78

The background of this question is the talk given by Kevin Buzzard.

I could not find the slides of that talk. The slides of another talk given by Kevin Buzzard along the same theme are available here.

One of the points in the talk is that, people accept some results but whose proofs are not publicly available. (He says this leads to wrong conclusions, but, I am not interested in wrong conclusions as of now. All I am interested is are results which are accepted as true but without a detailed proof, or with only a partial proof.)

What are results that are widely accepted to be true with no detailed proof, or only a partial proof?

I am looking for situations where $A$ has asserted in print that he/she has a proof of $X$, but hasn't published a proof of $X$, and then $B$ publishes a proof of $Y$, where the proof depends on the validity of $X$. For example as on pages 20,21,22 of the slides mentioned above.

Edit: Please give reference for the following:

  1. Where the result is announced?
  2. Where the result is used?

Edit (made after Per Alexandersson's answer): I am not looking for "readily available but not formally published". As mentioned by Timothy Chow, "there are many more examples if "readily available but not formally published" counts.".

ARA
  • 739
Praphulla Koushik
  • 5,921
  • 7
    This question is not intended to be a debate on whether some result is true or not :) I am only looking for results whose proofs are not published.. – Praphulla Koushik Apr 13 '20 at 06:09
  • 14
    I might object that a result without a proof cannot be known to be true as a matter of principle. What other method of verification do we have, other than a proof? Divine revelation? (A previous version of the question asked for results that were "believed" to be true, rather than "known" to be true, then I could generate a vast list from all theorems that assume the Riemann hypothesis.) – Carlo Beenakker Apr 13 '20 at 07:18
  • 1
    @CarloBeenakker Is the question in its present form looks ok for you? “I might object that a result without a proof cannot be known to be true as a matter of principle. What other method of verification do we have, other than a proof? Divine revelation?” Please have a look at the slides in the question for the relevance of this question... – Praphulla Koushik Apr 13 '20 at 07:25
  • 4
    Everybody knows that Goldbach's Conjecture is true, that the Twin Prime Conjecture is true, that there are infinitely many primes of the form $n^2+1$, and so on, and so forth, even though there are no (credible) proofs. Is that the kind of result you are asking for, Praphulla? Or do you mean results which do have a proof, it's just that no one has bothered to write a proof out? Probably no one has written out a proof that $9876543\times9638527=95195326372161$, but I trust that it's true because my calculator says so. Is that the kind of result you mean? – Gerry Myerson Apr 13 '20 at 07:41
  • I am looking for statements which people use to create new mathematics, assuming that statement is true, but there is no proof written.. For example as in page 20,21,22.. @GerryMyerson Does it makes it any clearer? – Praphulla Koushik Apr 13 '20 at 07:51
  • 3
    Maybe reword it as "accepted to be true" but "no publicly available proof"? – arsmath Apr 13 '20 at 08:16
  • 1
    Personally I do not understand why people do mathematics without understanding fully what they are doing. – Monroe Eskew Apr 13 '20 at 09:03
  • 3
    OK, so it looks like you are asking for situations where A has asserted in print that he/she has a proof of X, but hasn't published a proof of X, and then B publishes a proof of Y, where the proof depends on the validity of X. Is that it? – Gerry Myerson Apr 13 '20 at 09:06
  • 1
    @GerryMyerson thank you. Edited :) – Praphulla Koushik Apr 13 '20 at 10:03
  • 11
    I think the intent of the question would've been clearer if it asked (explicitly, especially in the title) for results which were announced to be true, but for which but whose proof has never appeared (yet). Anyway, clarifications considered, interesting question and +1 from me. – Wojowu Apr 13 '20 at 10:34
  • I was unclear on one thing myself. Do you want to include partial proofs as an example, or exclude? It now reads like you want to exclude them. – arsmath Apr 13 '20 at 11:08
  • I wanted to include partial proofs as example but that is fine.. :) The question looks more clearer now.. thank you arsmath – Praphulla Koushik Apr 13 '20 at 11:11
  • Happy to do it. I am very interested in the question. I edited it again to clarify the partial proof point. – arsmath Apr 13 '20 at 11:20
  • 3
