Wednesday, April 28, 2010

Perfect Rigor

Some time ago there was this Science cover plainly announcing that one of the famous and long standing problems - Poincare conjecture is solved. First there is a feeling of joy that human mind has conquered another victory. This problem became another type of problem after the original one was solved. The reason? There were $1,000,000 prize attached to it. Mathematicians almost never come up nowadays with purely original solutions - they are using ideas and methods which they learn from others, even mistakes are useful because they show where not to go in your thinking. 
No surprise there came a controversy - who should receive money - just this one strange guy from St. Petersburg who even did not care about a requirement to publish his proof in "peer reviewed scientific journal" - he simply deposited his proof in - open access to 600,550 e-prints in Physics, Mathematics, Computer Science, Quantitative Biology, Quantitative Finance and Statistics hosted by Cornell University Library on 11 November 2002 with a title: The entropy formula for the Ricci flow and its geometric applications , after this post over the period of 8 months two more posts followed completing the proof.

After this submission mathematicians carefully started to check the proof to make sure that this indeed is the proof - it was not the first attempt to prove geometrization conjecture from which Poincare conjecture would follow but all previous proofs failed to give a complete answer. In the abstract to his paper Perelman already said that he is giving a sketch of the proof - would this should count as a complete proof? The following events made a story to appear in The New Yoker Manifold Destiny - it was prompted by Perelman refusing to accept Fields Medal - highest honor any mathematician could dream off.
Sylvia Nasar, who already had written a book about a strange mathematician John Nash, went to St.Petersburg and tried to interview Grisha Perelman, who was and still is avoiding to talk to press or generally stopped communicating with people except his mother.
This year finally Clay Institute made a decision that the prize $1,000,000 should be awarded to Perelman... but he refused to accept money.

Of course, such story is worth an investigation - as predicting this outcome(not too hard task in his case!) Masha Gessen has written a book /Perfect Rigor: A Genius + The Mathematical Breakthrough of the Century

No, she did not get to talk to Perelman himself or to his mother but she has interviewed number of people who knew Grisha Perelman since he was a child and first exhibited his extraordinary skills in math problem solving. These interviews and author's personal experience paints a picture not only of the strange mathematician but also about the way Soviet mathematical world worked.
This is a book I am reading now. Already on a second page I read a description why young people chose to study mathematics - Gessen writes:
"Mathematics was antithetical to the Soviet way of everything. It promoted argument; it studied patterns in a country that controlled its citizens by forcing them to inhabit a shifting, unpredictable reality; it placed a premium on logic and consistency in a culture that thrived on rhetoric and fear; it required highly specialized knowledge to understand, making the mathematical conversation a code that was indecipherable to an outsider; and worst of all, mathematics laid claim to singular and knowable truths when the regime had staked its legitimacy on its own singular truth. All of this is what made mathematics in the Soviet Union uniquely appealing to those whose minds demanded consistency and logic, unattainable in virtually any other area of study."

Yes, this is exactly why I chose to study mathematics.

I have made it approximately half of the book so far. I have some different views of the picture of Soviet mathematics painted in this book. Author at the beginning (p.12) says that "..and then there were those who almost never became members of the establishment: those who happened to be born Jewish or female..."
Actually it was more than that. For example, I could not defend my PhD thesis in Riga because it was not allowed by Soviet authorities that there would be appropriate scientific board allowed to award scientific degrees. Anybody who did graduate work in mathematics or computer science in Latvia up to 1992 had to find a scientific board which would accept thesis for the defense. Then one should find an opponent - a person who would read carefully your thesis, will try hard to find any mistakes, gaps in your proofs etc. To find an opponent usually was up to the thesis advisor who had stronger connections with respectable scientists in your area of research. It took 3 years(1987-1990) for my thesis advisor to find the scientific board which would accept my thesis for a defense. It was nothing to do with my work itself - it was a time when everything was cracking in Soviet Union, you may find the place you could defend your thesis but then next day this board could be dissolved and your thesis defense is no more valid. That is why I ended up with a degree from a place which I had actually visited twice - once to give a talk in the seminar when they decided to accept my work and the second time during the actual event. But officially my degree states that it is awarded by the Institute of Mathematics of the Academy of Sciences of Belorussia. I had as an original opponent Dima Grigoriev - he was the head of Laboratory of algorithmic methods Leningrad Department of the Steklov Mathematical Institute - form the same Institute Steklova that is mentioned in the book about Perelman. The day before I had to leave for Minsk my advisor Prof. Freivalds called Grigoriev to St. Petersburg to ask in which hotel in Minsk he will be staying. Grigoriev had just returned back from France (he was at that time already privileged to have international contacts and was well known, since 1998 he is the Research Director at CNRS, Lille, France). My case was so unimportant for him (understandably - it is just one unknown graduate student - who cares!), that he had forgotten all about it and said that he is too tired to go to Minsk, and he suggests to postpone it till September, not to mess up nice White Nights in St. Petersburg in June. I was at that time expecting my second child in September, so September was not an option for me. I was standing there, in the office of my advisor - pale and and almost fainting. He had to rush to a meeting, so he told me to sit down, gave me bad quality photocopy (brought from Moscow, the only library you could get access to at least some publications in English) of something to read and told to wait for him and he will sort things out. And then he left, locking the door behind him. Years later he told me that he was afraid I might go out and jump into the river (I did not have such thoughts).

What he gave me to read were extracts from Feynman's "Surely you are joking, Mr. Feynman?. When he returned about two hours later he had found another professor who had agreed to save me and to be my opponent. My thesis were saved.

My daughter is now 19 and likes Feynman's books very much.
Last Saturday I told her what happened just before she was born.

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