(mirror site from original created in 2002-10)
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.
One of the oldest and most simply stated problems in topology is the Poincare Conjecture. This conjecture states that the only compact three dimensional simply connected manifold is a three dimensional sphere. While most senior undergraduate math majors can understand the statement of this conjecture the problem has baffled mathematicians for over a century. In recent years Hamilton had been investigating an approach to solve this problem using the Ricci Flow, an equation which evolves and morphs a manifold into a more understandable shape. Then in late 2002, after many years of studying Hamilton's work and investigating the concept of entropy, Perelman posted an article which combined with Hamilton's work would provide a proof of Thurston's Geometrization Conjecture and, thus, the Poincare Conjecture. Since then many experts have added necessary details to Perelman's ideas, some providing short cuts which would prove the Poincare Conjecture directly without the difficulties involved in the complete proof of Geometrization.
Some links to this mathematics described for the layman are
wikipedia article (I started in June 2007) which has this information and links to other excellent articles by other mathematicians.
Clay Mathematics Institute has a description of the Poincare Conjecture as a Millenium Problem by John Milnor (SUNY Stony Brook).
Taming the fourth dimension, by B. Schechter, New Scientist, 17 July 2004, Vol 183 No 2456
CNN: Russian may have solved great math mystery
Major math problem is believed solved, by S. Begley, Wall Street Journal, Jul. 21, 2006 explains the current Millenium Prize situation.
The Shapes of Space , by G. P. Collins, Scientific American, 2004 July, pp. 94-103
Math Fiction: Perelman's Song, by T. Chang, listed on Kasman's Mathematical Fiction website, with an intro by Kasman, available here, published by Math Horizons. See also here.
If it looks like a sphere..., by E. Klareich, Science News Online, June 14, 2003, Vol 163, No. 24, p 376.
Elusive Proof, Elusive Prover: A New Mathematical Mystery, by Dennis Overbye, New York Times, Science, August 15, 2006.
Structures of Three Manifolds, for the scientificly inclined audience by S.T. Yau, June 20, 2006.
A sensationalist article which divides the mathematics community has appeared in the New Yorker. See Hamilton's Response
Who cares about Poincare?, by Jordan Ellenberg, Slate, Aug 18, 2006, is for the layman
The Work of Grigori Perelman, talk by John Lott, ICM 2006 Presentation, for mathematicians in all areas, nice graphics
I have not read The Poincare Conjecture: In Search of the Shape of the Universe by Donal O'Shea. Anyone who has, send me a note to add here.
Poincare's Prize by George Szpiro. This is a book I editted for its mathematical content and I believe brings things to a level for the general audience. It has lots of mathematical politics and covers the past attempts towards a proof of the Poincare Conjecture in quite some detail. Anyone interested in more mathematical details should feel free to contact me. I might create a webpage with answers to people's questions.
Transparencies:
Here are some transparencies I made to give math club talks at Williams College, Wellesley College, Lehman College and Tufts University on the Perelman-Hamilton approach to the Poincare Conjecture. They are sketches of the main concepts including a discussion of what a three dimensional sphere is and what it means to be simply connected and compact. The key ideas in Hamilton and Perelman's work are described with sketches.
Transparencies (page by page in jpg created using Corel Draw):
page a, page b, page c, page d, page e, page f, page g, page h, page i, page j, page k.
If you have any difficulty downloading these jpg files. 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. Here they are a single pdf file.
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:
"Geometrization of 3-Manifolds via the Ricci Flow" by Mike Anderson
Structures of Three Manifolds by S.T. Yau,
The Work of Grigori Perelman, a talk by John Lott,
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):
Perelman had been a postdoc at Courant Institute (NYU) working with my thesis advisor, Jeff Cheeger, when I was a graduate student there. A postdoc is someone who has completed a PhD and is continuing to focus on research for a few years prior to settling into a tenure track position as a professor. Often graduate students talk to postdocs when they have questions about math articles and are nervous to approach their advisors. I recall Perelman was quite helpful on a few occasions describing the key ideas in a paper. He described ideas in pictures and would explain the overall concept as well as the details. He was quite tolerant of graduate students and our often naive questions.
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.
The Verification Process:
Since Perelman first posted ``The entropy formula for the Ricci flow and its geometric applications'' and "Ricci flow with surgery on three-manifolds'' , mathematicians have been checking his work in detail and some have posted papers which clarify certain aspects of his proofs. See here for work by Kleiner-Lott and others. Kleiner and Lott explain that Perelman's papers "contain some incorrect statements and incomplete arguements " but that "they did not find any serious problems, meaning problems that cannot be corrected using methods introduced by Perelman" in their completed Notes on Perelman's Papers. Before this conclusion was reached by these authors, significant work had been completed.
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:
On March 18, 2010, the Clay Mathematics Institute has determined that Perelman should be awarded a Millenium Prize for his solution of the Poincare Conjecture. Their press release provides yet another resource for those interested in reading about the mathematics and the verification process. Regretably this award has lead to an intrusion on Perelman's privacy once again by the press. For Perelman, the true award for solving this problem was the act of discovering the solution himself. People do not become mathematicians out of the hope of winning a million dollar prize someday. Please do not encourage the press to pursue him further.
Webpage written by Christina Sormani, CUNY Graduate Center and Lehman College