Three independent working groups of mathematicians claim to have completely proved Poincare hypothesis, which is known to be one of the most difficult mathematical problems of the XXth century. The final verdict will soon be announced at the International Congress of Mathematicians.

The hypothesis, which was formulated by Poincare in 1904, states that any three-dimensional surface in four-dimensional space, looking like a sphere, will stay a sphere after being expanded. More details about this hypothesis you can find in scientific journals.

Clay Mathematics Institute has established an award, worth $1 million, for proving said hypothesis; the fact, which may sound strange, because the hypothesis covers a special case, moreover, a not very interesting one. The real facts are that mathematicians care more about the difficulty of the proving process, rather than about properties of a three-dimensional surface. This problem concentrates everything, which mathematicians failed to prove with methods and ideas of geometry and topology, being at their disposal before. The problem allows looking in the depths of problems, which could be solved only by means of ideas of the “new generation”.

Like another well-known mathematical head-ache, Fermat’s theorem, Poincare hypothesis appeared to be the special case of one general statement about geometrical properties of random three-dimensional surfaces – Thurston's Geometrization Conjecture. That is why mathematicians have been concentrated on developing new mathematical method, which will help to solve such kind of problems, but not on proving the hypothesis itself.

Russian mathematician Grigory Perelman made a breakthrough in this field in 2002-2003. In his papers he suggested several new ideas together with developing and accomplishing the method, suggested by Richard Hamilton in the eighties of the last century. In his publications Perelman claims that he succeeded in proving not only Poincare hypothesis, but also geometrization conjecture.

The essence of the method is that there exists some equation of “gentle evolution”. During said evolution the surface will be deformed, finally turning into a sphere, as shown by Perelman. Method’s advantage is the chance to avoid any interstages and to look straight to the end of evolution to find a sphere there. Perelman’s papers launched an intrigue. He developed the general theory and mentioned key proofs of both Poincare hypothesis and geometrization conjecture. Russian mathematician didn’t publish the complete proof, but claimed he had proofs for both problems. In the year 2003 he traveled across the United States with lectures, during which he gave complete and accurate answers to any technical question of his audience.

Straight after Perelman’s preprints publication experts started to verify key moments of his theory and still not a single mistake has been found. Moreover, the years passed, and several groups of mathematicians “digested” Perelman’s ideas and started to write the final version of the complete proof.

May 2006 saw the paper of B. Kleiner and J. Lott where the authors gave full and detailed conclusion of all missing in Perelman’s proof aspects.

Then, in June 2006, the journal Asian Journal of Mathematics has published 327 pages of Chinese (Huai-Dong Cao and Xi-Ping Zhu) mathematicians’ paper, which proves that Perelman’s ideas work.

The most recently published paper (or may be a book?) of J. W. Morgan and G. Tian contains 473 pages and reveals another proof of said Poincare hypothesis (following Perelman’s ideas, of course).

Time will show whose development of Perelman’s proof is more accurate and clear. It even may become simpler, as it happened with the Fermat theorem. Now we all are waiting for the International Congress of Mathematicians, which is held in Madrid this August, where the official announcement of the proof of the hypothesis will take place, and possibly the name of the Clay Institute Award laureate will be revealed.

Source:

    elementy.ru
    www.icm2006.org
 

Anna Kizilova