Grigori Perelman and the Poincaré Conjecture

In 1904, the famous mathematician Poincare made a reasonable conjecture:  If a 3-dimensional geometric object had the same homology groups as a 3-dimensional sphere and, and is compact, without boundary, and if every loop in the object can be squeezed down to a point (just like the 3-dimensional sphere), then the object must be the 3-dimensional sphere.

It’s a bit technical to describe homology, but beyond that Poincare’s conjecture is pretty simple to understand:  It’s the mathematical version of “If it looks like a duck, then it’s a duck.”

But, just like Fermat’s Last Theorem and the Riemann Hypothesis, being easy to state often means it’s hard to solve.  So much so that it was one of the problems with a $1,000,000 bounty on its head as part of the Millennium Prize Problems!

Grigori Perelman

In 2002/2003 Grigori Perelman posted a sketch of a solution to Poincare’s Conjecture (and much more!) on the ArXiv using Ricci Flow (part of math having to do with differential equations).   Although Perelman was known to be a very good mathematician, his solution came out of the blue.  Although his work builds on the work of others, he made a number of important advances entirely on his own.

Terence Tao (who knows a thing or two about solving hard math problems) spoke about Perelman’s work and said:

They [the Millennium Prize Problems] are like these huge cliff walls, with no obvious hand holds. I have no idea how to get to the top. [Perelman’s proof of the Poincaré Conjecture] is a fantastic achievement, the most deserving of all of us here in my opinion. Most of the time in mathematics you look at something that’s already been done, take a problem and focus on that. But here, the sheer number of breakthroughs…well it’s amazing.

–Terence Tao

It took several years for the best mathematicians in the world to go through Perelman’s proofs carefully and check all the details.  At the 2006 International Congress of Mathematicians in Madrid, Spain it was declared that Perelman’s proof was correct.  At that time he was offered the Fields Medal (which is arguably the most famous prize in math and we’ve discussed it here).  Amazingly, Perelman declined the award!  He has since withdrawn from mathematics.  If and when he is offered his share of the $1,000,000 Millennium Prize, it’s not clear that he’ll accept that either!

If you’d like to read more about Perelman and the Poincare Conjecture, or if you’re looking for a gift for someone who might be interested, then you should take a look at Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century by Masha Gessen, a new book about the human side of this story.  You can read a review of it here. Or, if you’d like to read more about the math, then check out The Poincare Conjecture: In Search of the Shape of the Universe by Donal O’Shea.

Or, you can just rap:

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9 thoughts on “Grigori Perelman and the Poincaré Conjecture

  1. Pingback: A Double Header « OU Math Club

  2. “he’s just an unemployed don living with his mom…but hes got it going on”
    A breakthrough in its own right, “math geek” meets rap shattering barriers of stereotype. Play that funky music.

  3. I read this article from new yorker

    http://www.newyorker.com/archive/2006/08/28/060828fa_fact2

    That’s sad story for the math community that Perelman declined the award. According to the article, the reason was probably he didn’t like Shing-Tung Yau who was the 1982 medalist.

    Perelman is undoubtedly one of the great mathematicians. However, why he withdrew from mathematics which is probably the biggest part of his life. In math or any fields, there are some people who do the wrong things, but most of people do the right things. Hopefully, Perelman will post an interesting paper soon.

    • Phuong,

      It’s our understanding that Perelman is still doing mathematics and is friends with a number of mathematicians, he’s just not interested in the award or being famous. And you may well be right that he is unhappy with certain mathematicians. But hopefully he will continue doing research and publishing it.

  4. Obviously Perelman hasn’t seen the mathematical proof of how Money equals happiness.

    Time = Money

    Also, Time = Happiness

    Therefore, Money must also equal Happiness

  5. Pingback: Dr. Grigory Perelman – The World’s Cleverest Man Who solves One Of Mathematics’ Greatest Puzzles…WoW !!! « This is our site

  6. Pingback: Holiday Gift? « OU Math Club

  7. Pingback: What’s Grigori Perelman Up to These Days? | OU Math Club

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