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Grigori Perelman: The Mad Monk of Math

Grigori Perelman: The Mad Monk of Math. by Michael Williams. Introduction.

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Grigori Perelman: The Mad Monk of Math

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  1. Grigori Perelman: The Mad Monk of Math by Michael Williams

  2. Introduction • In November 2002, Russian mathematician Grigori Perelman posted online a highly unorthodox proof of a world famous problem, ignoring all of the rules that modern mathematicians follow to ensure credit for their work. • There were very few mathematicians available experienced enough to evaluate and vet the proof. It would take years for two teams of experts to confirm that it was logically consistent and error-free. • In the meantime, in 2003 Perelman gave a series of lectures about his work. He then returned to Saint Petersburg, gradually ceasing all communication with his colleagues. • By 2006, when his work was shown to prove the notoriously difficult Poincaré Conjecture, he had entirely retreated from the world of mathematicians. • Since Perelman is extremely reclusive and still relatively young, there are no official records of his life, and his single biographer has never personally met him.

  3. Early Life • Grigori “Grisha” Perelman was born on June 13, 1966 in Leningrad (now Saint Petersburg), Russia to Jewish parents. • His father was an electrical engineer, and his mother quit her graduate work in mathematics to raise him. • When Grisha was ten, his mother put him in an afterschool math club run by Sergei Rukshin, a famous math coach with whom many of the Russians entering the International Mathematical Olympiad in the last 20 years have studied. • Rukshin took a personal interest in Grisha, tutoring him until he was admitted to Leningrad’s Specialized Mathematics School Number 239 in 1980. Grisha

  4. Education • In 1982 Perelman won a Gold Medal with a perfect score in the International Mathematical Olympiad, earning his admission to Leningrad University where he studied under many great Russian mathematicians including the eminent geometer Viktor Zalgaller. • In the late 1980s and early 1990s, he worked as a postdoctoral fellow at NYU, SUNY Stony Brook, and UC Berkeley. • In 1994, after “succinctly and elegantly proving a topological theorem called the Soul Conjecture” (Paulos), he rejected a position at Stanford University because they asked him for a C.V. “If they know my work, they don’t need my C.V.,” he said. “If they need my C.V., they don’t know my work.” (Nasar) • Instead, he returned to Saint Petersburg in 1995, moved in with his mother and went back to work at the Steklov Institute, where he’d had his original postdoctoral position.

  5. ThePoincaré Conjecture • The Poincaré Conjecture was formulated in 1904 by French mathematician Henri Poincaré. It is a fundamental question in the mathematical discipline known as Topology, specifically concerning three-dimensional shapes resting in four-dimensional space. • The Poincaré Conjecture “asserts that if any loop in a certain kind of three-dimensional space can be shrunk to a point without ripping or tearing either the loop or the space, the space is equivalent to a sphere.” (Overbye) • What this amounts to is that any shape without holes can be mathematically reduced to the shape of a sphere, such as a cigar or a cup. An example of a topological object with a hole is a bagel or a coffee mug with a handle. “Poincaré found the right test. However, no one before Perelman was able to show that the test guaranteed that the given shape was in fact a three-sphere.” (Carlson)

  6. Topology & The Ricci Flow • A useful aid for visualizing a three-sphere (a Topological reference for a 4th-dimensional sphere), mathematicians will sometimes contemplate the tesseract, which is a 4th Dimensional cube as represented here. • In 1982, the American mathematician Richard Hamilton published a paper concerning a differential geometry equation known as the Ricci Flow. He suspected that it might be a useful tool in solving the Poincaré Conjecture, though he was never able to prove this himself. • Perelman became fascinated with the Ricci Flow; in later years he even referred to Hamilton as a mentor, though they never formally worked together. • In 1995, Hamilton published a new paper which further discussed solving the Poincaré Conjecture with the Ricci Flow. Seeing no evidence of new ideas in this paper, Perelman came up with a few thoughts of his own and emailed Hamilton outlining them, probably hoping to collaborate. He never received a reply. Tesseract

  7. Mathematical Work • By 1995, Perelman had proven himself to be a formidable mathematician, and moreover had rid himself of the responsibilities that come with professorships. He didn’t care about making money or furthering his reputation. Disappointed with the territorial squabbling of the mathematical community, he decided that he preferred solitude; the math was what mattered to him. • Perelman faded from view, communicating with no one. He continued to earn a small salary from the Steklov Institute, but nobody knew what he was working on. It is now reasonable to assume that until 2002, when he began posting his work on arXiv.org, he was working out the solution to the Poincaré Conjecture using the Ricci Flow. • Essentially, Perelman proved the Poincaré Conjecture by showing that “surgeries” (to use the topological meaning of the word) during the Ricci Flow process would smooth out the errors, or singularities, that constantly crop up when trying to identify three-spheres.

  8. Mathematical Work (Continued) • Perelman took an unusual step in posting his paper so informally online. He hadn’t told anyone what he was working on and had shown no one his results. If he had made a mistake, he risked ridicule from the community. Furthermore, nothing would prevent opportunistic mathematicians from taking his work, tidying it up, and claiming the proof as their own. • True to form, however, Perelman wasn’t worried about such trivialities. In his own words, “My reasoning was: if I made an error and someone used my work to construct a correct proof I would be pleased,” he said. “I never set out to be the sole solver of the Poincaré.” (Nasar) • Perelman was asked to lecture at several universities about his proof and began a month long tour in April of 2003. Shockingly to many who heard him speak, Perelman stated key points and answered questions about his proof, but never mentioned the Poincaré Conjecture directly.

