Controversy over Yau-Tian-Donaldson

The last posting here was about an unusually collaborative effort among mathematicians, whereas this one is about the opposite, an unusually contentious situation surrounding important recent mathematical progress.

What’s at issue is the proof of what has become known as the “Yau-Tian-Donaldson” conjecture, which describes when compact Kähler manifolds with positive first Chern class have a Kähler-Einstein metric. This is analogous to the Calabi conjecture, which deals with the case of vanishing first Chern class. Progress by Donaldson on this was first mentioned on this blog here (based on his talk at Atiyah’s 80th birthday conference in 2009). Last fall a proof of the conjecture was announced by Chen-Donaldson-Sun, with an independent claim for a proof by Gang Tian, see here. I wrote a bit about this last winter here, after the details appeared of the Chen-Donaldson-Sun proof, and that posting gives some links to expository articles about the subject.

I had heard that there were complaints about Tian’s behavior in this story, including claims that he did not have a complete proof of the conjecture and was not acknowledging his use of ideas from Chen-Donaldson-Sun. Recently this controversy has become public, with Chen-Donaldson-Sun deciding to put out a document (linked to from Donaldson’s website) that challenges Tian’s claims to have an independent proof. The introduction includes:

Gang Tian has made claims to credit for these results. The purpose of this document is to rebut these claims on the grounds of originality, priority and correctness of the mathematical arguments. We acknowledge Tian’s many contributions to this field in the past and, partly for this reason, we have avoided raising our objections publicly over the last 15 months, but it seems now that this is the course we have to take in order to document the facts. In addition, this seems to us the responsible action to take and one we owe to our colleagues, especially those affected by these developments.

I should make it clear I’m no expert on this mathematics, so ill-equipped to judge many of the technical claims being made. The Chen-Donaldson-Sun document is giving one side of a complicated story, so it would be useful to have Tian’s side for comparison, but I have no idea if he intends to respond.

On a more positive note, perhaps this controversy will not interfere much with future progress in this area, as Donaldson and Tian are jointly organizing a Spring 2016 workshop on this topic at MSRI.

: I hear from Tian that he has recently written a response to the Chen-Donaldson-Sun document, which is available here, and he may at some point write some more about this. Anyone who has read the CDS side of this should also take a look at what Tian has to say in response.

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19 Responses to Controversy over Yau-Tian-Donaldson

  1. An. says:

    If you carefully read this linked document “On some recent developments in K\”ahler geometry”, the accusations sound pretty serious. Also, they did a great job presenting the evidence, which all looks very convincing. Basically, they claim that most key arguments were copied by Tian without any reference after their preprints were posted. From first to second version of Tian’s paper on arxiv, several missing or wrong arguments were replaced with the ones almost identical to theirs, and still there are some gaps left (probably because he can not continue copying ALL their arguments). Why would somebody who had such a great career do something to taint it like that?

  2. srp says:

    In Hollywood, entities suing one another on one issue often collaborate simultaneously on others. Let’s hope similar compartmentalization works for the jointly organized workshop mentioned in the post.

  3. An. says:

    Of course, if you look at Tian’s let’s say top ten most cited papers, it is clear that this was probably one of his favorite pet problems. This might somewhat explain his behavior. Also, wouldn’t it be better to call it in the post the Yau conjecture, since this is primarily how CDS call it in their document (they mention that sometimes it is called YDT conjecture). Well, at least one good side effect of this controversy is that we can safely assume that there will never be a paper by the trio Sun-Tian-Donaldson.

  4. Peter Woit says:

    From what I can tell “Yau-Tian-Donaldson conjecture” is pretty standard terminology. I suspect the reason CDS are avoiding it is just Donaldson’s adherence to standard academic practice of authors avoiding use of their own name to describe things in their papers.

  5. An. says:

    Peter, this doesn’t stop Tian from using it, as well as the Cheeger-Colding-Tian, in his paper. I noticed that he also mentions Hermitian a lot.

  6. Daniel Mathews says:

    As mentioned in CDS document, Tian posted his paper in two versions. The first one posted on Nov. 20, 2012 has only 32 pages. The sencond one on Jan. 28, 2013 has 47 pages. What they didn’t mention is there is a third version posted quite recently on Oct. 18, 2013 at, which has 61 pages. The fact is, Tian has been correcting and adding more details for his paper in the past year, while CDS has not changed a word in their papers. What can one conclude from this simple fact?

  7. Kent says:

    One can’t conclude anything at all from that simple fact. He could be adding and correcting his papers because they were incorrect or incomplete at first. Or, perhaps his paper was correct and complete at first, but he is making additional progress. But the fact that there are multiple versions of his paper, on its own, proves nothing. It all depends on the details.

  8. theoreticalminimum says:


    Hermitian is absolutely fine. French mathematicians & physicits use hermitien. Apparently, this whole hermitian-vs-hermitean pseudo-pedantry is only an issue in some English-speaking milieu, which is rather funny.

  9. Mathematician says:

    Tian’s reply can be found here:

  10. An. says:

    theoreticalminimum, it was a joke (about Hermitian).

    Tian’s response is reasonable overall, but the response to accusations in Section 3.2 avoided the main issue, namely, why did the proof of Lemma 5.8 went from 25 lines in the first version of his paper to 10 pages in the second version, with the proof now identical to CDS (even though Tian claims he never read it).

