilton, but it never took place. Returning to Russia, the scientist continued to answer questions from mathematicians by e-mail. In 2005, tired of the atmosphere of publicity, intrigue and endless explanations associated with the protracted verification of his calculations, Perelman resigned from the institute and virtually cut off professional ties.

In 2006, all three groups of experts recognized the proof of Poincaré's conjecture as valid, to which Chinese mathematicians, led by Yau Shintun, whose surname flaunts in the name of an entire class of manifolds (Calabi – Yau spaces), responded with an attempt to challenge Perelman's priority. True, the toolkit chosen for this turned out to be unsuccessful: it strongly resembled plagiarism. The original article by Yau's students, Cao Huidong and Zhu Xiping, which occupied the entire June issue of The Asian Journal of Mathematics, was annotated as the final proof of the Poincaré conjecture using the Hamilton-Perelman theory. If you believe the journalistic investigations, then even before the publication of this article, openly curated by Yau, the latter asked 31 mathematicians from the editorial board of the journal to comment on it as soon as possible, but for some reason he did not provide the article itself.

Yau Shintun not only knew Hamilton very well, but also collaborated with him, and Perelman's statement about the successful solution of the problem came as a surprise to both scientists: after many years of working on it, they hoped, despite a temporary hitch, to come to the finish line first. Subsequently, Yau emphasized that Perelman's preprints looked sloppy and indistinct due to the lack of detailed calculations (the author cited them as necessary in response to requests from independent experts), and this prevented him and everyone else from fully understanding the evidence.

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An attempt to belittle the merits of Perelman - and Yau even kindly calculated them in percentage terms - failed, and soon the Chinese scientists corrected the title and annotation of their article. Now it had to be perceived not as evidence of the "crowned achievement" of Chinese mathematicians, but as an "independent and detailed exposition" of the proof of Poincaré's hypothesis, produced by Hamilton and Perelman - without encroaching on someone's priority. Perelman commented on Yau's actions as follows: “I can't say that I am outraged, the others are doing even worse …” Indeed, the Chinese mathematical genius can be understood: Yau later explained his zealous support for the article of his students by the desire to present the final proof in a digestible, understandable form and to consolidate in history the merits of compatriots in solving this millennium task - and in fact they cannot be denied …

Meanwhile, in August 2006, Perelman was awarded the Fields Prize "for his contributions to geometry and his revolutionary ideas in the study of the geometric and analytic structure of Ricci flow." But, like ten years ago, Perelman refused the award, and at the same time announced his unwillingness to continue to be a professional scientist. In December of the same year, Science magazine for the first time recognized the mathematical work - the work of Perelman - as the "Breakthrough of the Year". At the same time, the media erupted in a series of articles highlighting this achievement, albeit with an emphasis on the conflict that accompanied it. To defend his position, Yau turned to lawyers and threatened the journalists who "defamed his name" by the court, but he did not carry out the threat.

In 2007, Perelman was ranked ninth in The Daily Telegraph's "One Hundred Living Geniuses" ranking. And three years later, the Clay Mathematical Institute awarded the Millennium Prize for solving the Millennium Problem - for the first time in history. At first, Perelman ignored the one million dollar prize, and then officially rejected it: “In short, the main reason is disagreement with the organized mathematical community. I do not like their decisions, I consider them unfair. I believe that the contribution of the American mathematician Hamilton to the solution of this problem is no less than mine."

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In 2011, the Millennium Prize, which Perelman refused, the Clay Institute decided to send young, promising mathematicians to pay for the work of young, promising mathematicians, for whom a special temporary position was established at the Henri Poincaré Institute in Paris. At the same time, Richard Hamilton was awarded the Shao Prize in Mathematics for creating a program for solving the Poincaré conjecture. The million dollar bonus that year had to be split equally between Hamilton and the second math laureate, Demetrios Christodoulou.

Perelman retained a good attitude towards Hamilton, despite the failed dialogue and the obvious dissatisfaction of his senior colleague with the finale of this scientific story. And this says a lot about a person. According to rumors, Grigory Yakovlevich continues to live in St. Petersburg, periodically visiting Sweden, where he collaborates with a local research company. Well, the six millennium problems are still waiting for their genius.

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