I heard a quite knock on my door last week. It was Yunqiu Shen, a Professor in our Math Department, stopping by to tell me that the Poincaré Conjecture had been solved. No big deal...it just meant that someone had earned a million dollar prize!
In a nutshell, the Poincaré Conjecture is about the relationships of 3-dimensional space and surfaces. It claims that if all closed curves on a surface can shrink to a point continuously (imagine moving a rubber band around on a beach ball and then retracting it a single point held by your finger), this space can be deformed into a sphere as its topological equivalent. (To read more, consider the Wikipedia version).
Since first posed in 1904 by Henri Poincaré (a French mathematician and physicist), the century-old conjecture has become one of the most important questions in topology. The Clay Institute (Cambridge, MA) even lists it as one of the seven Millennium Prize Problems and offered a one million dollar prize for a correct solution.
On June 17, 2002, at a conference in Trieste, Italy, Russian mathematician Grigori Perelman offered the first proof of the conjecture....but it was never fully confirmed by other mathematicians. Also, the prize was not awarded.
And now, Professors Cao Huaidong (Lehigh University in Pennsylvania) and Zhu Xiping (Zhongshan University in South China) have co-authored a 300-page paper, "A Complete Proof of the Poincaré and Geometrization Conjectures - Application of the Hamilton-Perelman Theory of the Ricci Flow." It seems to be the final necessary step in the "proving/verification" process, as the paper is being published in the June issue of the highly-respected Asian Journal of Mathematics.
So who gets the prize? The proof by Perelman has been around for a couple of years, but this is the first time that mathematicians appear to consider the conjecture fully proven...or so says Professor Shing-Tung Yau, a Harvard mathematician. The expectation is that Grisha Perelman will get most of the credit for the solution...but will he also get most of the money?
And for me...another unsolvable problem bites the dust. First it was Appel/Haken's resolution of the Four Color Problem in 1976, Wiles' proof of Fermat's Last Theorem in 1995, and then Hales' solution to Kepler's Sphere-Stacking problem in 1998. And now its the Poincaré Conjecture being solved! What great problems remain...that I can understand?