    On the other hand, after reading Tian’s response, I feel that the CDS document focuses on minor details and does not explain what was the BIG NEW IDEA that Tian stole from them. After all, it is hard to imagine that Tian didn’t know some minor details about the topic he has been working on for decades.

  11. Phil says:

    Maybe there is not a single big new idea. The obstacles have been there and been known to experts for many years. There are many possible ways to solve the problem, and none of them seems to work at first sight. You have to try and try, test and test. The new idea is that some guy Jimmy finally find one of the ways works, and if he told the outline to an expert Tom who had worked on this problem for many years, Tom could fill in most of the details, if not all. But you cannnot say that Tom didn’t know this outline. He just didn’t know that this outline works.

  12. Curious says:

    There is another post written by a third person:

  13. Peter Woit says:

    I saw the document from Sen Hu you link to a few months ago, decided at the time that it wasn’t a good idea to mention it on the blog. This was because it seemed rather one-sided, and it was not clear at all that it was written by someone expert enough in the technicalities and their history.

    The Chen-Donaldson-Sun document did seem worth mentioning, since, while it may be one-sided, it’s an authoritative representation of the views of some of the principals in the the story. I’m glad to see that Tian has also given his point of view. Hearing what they have to say directly from both sides involved in this is a lot better than getting a one-sided argument from some third party. So, at this point I don’t see anything particularly useful about the Sen Hu document.

  14. Peter Woit says:

    I’ve deleted several comments here, in one case at the request of the original commenter. Please do not use this comment section to engage in attacks on people, especially not taking advantage of anonymity.

    People arguing over priority issues is not very edifying, but at least in this case we’re hearing directly from the people involved giving their point of view on these issues.

  15. Grinch says:

    It is too bad to see such wonderful mathematicians spatting in public about priority.
    Indeed, we are blessed in many unusual ways in life; it can be regarded as a real blessing to be neither smart nor knowledgeable enough to work in certain areas!

    More seriously, a practical way to avoid this sort of problem is to avoid “problem-solving” altogether in favor of posing new questions which you try to answer at least partially yourself before someone else can jump onto it. But once you do publish or speak publicly about it, you have to let go of it and let your “baby” walk on her own feet! And part of the problem here seems to be Tian’s unwillingness to let his beautiful baby go off on her own….

    Another related approach is to try to come up with new theories.

    A third is attitude: to not be so worried if someone else does have “priority”, if you do “rediscover” something. This is inevitable in today’s world. Try to unhook yourself from the disappointment; if you reinvented it, it means you are on the right track!
    Furthermore, don’t be discouraged: inevitably you came up with some new angle, you just have to see what it is! Of course, be generous in your citations. No one ever lost anything by over-citing others. Err in that direction! Upsetting someone is stupid and unnecessary- and unjust. Be bigger than that….

    And most importantly: recognize that
    the process was just as much fun, and having fun with it should be the first and ONLY point of doing mathematics. (However, never, ever admit that truth publicly or to a granting agency-this is a closely guarded secret of mathematics!)

    The best mathematicians I have known- coincidentally the ones most universally loved- focused on the beauty of the math itself, and also realized the beauty of the human mind, and human personality, in its many and varied (and often difficult) forms.

    There is a caveat to this idealism: we all need a job, we all need grants. But one hopes that is a corollary of good work and good ideas and that it doesn’t become the main point. Too much focus on credit, money, power leads to self-destruction and damage to a community one loves, and which one should be working to build. Life is a rich mixture of beauty, ugliness, generosity, selfishness, jealousy, compassion, justice, unfairness. We can choose what to focus on, which tendencies in ourselves and others to encourage. If we get paid to think, to do something we love, we should never forget how lucky we are, compared to the vast majority. And we should also remember that the mathematics itself doesn’t care one iota: it will go on being beautiful, despite the flaws of its discoverers.

  16. Noah Smith says:

    Sad to see this kind of priority dispute still cropping up in math.

  17. Mitchell Porter says:

    Noah: how does the better world look and work – it’s always clear who was first? no-one cares who was first?

  18. Noah Smith says:

    Well, we seem to have a lot fewer of these disputes in recent decades. Is that an illusion? If it’s real, it’s probably due to better information dissemination making it clearer who was first. But also probably due to more credit-sharing (i.e. fewer people caring who was first past the post, as long as people did the work independently).

    So, both.

  19. Mathematician2 says:

    “Grinch”, your remarks are exceedingly cruel. The reality is that altruism (in mathematics research and elsewhere) sometimes has very negative consequences for the altruist, and comments like yours serve only to throw salt into the wound. I used to talk openly about my ideas, was generous with co-authorships, etc., but started realizing that I kept getting burnt. In one case, after a talk by a former collaborator, I explained to him (and his collaborator on that talk), how to generalize their result. I was simply being generous. I was shocked when I later got a paper to referee by those two guys, with the result I told them, but not even an acknowledgement that I told them. When I discussed the situation with a senior colleague, they berated and belittled me with the implication, like “Grinch”‘s, that it would be contemptible for me to finally stick up for myself for once, and that I should shut up and accept the paper, which is exactly what I did, I was so browbeaten.

    I really wish that mathematics research (and other activities) were a utopian, happy cooperative endeavor that people did for the pure joy of it. But it’s just not that way. There are too many aggressive ambitious people out there, and if you are the generous altruistic type, you just get walked all over. Personally, the only way I could deal with that type of environment, was to avoid it completely and work in isolation. I would have loved to have been part of a happy cooperative mathematics research community, but I found that in my area, it doesn’t exist.

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