A new 473-page paper by Gang Tian and my colleague John Morgan that gives a complete proof of the Poincare conjecture based upon the argument outlined by Grigori Perelman (which carries out the program of my other Columbia colleague Richard Hamilton) is now available as a preprint on the arXiv entitled Ricci Flow and the Poincare Conjecture. This paper is in the process of being refereed and should ultimately appear as a book in the monograph series that the Clay Math Institute publishes with the AMS.
Morgan and Tian just provide a proof of Poincare, not the full geometrization conjecture. Other sources for worked out details of Perelman’s argument are the notes by Kleiner and Lott, and the recent paper by Cao-Zhu that appeared in the Asian Journal of Mathematics. Cao-Zhu provide fewer details than Morgan-Tian, but do give a proof of geometrization. Until very recently the Cao-Zhu paper was only available in the paper version of the journal, for sale by International Press for $69.00. Yesterday the journal put the full paper on-line, and it’s available here.
Latest rumor I hear is that the Fields Medal committee has definitely chosen Perelman as a Fields medalist, with the appearance of these detailed proofs using his arguments clinching the deal. However it remains unclear whether he’ll show up in Madrid, or even actually accept the honor being offered him.
Update: There’s an article about this in this week’s Nature.
Update: The September issue of the Notices of the AMS has an excellent article by Allyn Jackson about this. Next week’s Science Times is supposed to have an article by Dennis Overbye.
I used to wonder: what would happen if mathematicians received the solution to a centuries-old open problem in the form of a coded message from an extraterrestrial intelligence? Would they put their other projects aside, and work together to decode the message? Would they understand the solution? Would they try to start a dialogue with the intelligence? Would they credit it (maybe even offering it a Fields medal), or would they treat it roughly like astronomers treat a gamma-ray burst? While we still don’t know the answers to these questions, it’s possible that we have more insight than we did a few years ago.
Scott,
A weaker version of this has already happened, with mathematics getting unexpected solutions to problems via coded messages from Edward Witten, who some have suspected of an intelligence extra-terrestrial in origin. Many mathematicians did put their other projects aside and worked together to decode the message. They did start a dialog with the intelligence, but it remains unclear whether they understood what he was telling them (see the two-volume IAS set on QFT that this dialog inspired, for evidence to argue the case either way).
They definitely did credit the intelligence, and gave him a Fields medal.
Peter,
Could you elaborate on what you mean by “Cao-Zhu provide fewer details than Morgan-Tian” and on what basis you make this judgement? If it’s from reading the papers themselves, could you point to where this is apparent?
Deane,
I haven’t done more than skim the papers myself and I’m no expert. That comment was based largely on the fact that the Morgan-Tian paper is much longer than Cao-Zhu, and yet doesn’t cover as much (no geometrization), together with impressions I got (which may be mistaken), from talking to people who have looked more carefully at both papers. I’d love to hear here from an expert who could more accurately compare the two papers.
Pingback: Ars Mathematica » Blog Archive » Poincare Conjecture Settled?
I’ve heard it suggested that both John von Neumann and Alexander Grothendieck were aliens, so Scott, I think your hypothesis has been amply tested. The real question is why are aliens interfering with the progress of mathematics? What is their purpose? Is it benign, or (as I suspect) sinister? What mathematical truths are they preventing us from discovering by distracting us in this way?
Peter,
Using layman’s language, would you briefly explain how the Poincare Conjecture might spill over into the foundations of theoretical physics?
I would greatly appreciate your input/thoughts on this matter…
Best,
So Perelman can receive a Fields medal without actually writing up a proper proof? Just an outline that others fill in? Could Wiles have won a FM if he had said (while age
There’s a fun Wall Street Journal article on Perelman and the Clay Mathematics Institute’s million-dollar prize for proving the Poincare conjecture. It’ll be interesting to see how they deal with the complicated mess that’s been brewing. I don’t think they expected someone working on such an important conjecture to not bother publishing in a refereed journal!
Cynthia,
Sorry, but I don’t really see any relevance of the Poincare conjecture to fundamental theoretical physics. With some effort one might come up with some very speculative idea along these lines, but there’s nothing I’m aware of of this kind that there’s any evidence for.
It is kind of amusing that Ricci flow, the basis for Perelman’s proof, shows up in the RG equations for a 2D sigma model, but as far as I know nobody has done anything with that.
