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.

Update: 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.