PROOF OF THE POINCARE CONJECTURE

Russian mathematician Grigory Perelman has solved one of the “Millennium Prize Problems” by proving the Poincaré conjecture, formulated a hundred years ago. The proof of the conjecture makes it possible to understand what shape our Universe has, reasonably imagining that the shape of the Universe is a 3-dimensional sphere. If the Universe is a “figure” which can be shrunk to a point, it is probably possible to stretch from a point, which serves as an indirect confirmation of the Big Bang theory which states that the Universe just originated from a point.

“Millennium Prize Problems” are seven mathematical problems considered to be important classical problems that have remained unsolved for many years. For solving each of them, the Clay Mathematics Institute has set a reward of $1 million. Each of these problems had a very long history, and their solutions led to the emergence of entire new scientific fields, but the only correct answers to the questions posed were not found. Insightful people said that the Clay’s money was safe, but it was until 2002 when he proved the Poincaré conjecture. However, he did not take the money.

The Poincaré conjecture (proved) states that if a three-dimensional surface looks something like a sphere and if it continuously tightened to a point, it will turn exactly into a sphere.
Jules Henri Poincaré (1854-1912) was the President of the Paris Academy of Sciences and was elected as a member to the scientific academies of 30 countries. In 1904, the scientist published a work containing, among other things, a hypothesis called the Poincaré conjecture. It took about a century to find a proof of this conjecture.
In 2002-2003 it was proved by the Russian mathematician Grigory Perelman. After that, the Poincaré conjecture became known as the Poincaré-Perelman theorem. It took several years for the Russian scientist’s colleagues to verify the proof and to recognise the discovery. So far, only one of the “Millennium Prize Problems” has been solved.

Grigory Yakovlevich Perelman was born on 13 June 1966 in Leningrad (now St. Petersburg) to an intelligent family. His father, an electrical engineer, emigrated to Israel in the early 1990s, and his mother taught mathematics at a vocational school. Apart from a love of classical music, she instilled in her son a passion for solving problems and puzzles. In 9th grade, Grigory transferred to Phys&Math Lyceum No. 239, renowned for its top-notch teaching of physics and mathematics. Besides, since the 5th grade, as an additional extracurricular education, he attended the Mathematics Centre at the Pioneers Palace. Winning all-Union and International Olympiads allowed Perelman to enter the Leningrad (now St. Petersburg) State University without exams. Many specialists, especially Russian ones, note that Grigory Yakovlevich was prepared for his unprecedented rise by the top-notch teaching of the Leningrad (now St. Petersburg) Geometric School, which he attended at the School of Mathematics and Mechanics of the Leningrad (now St. Petersburg) State University and by the postgraduate course at the Steklov Institute of Mathematics, of which he became a member after defending his doctoral thesis. Currently, the Institute conducts basic research on problems in fundamental mathematics and a number of its applications, as well as on important problems in mechanics, mathematical and theoretical physics. The Institute staff actively participates in the popularization of science and the development of educational materials for schools and universities. The Institute publishes the magazine “Kvant”. In August 2010, the Institute established the Laboratory for the Popularization and Promotion of Mathematics, which promotes and popularizes the achievements of the national mathematical school, the subjects of research conducted at the Mathematical Institute; scientific results obtained by the Institute staff.
Difficult 1990s forced the young scientist to go to leave for work in the USA. Those who knew him then noted his austerity in everyday life, enthusiasm for his work, excellent training and high erudition, which were the key to Perelman’s proving the Poincaré conjecture. He took up this problem closely after returning to St. Petersburg in 1996, but he began working on it in the USA.
Grigory Yakovlevich notes that he has always been fascinated by complex problems such as the Poincaré conjecture. Perelman began to look for a proof in a direction derived from the conversation with the Columbia University Professor Richard Hamilton (born in 1943). While in the United States, he would come from out of the city to attend lectures by this extraordinary scientist. Perelman notes the friendly attitude of the Professor to the young mathematician from Russia. In their conversation Hamilton mentioned the Ricci Flow, a certain partial differential equation for a Riemannian metric, as a way to solve geometrization conjectures.
In 2002-2003 the Russian scientist, having proved the conjecture, “climbed Everest” as the Poincaré conjecture recognised by mathematicians. Perelman posted the proof on the Internet in the form of three small articles. These immediately caused agiotage, although the Russian mathematician did not follow the scientific way – to publish with professional reviews in a specialized journal. Grigory Yakovlevich spent a month explaining the essence of his discovery at the USA universities, but the number of people who understood the course of his thought was increasing very slowly. Only four years later came the conclusion of the greatest authorities: the proof of the Russian mathematician is correct. Thus, one of the “Millennium Prize Problems” was solved.

This proof by the Russian mathematician led to some very interesting conclusions in terms of our understanding of the world.
In terms of astronomy, this conjecture assumes that if our Universe has the characteristics of a simply connected, closed 3-manifold, therefore it is homeomorphic to the 3-sphere, whereas previously it was believed that the Universe was infinite (i.e. had the form of a 3-dimensional Euclidean space).
The Poincaré-Perelman theorem, especially the way it was proved, is also of great importance to mathematics. This theorem is considered to be the mathematical equation of the Universe. It describes our world, which is a smooth three-dimensional manifold. The proof of the conjecture allows to understand the shape of our Universe, reasonably imagining that the shape of the Universe is a three-dimensional sphere. If the Universe is a “figure” which can be shrunk to a point, it is probably possible to stretch from a point, which serves as an indirect confirmation of the Big Bang theory which states that the Universe just originated from a point.

Perelman’s proof is of great importance also for quantum mechanics. When quantum states are realized in the macrocosm, a huge number of systems interact, connected to each other without breaks in geometric integrity.
G.Y. Perelman was offered the Fields Medal, the Clay Mathematics Institute was going to give him a million dollars, but the mathematician rejected the award and money. Sir John Ball, then president of the International Mathematical Union, went to St. Petersburg to persuade the scientist to take the money, but Perelman said:
“The prize was of no importance to me. Everyone understood that if the proof was right, no other recognition was needed. I am not interested in money or fame”.
The famous “Forbes” magazine put Grigory Perelman on the list of the People of the Century.