A paper by Yuhan Zha, posted on the arXiv yesterday under the unassuming title “A height inequality,” claims to prove the ABC conjecture via a notion of “quasi-arithmetic differential,” some kind of Arakelov-theoretic gadget which apparently allows you to mimic complex differential geometry well enough to imitate the proof of the function field case. Zha was a Ph.D. student of Fulton and a Harvard postdoc, so this presumably merits serious consideration. Has anybody out there given this a real read yet?

## Claimed proof of the ABC conjecture

**Tagged**number theory, abc, abc conjecture, arakelov, arakelov theory, yuhan zha

Isn’t it a bit elitist of us (as a community) that whenever a proof of a longstanding problem is published we try to lend/detract to its credibility by saying things like: “… was a Ph.D. student of Fulton and a Harvard postdoc, so this presumably merits serious consideration.” There are a lot of cranks out there, and it’s been a while since a longstanding problem has been solved by anyone other than a professional mathematician. On the other hand, what would have been of Ramanujan if Hardy was this stuck up.

I also have some issues with the implication of the sentence. If it was “just” postdoc y who was a student of Prof. Y at State university Z, would it just merit some consideration?

That all said, I hope it all works out!

In my opinion this is not at all elitist. Obviously some expert will look at any manuscript which makes a serious attempt to solve a conjecture of this caliber. Reminding people of the pedigree of one author does not change that for other authors. But the reminder may help raise awareness among non-experts.

I remember a while ago there were rumors about a solution of the abc conjecture by a famous number theorist in the New York area. I heard that there was a problem with that approach. But I didn’t hear if the mistake was something people expected might be fixed, or if some other progress came out of that approach. Do you know what was the outcome?

Jason, the incident you are referring to happened in May of 2007. After appropriate scrutiny of the (never publicly released) preprint, I think the expert consensus was the method was fatally flawed.

David, thanks for the information.

Listing the author’s academic pedigree could be construed as informing my beliefs about any of the following:

(1) my initial hunch as to whether or not the proof will turn out to be correct

(2) my decision as to whether or not I personally should invest time looking at and evaluating the proof

(3) whether or not the proof is actually correct in the end

Allowing the pedigree to affect (1) could certainly be called elitist; allowing it to affect (2) is mostly just good personal economics. I very much doubt, however, that any self-respecting mathematician would ever use the academic pedigree of the author as a factor in his or her beliefs about (3); instead, he or she would wait for experts in the field to come to a consensus about the validity of the proof.

So if by “lend/detract to its credibility”, 1-Anonymous is referring to (3), then I don’t think that actually happens. Of course we have to watch out that our snap judgments about (1) don’t leak into our beliefs involving (3) – but I think it’s consistent to have a hunch about whether the proof is correct yet still remain agnostic about whether the proof is actually correct.

Let me proclaim my opinion, boldly (and anonymously) that the proof is not correct.

Two reasons, both independent of the “pedigree” of the author.

The abc conjecture is clearly a deep problem. Consequently, the proof should contain deep ideas. If you look at the paper, it seems more like a sequence of calculations rather than anything else (cf. a recent arXiv paper on the Gauss circle problem). This is especially true in light of the fact that these methods have not produced any significant arithmetic consequences since Vojta’s proof of Falting’s Theorem (as least as far as I’m aware). Usually, when a field gets “stuck” in this way it takes a bold new idea rather than a finessing coupe de grâce. Of course, maybe the paper has that bold idea and I just didn’t notice it — always a possibility.

Secondly, didn’t Yoichi Miyaoka claim to have proven Fermat’s Last Theorem in 1988 using a very similar sounding idea (an Arekelov height inequality)? Unfortunately, 1988 predates the arXiv so I can’t check the details.

(note that the two Anonymi above aren’t the same.)

Let me hasten to clarify that I don’t think people who work at state universities don’t deserve serious consideration when they claim a proof of a big conjecture — that would cut against my self-interest. What I meant was something much more like Greg’s option (2).

Speaking of claims, I can’t help being impressed by Renyi Ma, who has posted on arXiv since September 2008 rather short preprints proving:

* the Arnold conjectures about Lagrangian intersections and such (8 pages)

* the Hodge conjecture (7 pages)

* homotopy invariance of higher signatures (3 pages)

* and reproved the Poincaré conjecture (7 pages)

and a few others.

Has Yuhan Zha published anything? I couldn’t find anything at Math Reviews online, though perhaps this reflects my lack of search expertise.

Of course, whether someone has or hasn’t published anything before has no effect on whether or not what’s currently published is correct or not. I’m curious: has anyone gone ten years from getting the PhD without publications, and then published something really good?

The following blog post is very relevant to one part of that discussion:

http://scottaaronson.com/blog/?p=304

Gregory Perelman didn’t publish anything after 1994, although he did upload a couple preprints to the arXiv in 2003 and 2004 that attracted some interest.

Andrew Wiles had no publications between 1990 and 1995, although some of this can be attributed to the time it takes to push a paper (and particularly

thatpaper) through the Annals.But, of course, both of these mathematicians had a superb track record of non-trivial results in relevant fields before achieving their most famous breakthrough, and in retrospect one can see why there was such a lengthy gap in their publications. And such a track record is certainly relevant for estimating the prior probability that a claimed proof is correct, though of course if one has decided after all to invest the time to look at it more carefully, then the nature of the proof itself is obviously the more relevant factor.

Academic credentials are a weak form of track record, and so have some limited utility for this purpose, but are basically only useful when no other pertinent information is available to be competently and readily evaluated, and when the credentials are in a clearly relevant area. (But when a renowned expert in field X suddenly announces a breakthrough in an apparently unrelated field Y, one should however be cautious, unless the breakthrough also clearly provides some rich new connection between the two fields.)

Arithmetic geometry is not my own field of expertise, so I can’t evaluate the paper in any great depth, but on a cursory reading I was not able to see what the key new insight, idea, or strategy is, what the key difficulty to be resolved was, or what the key step is which distinguished the argument from previous attempts at establishing these sorts of arithmetic intersection inequalities (more generally, the fact that there are not extensive comparisons with the prior literature and partial results, discussions of key special cases, remarks or speculations about how the method applies to simpler or harder problems of this type, etc. is also somewhat worrying). So my own inclination is to wait for the experts in the area to weigh in before getting too excited at this stage.

> I’m curious: has anyone gone ten years from getting

> the PhD without publications, and then published

> something really good?

Gerry (#9),

William H. Young might be a very strong example of this,

except (1) I’m not certain he actually obtained a Ph.D.

and (2) this was a long time ago and things are much

different now. In any event, Young is probably the most

singularly unique example of a mathematician who did

no research until almost 40 and then, afterwards, published

several thousands of pages of high quality and uniquely

original mathematics.

Since this has been “out there” for about two months now, I was curious if there is any expert feedback yet? Unlike the recent case of Li’s paper on the Riemann hypothesis, this hasn’t been updated or retracted since the initial upload. Presumably, someone should have been able to take a careful look by now.

An error has been found but again he claims he has fixed it. http://arxiv.org/abs/0901.3042