The Poincare Conjecture says "hey, you've got this alien blob that can ooze its way out of the hold of any lasso you tie around it? Then that blob is just an out-of-shape ball". Perelman and Hamilton proved this fact by heating the blob up, making it sing, stretching it like hot mozzarella and chopping it into a million pieces. In short, the alien ain't no bagel you can swing around with a string through his hole.
Transparencies (page by page in jpg)
If you have any difficulty downloading these jpg files, please email me and I will send them to you (sormanic at member.ams.org). If an artist would like to help me render these images in a more beautiful manner, please contact me. They are drawn using a tablet PC bought with a PSCCUNY grant using Corel Draw software.
Mathematicians may feel free to use these transparencies when giving undergraduate presentations. Keep in mind it is designed to be a 40 minute talk for math majors so each transparency should be dwelled on until the audience shows understanding. The remaining 10 minutes can be spent answering questions about being a mathematician.
Before giving a presentation, one might wish to read one of the following:
See also the wikipedia article I started in June 2007 which has this information and links to other excellent articles by other mathematicians.
Note that I do not clarify on the transparencies exactly what Perelman did and what Hamilton did and how Perelman dealt with certain difficulties with Hamilton's original plan. This research is the combined effort of many mathematicians and one might say Perelman's crucial contribution was the introduction of an entropy estimate and a monotonicity formula which essentially says that a special eigenvalue of the manifold undergoing the Ricci flow and cut/paste process is increasing. Although the talk doesn't discuss this deep insight, it is necessary to understand what happens when the Ricci flow doesn't just approach a round sphere. One may describe an eigenvalue to the audience by saying that it is essentially a note produced by a manifold when it resonates.
The Human Side (please check out the transparencies above first):
He later became a professor at the Steklov Institute Leningrad, choosing this purely research professorship over quite a few options here in the United States. Although I didn't keep in touch with him personally, my advisor sent me copies of unpublished work Perelman would send to him. I used them for my research and went on to become a postdoc myself. During a postdoc with Shing-Tung Yau, I met Hamilton who was regularly visitting Yau and was intensely working with him to solve Thurston's Geometrization Conjecture. The Poincare Conjecture would be a simple consequence of Thurston's Geometrization.
Hamilton was a dynamic professor. He was not as geometric in his examination of mathematics as Perelman. He was an expert with formulas and in this sense he was more of an analyst than a geometer. However, once his proofs were completed with analysis (the construction of crucial inqualities involving derivatives), his results were geometric. Hamilton was very friendly, loved horses and wind surfing as well as mathematics. He gave fun talks with plenty of pictures and, when relevant, he even liked to draw bunnies.
While Hamilton had completed many steps towards proving Thurston's conjecture, he was stuck on an issue involving the development of a singularity (a place where the derivatives are not defined) and something called the cigar soliton. He openly admitted that this soliton was the key difficulty hindering his program and welcomed others to tackle the question.
When Perelman posted his paper ``The entropy formula for the Ricci flow and its geometric applications'' on the mathematics arxiv, it immediately looked like he might have overcome Hamilton's issue using a brilliant new concept. However, unlike a typical paper submitted for publication, Perelman's posted article was more of a sketch than a detailed proof. While the work did give convincing arguements eliminating the cigar soliton, Perelman did not claim to have proven the Thurston Geometrization Conjecture until the end of a second even more difficult paper, "Ricci Flow with Surgery on Three Manifolds". he posted soon after.
Perelman then came to the United States to give talks on his results and I was able to attend a week's worth of talks he gave at Stony Brook in April 2003. His talks were very geometric, full of pictures, and he answered everyone's questions in quite a bit of detail. In fact, he allowed himself to be grilled by experts for hours every afternoon. He was apparently not happy to have the attention of the press and soon returned to Russia.
Finishing Poincare: A third paper posted by Perelman and a paper by Colding-Minicozzi provide descriptions which make short cuts to prove Poincare's Conjecture without proving all of Thurston's Geometrization. They use minimal surface or soap-film techniques. Morgan-Tian have written out an almost 500 page detailed proof of the Poincare Conjecture providing the background and details for Perelman's proof of the Poincare Conjecture.
Finishing Geometrization: To complete geometrization, in his second paper, Perelman develops a clever thick-thin decomposition of the manifold. He then describes how one can apply collapsing theory to understand the topology of the thin part. Shioya-Yamaguchi then wrote a paper with a necessary adaption of Cheeger-Gromov collapsing theory required to understand the collapsing of thin manifolds. Complete details of one way to divide the manifold into thick and thin parts was then provided by Cao-Zhu. The Cao-Zhu paper, originally published in Asian Journal of Mathematics (and later updated on the arxiv with a modified introduction and abstract) is an almost 300 page detailed proof of geometrization based on Hamilton-Perelman's work. Kleiner-Lott also have finalized their extensive notes providing "details" for "Perelman's arguements for the Geometrization Conjecture."
All of these extensive treatises on the subject require serious mathematics and insight needed to complete the proofs on a rigorous level.
It should be noted that all the mathematicians mentioned here are well known mathematicians in their own right with important results quite distinct from their involvement in the Poincare Conjecture, including Hamilton and Perelman. This is not the result of single dedicated mathematician working in a vacuum but rather of the combined efforts of many. It is the culmination of a century of research in Riemannian Geometry and in Analysis.
The Millenium Prize:
Webpage written by Christina Sormani, CUNY Graduate Center and Lehman College