  9. After-Math • It would take three years for several mathematicians to comb through Perelman’s proof and explain it in three book-length papers, verifying that the Poincaré Conjecture had indeed been solved. • There is no doubt now that Perelman’s proof will have a large impact not only in mathematics, but possibly someday in Physics and other areas as well. The trouble is that mathematicians are generally about a century in advance of physicists, and it is too early to make specific conjectures. • The two-dimensional version of Poincaré, for example, was proven in the mid-1800s by German mathematician Georg Friedrich Bernhard Riemann, but did not come into practical usefulness until Einstein‘s General Theory of Relativity in the early 20th century. It has also become important in the relatively new discipline of String Theory. • Many topologists believe Perelman’s proof will be an important tool for one day defining the largest known topological shape: the Universe itself.

  10. After-Math (Continued) • The International Mathematical Union awarded Perelman a Fields Medal in 2006, the most prestigious award a mathematician can win, equivalent to a Nobel Prize. He refused to accept it and did not attend the congress. He pointed out, “Everybody understood that if the proof is correct then no other recognition is needed.” (Paulos) • On March 18th 2010, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) offered Perelman a one-million dollar prize for solving The Poincaré Conjecture. • On March 23rd 2010, Perelman rejected the money and the prize. “I don’t want to be on display like an animal in a zoo. I’m not a hero of mathematics. I’m not even that successful. That is why I don’t want to have everybody looking at me.” (Paulos)

  11. Later Life • Part of the reason that Perelman refused all of the honors and money must be due to the disappointment he felt with the mathematical community itself. For example, one competitor did in fact try to claim credit for solving the Poincaré Conjecture by restating Perelman’s work. For another, the one person Perelman admired and attempted to collaborate with, Richard Hamilton, ignored him both before and after his success. • Perelman has severed all ties with his old profession, reportedly even leaving his position at the Steklov Institute. • He still lives with his mother, surviving on her modest pension and money sent by his sister. • No one knows if he still practices math. His neighbors say they see him every day walking to the grocery where he buys the same things: eggs, cheese, spaghetti, sour cream, bread, and a kilo of oranges. They believe he amuses himself by playing ping-pong against the wall of his apartment.

  12. Bibliography • Carlson, James. “First Clay Mathematics Institute Millennium Prize Announced Today; Prize for Resolution of the Poincaré Conjecture a Awarded to Dr. Grigoriy Perelman.” The Clay Mathematics Institute. 18 Mar 2010. Web. 26 Jan 2010. <http://www.claymath.org/poincare/>. • Nasar, Sylvia & Gruber, David. “Manifold Destiny.” The New Yorker. Aug 28 2006. Web. 25 Jan 2012. <http://www.newyorker.com/archive/2006/08/28/060828fa_fact2>. • Osborn, Andrew & Krepysheva, Olga. “Russian maths genius may turn down $1m prize.” The Telegraph.co.uk. 27 Mar 2010. Web. 25 Jan 2012. <http://www.telegraph.co.uk/news/worldnews/europe/russia/7530771/Russian-maths-genius-may-turn-down-1m-prize.html>. • Overbye, Dennis. “Elusive Proof, Elusive Prover: A New Mathematical Mystery.” The New York Times. Aug 15 2006. Web. 25 Jan 2012. <http://www.nytimes.com/2006/08/15/science/15math.html?pagewanted=1>. • Overbye, Dennis. “Ask Science: Poincaré’s Conjecture.” The New York Times. Aug 18 2006. Web. 25 Jan 2012. <http://www.nytimes.com/2006/08/18/science/18askscience.html>. • Paulos, John Allen. “He Conquered the Conjecture.” The New York Review of Books. 29 Apr 2010. Web. 25 Jan 2012. <http://www.nybooks.com/articles/archives/2010/apr/29/he-conquered-the-conjecture/?pagination=false>.

  13. Photo Credits • Title page, left to right: • Grigori w/ math - http://shcr.com.au/tendon-dr-grigori-perelman/ • Grigori writing on chalkboard - http://scienzaesalute.blogosfere.it/2007/06/grigori-perelman-paparazzato-da-un-blogger.html • Grigori Snarl - http://vassilio.livejournal.com/137363.html?thread=433043 • Introduction page, Grigori Sketch - https://www.facebook.com/people/Grisha-Perelman/1480359510 • Early Life page, Grigori with young friends - http://pavlopoulos.wordpress.com/2009/08/30/the-perelmans-story/ • Education page, B&W in front of chalkboard - http://www.telegraph.co.uk/news/1526782/Worlds-top-maths-genius-jobless-and-living-with-mother.html • The Poincaré Conjecture page, Grigori writing on chalkboard -http://www.guardian.co.uk/books/2011/mar/27/perfect-rigour-grigori-perelman-review • Topology & The Ricci Flow page, Tesseract gif - http://en.wikipedia.org/wiki/Tesseract • Mathematical Work page 1, Grigori - http://www.telegraph.co.uk/news/1526782/Worlds-top-maths-genius-jobless-and-living-with-mother.html • Mathematical Work page 2, Grigori in front of chalkboard - http://psymath.blogspot.com/2011/03/arxiv-et-hal-hyper-articles-en-ligne.html • After-Math page 1, Grigori in front of chalkboard - http://www.bocelek.com/tag/grigori-grisha-perelman-lavagna/ • After-Math page 2, Fields Medal - http://tvsh2004.narod.ru/science/physics3.html • Later Life page, Grigori walking- http://vassilio.livejournal.com/137363.html?thread=433043 • Closing Photo of Grigori- http://www.beyond-the-pale.co.uk/hero.htm • Bibliography page, Grigori sketch - http://elgalli.blogspot.com/2011/04/genio-ruso-resuelve-problema-matematico.html

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