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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 Poincaré 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.

Later: I realise now that this is a bit of a misleading question, as 'proofs' can in reality come in many different flavours and/or levels of rigour ie. all the i's dotted and t's crossed, or some things left unsaid or which still remain to be proved.

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    $\begingroup$ 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, 2019 at 7:19
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    $\begingroup$ 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… $\endgroup$ Mar 23, 2019 at 14:37
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    $\begingroup$ @AlecRhea I think questions about the validity of the proof of conjecture abc are referring to Mochizuki, not xyz... $\endgroup$
    – user1728
    Mar 23, 2019 at 15:48

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The positive mass theorem is more or less to do with the geometry of a type of positive scalar curvature condition.

Witten's work considers harmonic spinors, which are solutions to a certain linear elliptic system of partial differential equations. In his paper he presents a calculation which proves a rigidity theorem for harmonic spinors under a type of positive scalar curvature, directly comparable to Bochner's famous rigidity theorem for harmonic 1-forms in positive Ricci curvature. It is a little more complicated only since spinors are more complicated than differential forms.

Given the existence of a harmonic spinor with certain asymptotics, an integrated version of Witten's Bochner-type formula proves that the relevant positive scalar curvature condition implies the nonnegativity of the mass, now being expressed as the integral of the sum of squares of expressions built out of the harmonic spinor.

Witten's proof of the existence of such harmonic spinors is openly incomplete; he says "We have shown that the Dirac operator has no zero eigenvalue. Using this fact, we presume that standard methods can be used to yield" a key analytical step. The problem is existence of a Green's function for the relevant elliptic operator over noncompact spaces. Parker and Taubes gave a complete proof. I think it is not completely accurate to say that their proof only consists of "standard methods," since care is needed about weighted Sobolev spaces which can be somewhat delicate.

So I think it is inaccurate/misleading to put Witten's proof of positive energy theorem with some of his other works in terms of "inspiration and insight" for mathematics, or to just cite "origin in supergravity" (as Atiyah's laudatio does). His work here is a pretty direct and rigorous mathematical argument, in the vein of standard differential geometry. The gap is only due to his not being an expert in PDE methods. Even so he makes plausible that the relevant harmonic spinors exist. I think that a PDE expert reading his paper would likely even find it informally convincing.

As far as the Fields medal goes, I think some of his other work must have been more relevant. It's not hard to imagine someone else having discovered the main parts of Witten's proof, having to do with differential identities for spinors, with much less fanfare. The calculation relating the mass to the spinor is maybe the most striking part but I think many people (whether reasonably or not) would not regard it as a high point of mathematics (or whatever Fields medal is supposed to be about).

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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.

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    $\begingroup$ 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. $\endgroup$ Mar 23, 2019 at 14:55
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    $\begingroup$ 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. $\endgroup$ Mar 23, 2019 at 16:38
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    $\begingroup$ @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, 2019 at 22:29
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    $\begingroup$ @WillSawin : Thanks! That Wikipedia page's entry for Witten is strange. Witten's "citation" is placed in quotation marks, with a footnote directing the reader to Atiyah's laudatio, but Atiyah's laudatio does not contain the phrase in quotation marks. $\endgroup$ Mar 28, 2019 at 2:13
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    $\begingroup$ @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, 2019 at 10:21

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