I gather that the proof involves showing that the Ricci flow on any simply-connected 3-fold goes towards a fixed point which is precisely the 3-sphere.
Now, the plain 2D-sigma model on the 3-sphere is not conformal, since the 3-sphere is not Ricci flat. So are we really talking about the SU(2) WZW model?
There are paprs by I Bakas on Ricci flows from a 2D field theory point of view. My recollection is that he has to deal with gauge groups with generators that have continuous indices, or something that strange.
In reference to John Baez post; The Clay Comitee already made it clear in its rules to determine the prize winner that. “It is discretionary to the comitee to award the prize even if the proof has not been properly published in a refeered journal as long as it survived the scrutiny period of at least two years”.
I conjecture that Grigori Perelman will be the first awardee to turn down the Fields Medal. If we assume such an occurrence, then one can infer that he will be absent in Madrid, when the IMU disciples/attachés meet.
It will be interesting to observe how a few hundreds of thousand of dollars (amounting in toto to more than a million) from the Clay Institute and IMU safes will get distributed in the months to come, amongst the few people who have worked on the proof. After all, awarding prizes and money has always been a real fuss, but sometimes arguably useful.
However, we take the opportunity to applaud the great intellectual triumph of this little group of people whose work have, for decades, been directed towards engineering mathematical techniques to solve one of the greatest problems in the realm of reason. And bravo Perelman!
Waldron vs. Woit – this was a cakewalk. “Marketplace for ideas” – what idiocy. Peter, speak more slowly and don’t pull punches in these interviews!!
-drl
sorry the above was meant for the other thread
Why is everyone so sure Perelman will turn down the Fields? Did he say that?
Wow. The paper is in fact quite readable. I predict two things resulting from this readability. One, it will be denounced by mathematicians as inadequate, and two, the proof will stand the test of time.
I meant the Cao-Zhu paper, which contains clearer proofs, not the Tian-Morgan, whos prrofs seem much more reticent.
According to the ICM website
http://www.icm2006.org/press/bulletins/bulletin20/#poincare
“Although Perelman himself, reluctant at least until now to appear at public events, will be absent, the subject will undoubtedly be the highlight of the congress.“.
Perelman is obviously aware of the impact of his work as he met many people during his lectures in the US, and clearly for him the ICM would be a waste of time (lots of media attention and no new maths or contacts). So not coming there only means he’s not too much of a party animal, nothing wrong with that.
Walt said:
“Why is everyone so sure Perelman will turn down the Fields? Did he say that?”
To answer your second question, as far as I know, Perelman did not say anything about whether he would accept or decline the award of a Fields medal. If there’s a statement of his concerning this, to him at least, trivial issue, then you understand that I am unaware of it.
As far as I am concerned, I was just speculating his refusal of the Fields medal, and there are a few indications that he might actually do so.
1) He has already rejected at least one mathematical prize. In 1996, he refused to accept an award in Budapest, Hungary, from the European Mathematical Society.
2) I read on the wikipage about him that he is said to be “very unmaterialistic”.
3) In an article (“On the verge of a solution”) by Douglas Birch, which appeared in The Baltimore Sun (January 19, 2004), it is written (I quote):
“But the 37-year-old native of Leningrad, now St. Petersburg, doesn’t seem interested in money or acclaim. While he could probably get a far more lucrative job in the West, he earns only about $200 a month at the Steklov Institute of Mathematics in St. Petersburg.”
4) In a news announcement of The Abdus Salam International Centre for Theoretical Physics, I read (I quote):
“There is also some indication, as yet unconfirmed, that he will not accept the US$1 million prize from the Clay Mathematics Institute in Cambridge, Massachusetts, if it is offered.”
These might, or might not, be taken to be true (to the extent that they come from the mouth of the mathematician, if at all, undistorted). Some uncertainty indeed!
After all, we all share the opinion that Perelman doesn’t care much, if at all, about the prizes (Clay and Fields) and millions. He just needs to be left to do his work, like he’s been doing in the last decade or so. He proved Thurston’s Geometrization conjecture was true (with demonstration), that’s what really counts in history.
I think that one pertinent and actually significant reason, in the societal context at least, of his refusal would be that he may be concerned that the talk, and possible eventual acceptance, of the prize moneys could make him a target of the Russian underworld. I don’t think the society of mathematicians wants to risk this kind of thing to happen to one of its most able members.