    What's the relevance of the ag.algebraic-geometry, ct.category theory and dg.differential-geometry tags? Presumably examples can come from any area of mathematics. The big-list tag would certainly be appropriate, and the question should be community wiki. – HJRW Apr 13 '20 at 11:25
  • @arsmath thank you. – Praphulla Koushik Apr 13 '20 at 11:31
  • @HJRW Yes, examples can come from anywhere.. I am.in particular interested in examples coming from these tags.. Yes, big list tag is suitable... – Praphulla Koushik Apr 13 '20 at 11:33
  • I feel that questions about things "widely accepted to be true with no detailed proof" will end up being too much opinion-based. I am not sure that it is appropriate for MO. – Vladimir Dotsenko Apr 13 '20 at 11:56
  • 6
    @VladimirDotsenko If there is a result A published by some one which is used by others but there is no published proof of the result A, that will be an example for the question.. I do not see where opinion is coming here.. :) Please let me know if I am missing something.. I do not want anybody to spend time on something that is opinion based.. That is the whole point of this question.. :) Opinions are not appreciated.. evidence is appreciated.. – Praphulla Koushik Apr 13 '20 at 12:41
  • 10
    Several famous results of Hugh Woodin fit the bill. – Monroe Eskew Apr 13 '20 at 14:28
  • 4
    @MonroeEskew please feel free to make it as an answer.. I know nothing about them :) – Praphulla Koushik Apr 13 '20 at 14:43
  • 1
    @CarloBeenakker “I might object that a result without a proof cannot be known to be true as a matter of principle. What other method of verification do we have, other than a proof?” PA is consistent. It is known by intuition :) – user76284 Apr 13 '20 at 18:22
  • 4
    @MonroeEskew “Personally I do not understand why people do mathematics without understanding fully what they are doing.” Von Neumann would like to have a word with you. – user76284 Apr 13 '20 at 18:34
  • 1
    @Monroe Which ones do you have in mind? There is the fact that Turing determinacy implies Suslin-coSuslin determinacy (in the presence of DC?), which gives L(R)-determinacy. – Andrés E. Caicedo Apr 13 '20 at 19:47
  • @AndrésE.Caicedo Has the proof of that really not appeared anywhere (by Woodin or others)? I've seen at least "Turing determinacy implies L(R)-determinacy" quoted as well-established fact. – Noah Schweber Apr 13 '20 at 20:08
  • 1
    @Noah It may be in the process of being written up (by someone writing a book on AD+). Woodin has lectured on it at a seminar at Harvard, so there may be notes. There is no published account; I don't know the details. – Andrés E. Caicedo Apr 13 '20 at 20:14
  • I think all the (unsolved) Clay Mathematics Millennial Problems fit this question? The answers are widely believed to be known, and tons of papers rely on one of them, but most have not been proved. – BlueRaja Apr 13 '20 at 22:53
  • 3
    @Blue I think the distinction is between someone writing "if the Birch Swinnerton-Dyer conjecture is true, then....", and someone writing "since the Birch Swinnerton-Dyer conjecture is true ...." and citing a paper claiming, without proof, that it's true. – Gerry Myerson Apr 14 '20 at 00:20
  • 2
    I think it is widely accepted that Helfgott has proved the ternary Goldbach conjecture, although the proof has not yet been published. I don't know whether any published paper has relied on Helfgott's work ("relied", in the sense of stating a theorem and citing Helfgott's work as an essential part of the proof). – Gerry Myerson Apr 14 '20 at 00:38
  • Can only be a comment, but modern lattice coding techniques are based on intuitions whose proofs are at best obscure. For example what are upper and lower asymptotic bounds on the typical log-amount of points a (sequence of scale-normalized) construction-A lattice(s) puts in a ball with normalized volume $V$? Should be $\sim V$ but you never use this intuition directly. Instead you only talk about the amount of points in a shape of similar volume that covers a proportionally large part of the ball. Fine for many applications but inconvenient. – Christian Chapman Apr 14 '20 at 18:17
  • @enthdegree I have absolutely no knowledge of information theory so, I can not give any useful response... It would be nice if you can spend some time on your idea and turn it into an answer for this question.. – Praphulla Koushik Apr 14 '20 at 18:18

9 Answers9

37

I'm going to interpret this as a request for examples of results that were announced a while ago but whose proofs have not yet appeared. In other words, people don't doubt that the result is correct and that the author(s) can prove it, and there is an expectation that the current lack of a proof will not be a permanent state of affairs (i.e., a paper with the proof will be written and made public eventually).

One example of this is Rota's Conjecture on excluded minor characterizations of matroids representable over a given finite field. This was announced in 2014 by Geelen, Gerards, and Whittle, but apart from the sketch in that Notices article, no further details have yet appeared.

EDIT: An example of a paper that cites this unpublished work, and relies on it in an essential way, is The matroid secretary problem for minor-closed classes and random matroids by Tony Huynh and Peter Nelson. After stating Theorem 2, Huynh and Nelson write:

To be forthright with the reader, we stress that Theorem 2 relies on a structural hypothesis communicated to us by Geelen, Gerards, and Whittle, which has not yet appeared in print. This hypothesis is stated as Hypothesis 1. The proof of Hypothesis 1 will stretch to hundreds of pages, and will be a consequence of their decade-plus ‘matroid minors project’. This is a body of work generalising Robertson and Seymour’s graph minors structure theorem to matroids representable over a fixed finite field, leading to a solution of Rota’s Conjecture.

Another example is On the existence of asymptotically good linear codes in minor-closed classes by Peter Nelson and Stefan H. M. van Zwam, IEEE Trans. Info. Theory 61 (2015), 1153–1158. The results of Nelson and van Zwamm have in turn been used in an essential way to prove Theorem 1.4 of Girth conditions and Rota's basis conjecture by Benjamin Friedman and Sean McGuinness.

Zach Teitler
  • 6,197
Timothy Chow
  • 78,129
  • Thanks for your answer... Do we know by any chance what are the reasons for no news after 2014? – Praphulla Koushik Apr 13 '20 at 16:30
  • 2
    @PraphullaKoushik : I recently asked Geelen about this and he said that there's no specific reason, just that the proof is long and complicated and that the authors have had difficulty finding solid chunks of time to devote to writing it up. – Timothy Chow Apr 13 '20 at 16:40
  • Ok.. I am also looking for some papers which cite this announcement and use that the result is true...If I see anything I will leave as a comment here, you can add it if you think it fits in the answer.. – Praphulla Koushik Apr 13 '20 at 16:44
  • 3
    @PraphullaKoushik : I added an example. – Timothy Chow Apr 13 '20 at 17:06
  • 1
    Thank you, this is now a complete answer for the question :) – Praphulla Koushik Apr 13 '20 at 17:12
  • 6
    @PraphullaKoushik Something I only just learned is that Grace and Van Zwam found a counterexample to a statement that Geelen, Gerards, and Whittle had claimed was a theorem. This counterexample still leaves most of their other claims intact, but it does make me wish that GGW would write up their proofs ASAP. – Timothy Chow Jan 20 '22 at 00:07
22

Well, in some sense the Classification of Finite Simple Groups is in this state. It most certainly satisfies your second requirement: lots of papers have been published which rely on CFSG. However, a complete proof is (at least in some sense) still work in progress by Lyons, Solomon, Ashbacher, Smith and others.

Alon Amit
  • 6,414
  • 4
    This is the first example discussed in Buzzard's talk (though he doesn't go over where CFSG is applied) – Wojowu Apr 13 '20 at 21:07
  • Ah, didn't recall that. – Alon Amit Apr 13 '20 at 21:21
  • 1
    Isn't there a now complete proof after the publication of Aschbacher and Smith? The point of the book series is to have the proof in one place, and because it can be proven more efficiently once you already know the complete list of simple groups. – arsmath Apr 13 '20 at 22:48
  • 3
    A complete proof exists. Just not in one place. –  Apr 14 '20 at 01:28
  • 24
    Well, Ron Solomon says they’re still finding gaps. I don’t think anyone expects dramatic changes, but it’s hard to argue that the proof is as solid and final as the typical published paper. – Alon Amit Apr 14 '20 at 01:58
  • 1
    @MarkSapir would that contradict what Kevin Buzzard said in the slides? – Praphulla Koushik Apr 14 '20 at 03:36
  • 2
    An example of a (minor) gap that was filled after the Aschbacher-Smith books were published is a 2008 paper by Harada and Solomon. As the introduction to the paper explains, proofs of the theorems in question had been announced more than once before, but either had errors or were unpublished. – Timothy Chow Mar 25 '22 at 22:24
18

I think one example is given in this MO question of mine: a quartic in $\mathbb{P}^3$ with at worst Du Val singularities is a K3 surface (and similar statements for two types of complete intersections in higher-dimensional projective spaces).

Using the excellent answer and comments I was able to piece together a proof, but I could not locate one in the literature, whereas of course the result was "well-known to experts" (to such an extent that I even felt embarrassed for asking about the proof in the first place).

R.P.
  • 4,745
  • 23
    This kind of known but whose proof is not published and you hesitate to ask on MO are what I find useful to Grad students :) :) Please add an outline of the proof of this result when you write your next paper. :) +1 – Praphulla Koushik Apr 14 '20 at 02:48
16

The proof of the theorem of MacPherson that functors out of the exit path category are equivalent to constructible sheaves was not written down, just claimed. Others have since given much more general theorems, but whose reduction to MacPherson's result is not immediate.

David Roberts
  • 33,851
  • Is the theorem ofMacPherson used anywhere or only the general versions? – Praphulla Koushik Apr 15 '20 at 09:13
  • 3
    It's cited a lot, the massively general versions arrived decades after the original. As in, it's a key technical tool in intersection homology IIRC. – David Roberts Apr 15 '20 at 10:24
  • 3
    It is interesting how can some one cite a result more than once but there is no publication by the author where the result is mentioned... It must have been a very strong result :) :) – Praphulla Koushik Apr 15 '20 at 10:30
  • 2
    Well, there's a number of famous examples. The Grothendieck–Riemann–Roch theorem, due to Grothendieck, appeared in a paper by Borel and Serre after being communicated in a letter. Grothendieck's proof was published about 14 years later, in SGA6. https://en.wikipedia.org/wiki/Grothendieck%E2%80%93Riemann%E2%80%93Roch_theorem#History – David Roberts Apr 16 '20 at 00:21
  • 1
    Ok. So, Grothendieck mentioned a theorem (with out proof) in a letter to Serre, what is now considered as a GRR theorem. Then, Serre and Borel ran a seminar to understand this, and published (Is it their own proof?) as “Le théorème de Riemann–Roch”.. Then, after some years Grothendieck published his first proof in the “book” SGA 6... Is it correct? – Praphulla Koushik Apr 16 '20 at 04:06
  • 7
    Yes. It happened the other way around, too. For example: Grothendieck asked Serre a question, the latter proved it, then Grothendieck published the theorem and proof. This is not that unusual. It's the complete lack of proof of MacPherson's theorem by anyone that is weird, here. – David Roberts Apr 16 '20 at 05:19
  • 1
    "It's the complete lack of proof of MacPherson's theorem by anyone that is weird, here" Agreed... :) – Praphulla Koushik Apr 16 '20 at 05:21
  • Besides the 2-categorical version of Treumann's thesis, isn't this worked out in detail in the 1-categorical case in section 6 of this paper of Curry and Patel? – R. van Dobben de Bruyn Mar 17 '21 at 18:51
  • @R.vanDobbendeBruyn apart from being phrased in the dual setting, which is really a non-issue, I recall thinking that it's not clear that the definition of exit/entrance path category was quite the same. See the second dot-point on page 2. – David Roberts Mar 17 '21 at 22:34
16

I just realized that the OP links to a YouTube video and to some slides, but the two don't match—they're two different talks by Buzzard.

For completeness, let me therefore mention some results by James Arthur, which are mentioned in the linked slides but not the linked YouTube video. On page 13 of Abelian Surfaces over totally real fields are Potentially Modular by George Boxer, Frank Calegari, Toby Gee, and Vincent Pilloni, there is the following remark.

It should be noted that we use Arthur’s multiplicity formula for the discrete spectrum of GSp4, as announced in [Art04]. A proof of this (relying on Arthur’s work for symplectic and orthogonal groups in [Art13]) was given in [GT18], but this proof is only as unconditional as the results of [Art13] and [MW16a, MW16b]. In particular, it depends on cases of the twisted weighted fundamental lemma that were announced in [CL10], but whose proofs have not yet appeared, as well as on the references [A24], [A25], [A26] and [A27] in [Art13], which at the time of writing have not appeared publicly.

Arthur's (unavailable) references [A24] through [A27] are:

[A24] Endoscopy and singular invariant distributions, in preparation.

[A25] Duality, Endoscopy, and Hecke operators, in preparation.

[A26] A nontempered intertwining relation for $GL(N)$, in preparation.

[A27] Transfer factors and Whittaker models, in preparation.

Zach Teitler
  • 6,197
Timothy Chow
  • 78,129
12

In 1999, Robertson, Sanders, Seymour, and Thomas announced a proof of Tutte's "snark conjecture" (that every snark has a Petersen graph minor), but as far as I am aware the full proof still has not appeared: see this MO question. I don't know if this result has ever been applied anywhere, though. The proof was announced in "Recent Excluded Minor Theorems for Graphs" by Thomas (available as a preprint online here; with citation information at MR1725004): see Theorem 10.2 of that paper specifically. More information about the status of these results seems available on Thomas's webpage.

Sam Hopkins
  • 22,785
  • 3
    I just learned that Thomas died very recently (and tragically at age 57 following a long battle with ALS); see: https://mattbaker.blog/2020/04/19/colorings-and-embeddings-of-graphs/. – Sam Hopkins May 09 '20 at 02:57
10

A long time ago, M. Ajtai, J. Komlós, M. Simonovits, and E. Szemerédi announced a proof (for large $k$) of the Erdős–Sós conjecture that every graph with average degree more than $k-1$ contains all trees with $k$ edges as subgraphs, but the proof has not yet appeared as of this writing (2022).

What do I mean by "a long time ago"? Reinhard Diestel, in the notes to Chapter 7 of Graph Theory (5th edition), gives a date of 2009. But Václav Rozhoň, in A local approach to the Erdős-Sós conjecture, says that the result was announced in the early 1990's.

EDIT: I found another reference, Local and mean Ramsey numbers for trees, by B. Bollobás, A. Kostochka, and R. H. Schelp (J. Combin. Theory Ser. B 79 (2000), 100–103), which says, "It was announced recently that M. Ajtai, J. Komlós, and E. Szemerédi confirmed the Erdős–Sós conjecture for sufficiently large trees."
Timothy Chow
  • 78,129
  • 4
    Interesting to see from the answers that structural graph/matroid theory has many examples (perhaps because of how many technical cases must be addressed in the proofs?) – Sam Hopkins Mar 22 '22 at 16:54
5

The comments by Monroe Eskew and Andrés E. Caicedo concerning unpublished results of Hugh Woodin deserve to be made into an answer IMO. As a concrete example, Caicedo wrote:

There is the fact that Turing determinacy implies Suslin-coSuslin determinacy (in the presence of DC?), which gives L(R)-determinacy.

There are various other results by Woodin that may or may not fit the bill; in many (though maybe not all) cases, proofs have been provided by other authors. For more details, see Woodin's unpublished proof of the global failure of GCH and Unpublished works of Woodin on SCH and Radin forcing.

Timothy Chow
  • 78,129
2

The Schur positivity of LLT polynomials by I. Grojnowski and M. Haiman is widely accepted in the community of algebraic combinatorics, but their preprint has not been published.

It is still a major open problem to give a combinatorial formula for the coefficients in the Schur expansion, which is manifestly positive.

Per Alexandersson
  • 15,598
  • 2
    Your answer highlights the distinction between "publicly available" and "published." My interpretation was that the proof needed to be unavailable, not just unpublished, but reading the question carefully, I see that it is ambivalent. Probably there are many more examples if "readily available but not formally published" counts. – Timothy Chow Apr 15 '20 at 13:21
  • It is not clear how to respond to this answer... As mentioned by Timothy Chow, I am looking for "readily available but not formally published"... Thanks for your answer :) – Praphulla Koushik Apr 15 '20 at 13:31
  • @PraphullaKoushik : You are? Or is that a typo? In most of the other answers, the proof is not readily available. – Timothy Chow Apr 15 '20 at 16:22
  • 6
    @TimothyChow Sorry. It is a typo.. I am not looking for "readily available but not formally published". – Praphulla Koushik Apr 15 '20 at 16:32