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!
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.
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 He has since withdrawn from mathematics.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: