text:
Is it fair to desk-reject a manuscript because it breaks relativity?
Every so often a journal I edit receives papers with some kind of claim that, if true, would break mainstream physics. For example, a paper could claim that waves in dark matter can travel at v ~ 1014 m/s.
Because relativity has met every experimental test we've thrown at it, and because relativity has a hard limit on velocities at the speed of light (3×108 m/s), any paper making this kind of claim is immediately suspect. One doesn't need to know what else is in the paper to be pretty confident it is wrong.
However, it is also known that relativity is an incomplete theory because it is incompatible with quantum mechanics. It's conceivable that whatever theory eventually supplants these two theories could allow velocities greater than the speed of light.
Question: Is it fair to desk-reject the above paper based solely on this one sentence? Like, there's a 99.99% chance (maybe even higher) that the paper is wrong, but there's still a nonzero chance it could be correct. If it is unfair, what else do I need before it becomes fair to desk-reject?

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Perhaps you should change your example to something with mass actually exceeding the speed of light to avoid off-topic discussion.

@Allure I'd consider that posting quotes from a paper you actually have had submitted to you breaks the confidentiality that is assumed when one submits a paper.

@BryanKrause /wrist. Fortunately the paper is already available as a preprint ...

@Allure: how did this end up being resolved?

I removed a ton more discussion about physics. If you want to talk about physics, please use the chat; we can only move comments to chat once, so future comments about physics will be deleted.

That procedure would remove the ability to overturn incorrect mainstream results. But the author should explicitly address the issue
Whilst I appreciate the Bayesian reasoning involved here (high a priori probability of invalidity), the suggested process has the drawback that it could never allow publication of a correct idea that overturns an established (but false) rule of physics.  As you say, it is contrary to "mainstream" physics, which is usually, but perhaps not necessarily, synonymous with correct physics.  Ideally, the peer review process ought to operate on the merits of ideas and arguments rather than dismissing them without review if they are contrary to the "mainstream".  This would allow challenges to the mainstream view to be considered on the merits, and would mean that it is possible for incorrect positions in the mainstream view to be overturned with peer reviewed literature.  Moreover, even when the mainstream position is correct, exposure to a plausible alternative view or critique could sharpen the mainstream position.
Obviously this has a time cost, but hopefully a reasonably modest one.  Desk rejection requires a desk review, so I think it would be wrong to reject on the basis of that one sentence, notwithstanding that it makes an assertion that is contrary to mainstream physics.  Ideally, the initial editor should undertake an initial review of the paper to see if it is understandable, and may make a positive contribution to the literature.  If it provides novel results, is understandable, and you can't see any critical flaw that would invalidate it (even if you think there must be one, because the result is implausible), then it is probably a candidate to proceed to referee review, or at least a more comprehensive deep-dive review by the initial editor.  For a deeper review, obviously one of the things you are going to look for is for the author to provide an explanation of the implications and counter-arguments to their position.  If the result would indeed "break" mainstream physics, then it would behove the author to give some explanation of how to resolve the breakage, or in what position that leaves the field (e.g., are the empirical results in other literature still correct, but for another reason, or are they incorrect for some reason).**  It might be useful if you have a few referees available who are happy to take on reviews of "implausible results" and have the patience to look deeply into the arguments, notwithstanding the implausibility of the conclusion.
One reason that you might legitimately desk reject a paper of this kind is if a claim made tangentially in the paper so fundamentally contradicts an established law of physics that that claim ought to be the topic of the paper.  For example (hat tip to Peter Schneider in comments), for an assertion that a thing travels faster than the speed of light, that would be a major discovery in itself (with large implications), and it ought to be the topic of the paper.  If it is mentioned only tangentially as an aspect of study of something less important then that is a serious deficiency of the paper, and may warrant rejection (with an instruction to the author to check if the result is correct, and if so, write the ground-breaking paper focused on that result).
One of the reasons I take this position is that one of the critiques one sometimes hears of academic peer review is that it is a closed shop that assesses ideas based on popularity rather than merit.  Adopting the procedure you are suggesting seems to me to go down the avenue of validating this type of critique --- a paper is immediately dismissed because it makes a claim that is not "mainstream".  (This criticism of academia is not entirely without merit, but I'd hope that the procedures of peer review would mean it is a marginal phenomenon at most; we certainly shouldn't be adopting practices that explicitly incorporate an initial rejection criterion for results that contradict mainstream understanding.)