So far, we know from the announcement on the ICM 2006 website (thanks to nonblogger for the link) that Perelman will indeed be absent. But this does not exclude the fact that a possible Fields medal could be awarded in absentia, if this can happen at all. Well, we will know in a few days’ time!
ST Yau has a paper in the arxiv today in which it is pointed out that the Chinese were the first to invent pasta, gunpowder, magnetism, etc etc etc, and that it is long past time that they were accorded their rightful place at the head of etc etc etc.
The previous comment is a libelous mischaracterizatization of Yau’s talk. The commenter either (a) has an axe to grind against Yau, or (b) is a racist.
Unfortunately this is a subject where a lot of people have axes to grind. Please don’t do this here.
I would suggest to wait for the validation of the proof presented by Perelman, and hence the paper by Cao and Zhu. and others. Remember that The Pioncare Conjecture is notorious for its technicalities and has stopped many great mathematicians in the past.
British Prof. Dunwoody’s proof being the latest to come short.
Let be fair here: I respect Yau but not what he said in several of his interviews:
1) Someone won the Veblen under his influence on the committee;
2) in the Qiao Bao July 9 2006 issue, Chinese Weekend, he SAID THIS, “But the other group of mathematicians–Morgan and Tian Gan, they said they already had their paper, but it’s been two months from now, if they have it, why not publish it? They said, their 400 page manuscript had been submitted to America’s Clay Math Institute. I (this year) in late April asked this institute’s director, his name is Carlson, he said he doesn’t have this “manuscript”, but only an “introduction”.
That’s true. Professor Yau once criticized the Chinese people for envying their people’s accomplishment. He in that interview claimed that if these critics of Zhu and Cao’s work can understand the paper of Poincare Conjecture, he would award them with 10000 dollars (or yuans?) However, on the other hand, he repels the work of another Chinese mathematician–Tian, work done jointly with Morgan. So much contradictions in his talks and interviews. Hope he is not a hypocrite.
So whose work in Poincare conjecture is THE SINGLE MOST IMPORTANT? I heard that Freedman, Thurston, Hamilton, Perelman and Smale have all worked on it. It seems it’s just the Ricci Flow and Thurston’s Geometrization that play the role in solving the problem. IF the tools are handily ready, then solving the Poincare Conjecture isn’t like work on an extremely lengthy Olympiad math problem? Any ideas?
I think Smale’s work only applies to higher dimensions, and doesn’t help at all for the 3-d Poincare conjecture, but I could be wrong. Thurston sketched out a new way of looking at 3-manifolds, but his role somewhat analogous to Poincare’s. Hamilton invented the key technique to prove the conjecture, but if his idea by itself was sufficient to solve the problem, then he would have done it himself.
“A weaker version of this has already happened, with mathematics getting unexpected solutions to problems via coded messages from Edward Witten, who some have suspected of an intelligence extra-terrestrial in origin.” – Woit.
Susskind says something similar about Edward Witten in a recent invterview see http://www.thestar.com/NASApp/cs/ContentServer?pagename=thestar/Layout/Article_Type1&c=Article&cid=1154082909559&call_pageid=1105528093962&col=1105528093790 which is an article in The Toronto Star by Siobhan Roberts:
“… at a public lecture at the Strings05 conference in Toronto, an audience member politely berated physicists for their bewildering smorgasbord of analogies, asking why the scientists couldn’t reach consensus on a few key analogies so as to convey a more coherent and unified message to the public.
“The answer came as a disappointment. Robbert Dijkgraaf, a mathematical physicist at the University of Amsterdam, bluntly stated that the plethora of analogies is an indication that string theorists themselves are grappling with the mysteries of their work; they are groping in the dark and thus need every glimmering of analogical input they can get.
‘ “What makes our field work, particularly in the present climate of not having very much in the way of newer experimental information, is the diversity of analogy, the diversity of thinking,” says Leonard Susskind, the Felix Bloch professor of theoretical physics at Stanford, and the discoverer of string theory.
‘ “Every really good physicist I know has their own absolutely unique way of thinking,” says Susskind. “No two of them think alike. And I would say it’s that diversity that makes the whole subject progress. I have a very idiosyncratic way of thinking. My friend Ed Witten (at Princeton’s Institute for Advanced Study) has a very idiosyncratic way of thinking. We think so differently, it’s amazing that we can ever interact with each other. We learn how. And one of the ways we learn how is by using analogy.”
“Susskind considers analogy particularly important in the current era because physics is almost going beyond the ken of human intelligence.
‘ “Physicists have gone through many generations of rewiring themselves, to learn how to think about things in a way which initially was very counterintuitive and very far beyond what nature wired us for,” he says. Physicists compensate for their evolutionary shortcomings, he says, either by learning how to use abstract mathematics or by building analogies.
“Susskind, for his own part, deploys more of the latter. Analogy is one of his most reliable tools (visual thinking is the other). And Susskind has a few favourites that he always returns to, especially when he is stuck or confused.
“He thinks of black holes as an infinite lake with boats swirling toward a drain at the bottom, and he envisions the expanding universe as an inflating balloon.
“However, the real art of analogy, he says, “is not just making them up and using them, but knowing when they’re defective, knowing their limitations. All analogies are defective at some level.”
“A balloon eventually pops, for example, whereas a universe does not. At least not yet.”
The Chinese invented Pythagoras’s theorem hundreds of years before Pythagoras. They were knitting woollen sweaters 6000 years ago when Europeans were wearing skins. They invented tiramisu 10000 years before the Italians, and were watching TV 20000000 years ago when Europeans had not even evolved eyes yet. No grounds for surprise that they proved Poincare first.
Perelman certainly deserves the Fields, I think only he and Grothedieck are the only two great mathematicians who do not pay attention to fame and glory, as I know. Does the proof of the geometrization automatically include a proof of the Poincare conjecture?
Sarah,
I agree Perelman deserves the Fields. But there are plenty of other great mathematicians who don’t seem to be very interested in fame or glory. Grothendieck didn’t turn down the Fields medal, and his refusal to accept the Crafoord prize came when he was already isolating himself from the world in an eccentric fashion. In “Recolte et Semailles” he has harsh things to say about Deligne, which seem to be based on his feeling that Deligne was getting credit for his ideas.
Yes, geometrization includes Poincare as a special case.
Peter,
Grigori Perelman carried out the program of Richard Hamilton with some new methods. Cao-Zhu’s paper claims it grows out of the theories of Hamilton and Perelman and completes the proof of the Poincaré and the geometrization conjectures (from what I read, Perelman’s papers seem to grow out of the work of Hamilton and others too). Cao-Zhu’s paper is published in a refereed journal. Unless it is found that there are gaps or even errors in the Cao-Zhu paper, should the proof of the conjectures be considered as finished? If so, are the Kleiner-Lott paper and the Morgan-Tian paper appeared a little too late? Especially for the Morgan-Tian paper, it was posted on Arxiv one month after the publication of the Cao-Zhu paper.
Jeremy
As concerning if Morgan-Tian’s paper is too late, everyone knows they have been working on it and by the time Zhu and Cao were done with their paper, Morgan and Tian had their ready also. I think they were just trying to polish the paper but when they saw Cao and Zhu’s came out, they believe it was time to have theirs published. That’s my guess.
My guess is that since there’s a gigantic mess for who gets credit for what, that everyone will just sidestep the issue, and call it the Geometrization Theorem or something like that. The Clay Institute doesn’t have that out, of course.
I believe Morgan-Tian went to the referees in May, before the Cao-Zhu manuscript was available, so their work was completely independent. Most of Kleiner-Lott was freely available on the web quite a while ago, long before they posted a version to the arXiv, and Cao-Zhu refer to this in their paper. Kleiner-Lott worked out many details of Perelman, but didn’t write up a full proof of Poincare.
My own guess about how the math community will ultimately apportion credit for this is that the proof is based on Hamilton’s program, that important new ideas were required to make it work, and that those are due to Perelman. It seems that Perelman’s outline of a proof was essentially correct. Credit for working out the details of Perelman’s proof will go to a sizable group of people, including Kleiner-Lott, Cao-Zhu and Morgan-Tian, but I’d be very surprised if the Clay committee considers this kind of work something that should be rewarded with part of the million dollars.
Peter,
What happens if Hamilton had claimed that he had given outline of a proof to the Poincaré and the geometrization conjectures in 1982, when he first introduced the Ricci flow approach, then he would leave it to the others to fill in the details (gaps)? Of course, he didn’t. He went on to fill in the details and faced some serious difficulties. But if he did, would Perelman’s work also be considered as filling the details?
Hamilton started the program. He did not finish it, probably because he didn’t know how to fill in some the details. Is it possible that Perelman also didn’t know how to fill in some of the details of his outline?
