17
$\begingroup$

I noticed that the only official reason given for awarding Edward Witten the Fields Medal was his 1981 proof of the positive mass theorem with spinors, so I was assuming that the proof was fully rigorous.

However, I came across this paper https://projecteuclid.org/download/pdf_1/euclid.cmp/1103921154 by Taubes and Parker which claims to make Witten's proof 'mathematically rigorous' and to justify assumptions which Witten made about Dirac operators. Does this mean that the Witten proof is not rigorous, or is it just the case that there were some unjustified lemmas to clear up which do not affect the validity or rigour of the argument (similar to the case of Perelman's proof of the Poincare conjecture, where some lemmas and slight gaps had to be filled in)?

I am just curious as I have never really heard of the Taubes-Parker paper so I was assuming that the Witten paper was fully rigorous.

$\endgroup$
  • 2
    $\begingroup$ I think some clarification on what is meant by 'fully rigorous' would help, and possibly avoid some close votes -- questions like 'is xyz's proof of conjecture abc valid' are too opinion based, but if you could point to specific arguments/lemmas that appear nebulous for specific mathematical reasons it seems a good question. For example we could interpret 'fully rigorous' as 'being proved directly from the axioms of some set theory/logic' in which case the answer is obviously no, but this is not generally the bar for rigor in mathematics outside logic and set theory. $\endgroup$ – Alec Rhea Mar 23 at 7:19
  • 4
    $\begingroup$ Following @AlecRhea’s good points, I’d suggest the most productive way to frame this question isn’t “Is Witten’s proof rigorous?” so much as “How rigorous was Witten’s proof, and what is its relationship to later more rigorous elaborations like Taubes–Parker?” There is a spectrum different levels of rigour short of a full proof: a full sketch with some details missing; a sketch with most details omitted; a detailed outline; a heuristic argument which turns out to guide an eventual full proof; a heuristic argument that motivates the result but doesn’t form the basis of any proof… $\endgroup$ – Peter LeFanu Lumsdaine Mar 23 at 14:37
  • $\begingroup$ @AlecRhea I think questions about the validity of the proof of conjecture abc are referring to Mochizuki, not xyz... $\endgroup$ – user1728 Mar 23 at 15:48
11
$\begingroup$

You should probably read the conclusion (Section 6) of Atiyah's laudatio on Witten work during the 1990 ICM (specifically, the positive mass theorem is treated in Section 3).

6. Conclusion

From this very brief summary of Witten's achievements it should be clear that he has made a profound impact on contemporary mathematics. In his hands physics is once again providing a rich source of inspiration and insight in mathematics. Of course physical insight does not always lead to immediately rigorous mathematical proofs but it frequently leads one in the right direction, and technically correct proofs can then hopefully be found. This is the case with Witten's work. So far his insight has never let him down and rigorous proofs, of the standard we mathematicians rightly expect, have always been forthcoming. There is therefore no doubt that contributions to mathematics of this order are fully worthy of a Fields Medal.

$\endgroup$
  • 4
    $\begingroup$ I have read it and do not doubt Witten's contributions to physics or to mathematics: I was asking specifically about the proof of the positive mass theorem which was given as the official reason on the citation. $\endgroup$ – Tom Mar 23 at 14:55
  • 1
    $\begingroup$ The proof of positive mass theorem is treated in Part 3 of the laudatio. It is called a "outline", so actually some technical detail was missing, but the idea was completely correct. That said, I do not know why only that result has been given as official reason for the prize, since the laudatio contains much more. $\endgroup$ – Francesco Polizzi Mar 23 at 16:38
  • 1
    $\begingroup$ @TimothyChow The Wikipedia page en.wikipedia.org/wiki/Fields_Medal has all of them, but I don't know where they sourced them from. For all the years up to 1986, you can get the same information by clicking on the names. So 1990 - 2010 is the period it seems to be missing on the website. $\endgroup$ – Will Sawin Mar 27 at 22:29
  • 1
    $\begingroup$ I went to our library to look at the book "International Mathematical Congresses: An Illustrated History 1893-1986," by Albers, Alexanderson, and Reid. At the end of the book it gives photos of all the medalists up to 1986, with a little blurb underneath each photo. This appears to be the source for Wikipedia and the mathunion website. But the book gives no indication that these are "official citations." I would hazard a guess that one of the authors of the book just made them up. $\endgroup$ – Timothy Chow Mar 28 at 2:19
  • 2
    $\begingroup$ @TimothyChow In fact this column of the Wikipedia page seems to incorporate information from many different sources in a way that could be deceptive. Emmanuel Kowalski has a copy of the '94 proceedings and has verified that Bourgain's and Zelmanov's "citations" come from the first sentence of their laudatios, while Yoccoz's does not (and thinks Yoccoz's may not be a good summary). I think the Wikipedia page should be edited to explain the different sources - I might get started on this. $\endgroup$ – Will Sawin Mar 28 at 10:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.