Perelman is a great mathematician. So is Hamilton. Their work has shown that. But it does not mean that they know how to complete the proof of the conjectures.
Kleiner-Lott devoted much of their almost 200 pages paper to fill in Perelman’s gaps; Morgan-Tian devoted a large part of their more than 400 pages paper to fill in some of Perelman’s gaps (not including geometrization); Cao-Zhu devoted a large part of their more than 300 pages paper to fill in some Perelman’s gaps and some Hamilton’s gaps (Cao-Zhu do have some of their own ideas to fill in Hamilton’s gaps). Obviously, these are not small gaps. Are they?
Cao-Zhu, Kleiner-Lott and Morgan-Tian must have faced and solved many serious difficulties too. Kleiner-Lott did not write up a complete proof, but Cao-Zhu (for Poincaré and geometrization) and Morgan-Tian (for Poincaré) did. I believe that they all should share the credit with Hamilton and Perelman for completing the proof of the Poincaré conjecture. In fact, if there is nothing wrong in their papers. They are the ones who really completed the proof. Right?
As for the prize money. None of the U.S. based professors really need it. Do they?
Jeremy,
Hamilton never claimed that he had an outline of a proof of Poincare that could be completed by filling in “details”. The new ideas that Perelman came up with were not details, but original insights, mathematics not of a routine kind, but of the highest level. Perelman did claim that he had an outline that just required details to fill in, not any new insights. He could very well have been wrong. If so, Morgan/Tian and Cao/Zhu would have come to a point where standard techniques weren’t enough to fill in the outline. As far as I know, this didn’t happen, and Perelman’s claims turned out to be correct.
As for the money, most people are of the opinion that they could use more. Perelman seems to be an unusual case and the story is that he doesn’t want the money. I doubt Hamilton (or most other mathematicians) would turn it down.
Peter,
I cann’t remember which professor said this: “anything that is proved is obvious, nothing is obvious before it is proved”. Does this have any true in it? Can I replace “obvious” with your ” of a routine kind”?
Money is such a distraction. I thought that keeping mathematics out of Nobel Prize is to keep mathematicians concentrated.
Jeremy,
It’s certainly true that after someone has come up with a new idea, it often looks “obvious” and it’s hard to understand why it was so difficult to find it. But, in this case you can look at what people had to say before the ideas were there. As far as I know, before Perelman, Hamilton was not claiming he had an outline of a proof for Poincare that could be filled in with standard techniques. He and others could have pointed you to the difficulties with pushing through Hamilton’s program, and identified exactly the places where no one knew of an argument that would work. After Perelman, everyone seems to acknowledge that he found possible ways around these problems, although at first no one was sure whether the details could be successfully filled in, as Perelman claimed (and was willing to often back up by providing details when asked for them). A pretty good definition of a non-obvious (or not of a routine kind) argument is one that the leading expert on the field (Hamilton) wasn’t able to find despite working on it for quite a while.
Again, as far as I know, Cao-Zhu, Kleiner-Lott, and Morgan-Tian did not run into any problems in their work that they would describe as requiring the kind of original ideas that Perelman came up with.
Hi, Peter, saw the following question under your name:
Did Yau really claim that he was the one who solved the Poincare Conjecture? If so, that would be seriously misleading. # posted by Peter Woit : 20/6/06 12:14
http://rivelles.blogspot.com/2006/06/strings-2006-day-2.html
Here is my personal opinion and some facts plus other trivial interesting things.
There was an interview with ST Yau on Qiao Bao (Chinese Overseas Newspaper), the media has little or no comments on the Poincare Conjecture and its solvers. It was question and answer type. Yau did claim that he was ONE of those who contributed to the Poincare Conjecture. But he did NOT CLAIM that he solve it. However, in another interview and talk, in which he composed a very beautiful poem (at least it’s interesting poem) that describes his efforts and fascination in Poincare Conjecture. For contribution percentage, he modified a little in the QiaoBao, he probably didn’t make clear before that: CHINESE MATHEMATICIANS’S TOTAL CONTRIBUTION TO THIS WHOLE WORK [Poincare Conjecture? or Topology or Ricci Flow or Geometrization? I do not know] IS NOT LESS THAN 30%. Previously, I read on a newpaper that quoted as Cao and Zhu’s work takes about 35% of the credit. But this is probably a loose statement and might not be what Professor Yau meant–but I dont know; and I don’t know if he intended to mislead the Chinese media, to be fair. Well, Zhu and Cao, on the other hand, as well as many Chinese mathematicians are always low-pitched and being modest. Yau is the one who gave most if not all the talks and annoucements. in fact, what he says does not represent what many of the Chinese mathematicians think. And many of the Chinese Mathematicians do not even agree with them. And to those so called Chinese Academicians, as he calls them, Yau claimed that he would personally offered 10000 yuans or dollars to whoever is able to understand Cao and Zhu’s work and the Proof of Poincare Conjecture at that moment.
Some people say since Professor SS CHERN passed away, Yau is the single most influential Chinese mathematician nowaday. It seems true. Another mathematician, Tian Gang seems to be very prolific and able too. Out of Curiosity, I looked up Math Genealogy and found out Professor Yau’s students–most of them are very successful mathematicians on their own. I think Yau would probably get the Wolf Prize soon for this. After all, he’s got Fields, Crafoord, Veblen, MaCarthy Fellowship such big ones, Wolf may be coming too. who know. Among all the Fields Medal winners, only Yau have so many successful students so far. Mathematical Monster–an nick name I once heard given to Professor Thurston for he’s such a haughty genius and polymath that some, if not most, of us envy yet respect. Read this, ts interesting! http://www.news.cornell.edu/chronicle/02/11.21.02/Thurston_profile.html
It’s interesting to note that Thurston is one of the few who received the Alan Waterman Prize, the first one being Fefferman. Other receipients are Gang Tian, Edelsbrunner, Emmanel Candes, Friedman. But the only ones who are of the same rank as Thurston are perhaps the undisputed (genius and polymath): Bombieri, Charles Fefferman, (Bourgain?). However, all four of these geniuses do not yet have as many successful students as Yau, though Thurston has done way better than the other three genius-polymathes, and has Kerkoff, Gabai etc being the most outstanding ones. Probably geniuses are just like concentrating on their own work or maybe they have not met the right students and have high expectation for their students, who knows. Sometimes working with a genius can hurt one’s feeling of being an ordinary scholar. But being an ordinary math student is also fun–nevertheless it’s life that we live. But anyway, hey, less than 2 weeks left for ICM…I am so looking forward to it!!!Everyone says Terence Tao and Perelman are on the Fields Medal list, we’ll see.
by the way, now we have so many prizes in math: Abel, Fields, Wolf, Shaw Prize, Crafoord, Shock, Alan Waterman, the New Gauss Prize etc. Can you tell us a little more about the significance of each? Which do you think is the single most prestigious that every math department respects? Anyone like to rank them in order of importance? It’s nothing but some little fun. Thanks for your time.
William Thurston is the most approachable mathematician I’ve ever met, with the possible exception of John Horton Conway (when he’s in a good mood). They do both expect that their interlocutors make the best effort they can to understand the matter under discussion; it would not surprise me if it is somewhat hazardous to give Thurston the impression that you are mathematically able and then disappoint him. Thankfully I never had that problem. 🙂
Cheers,
– Michael
I read a survey article by Yau on arxiv (sorry, I don’t know the number) that seemed very level-headed, and did not in anyway exaggerate the contributions of himself, or Cao and Zhu. So I think there’s nothing to this.
Hi, Walt, I read that too. Unfortunately you will never find information like Janet posted in the Arxiv or any other Math Journal, Media etc in English. The talks and interviews are mostly in Chinese. In fact, even in Cao and Zhu’s paper, they are being very humble and very excited about the work they are undertaking. They are truly exceptional mathematicians who have done some very important and valuable piece of work. They shall be praised for their work. But someone, don’t remember whom, said: “(Yau) threw a monkey wrench into the question of who gets the credit.” One can not get a whole picture of what is going on if she doesn’t read a lot (not only in English) and follow closely. I don’t know about the truth but just share my information here. Thanks.
The September issue of the Notices is out, and has an article by Allyn Jackson about the Poicaré Conjecture (“Conjectures No More?: Consensus Forming on the Proofs of Poincaré and Geometrization Conjectures”). I haven’t had the time to read it completely yet, so will not comment on it.
And watch for the article in the New York Times on Tuesday.
Another pre-ICM hint: Tao has put a more recent picture of himself on his webpage these days, and I can’t imagine this is purely coincidental 😉
Allyn Jackson’s report is great!! Excellent reading!!