Update (10 April 2023): In relation to this topic, you might be interested in a recent case where a published paper asserted a result that called standard physics models into question, only to be subsequently questioned itself as potentially erroneous.  Aaltonen et al (2022) analysed a 2011 dataset from a large hadron collider at CDF to infer the weight of the W-Boson.  The paper found it to be substantially heavier than expected under the standard model of this type of particle (see related report here), which would have implications for whether there are additional hidden objects that are not in the standard model.  However, CDF researchers examining the same dataset with an alternative statistical model find that the inferred weight is in accordance with standard model, and think that the unusual result may be due to inadequate statistical modelling (see report here).  This episode is somewhat similar to the issue you raise here, and it looks like it is a case of inadequate statistical modelling causing a published "world breaking" result.  Even assuming that the original published paper turns out to be wrong, there may be some residual value from having both published analyses of the data, since they show the effect of better specification of the statistical model for the data.

** Don't be too demanding here on resolving the breakage --- finding an error in mainstream understanding of the discipline doesn't mean you know all the implications just yet, but the author should at least address the issue, outline their understanding of the implications, and respond to obvious counter-arguments.

The last paragraph is very important. If there is no acknowledgment and discussion of the non-mainstream aspects I would be very worried.

I agree in general but wonder whether the specific example -- a paper on dark matter which results in v > c -- may warrant the desk rejection. The speed of light and the restricted flow of information following from it is not arbitrary or a coincidence but a fundamental constituent and condition of our reality. A paper claiming v > c would have the revolutionary breaking of relativity front and center, with a careful discussion. Dark matter would be a footnote. If, by contrast, the paper arrives at v > c as a side effect and does not make that the center, the main argument is wrong.

i took liberty to move part of the final conclusion (** part) on top in order to give OP some actionable advice

I would disagree with the premise that we should send implausible results out for review just because we are worried that it will make it appear like peer review is a closed shop. If you can find one of these major problems (e.g. special relativity is wrong, we can violate the second law of thermodynamics, there is no such thing as evolution, the earth is flat) and the author doesn't seriously consider this as an issue, peer review will probably be just 3 reviews saying the same thing you would. (1/3)

If you desk reject saying this appears to violate solid research and you didn’t consider that either they will be serious and add an analysis, or they are a crackpot and will say you’re a closed shop. The issue here is only when dealing with crackpots, and 3 people not bothering to pay it attention due to lack of skill/experience will probably make peer review appear to be a closed shop quicker than a desk reject would. (2/3)

It would be even worst if one of your reviewers is sloppy/also a bit of a crackpot/inexperience and uncertain about rejecting/believe all research should be published and the community will work it out. The crackpot will focus on the bad review that says yes or is inconclusive and say that they this reviewer is the only one who understood their work. But an editor can just have a file for how to deal with these cases, while the entire community will not do that. (3/3)

@user253751 I'm not saying that the last word has been spoken on this or any other matter. That would be foolish, and feels systemically wrong. But because c is so foundational, any violation of it would be ground-breaking and revolutionary and would undergo a great deal of scrutiny, formal and informal peer review, corroboration attempts  and discussion. It would not be presented as a detail in a different treatise.

Just to be clear, the question here isn't whether or not to desk reject, but whether it is okay to desk reject after reading just one sentence.  My answer to this is no, but of course, desk rejection after a proper desk review is perfectly reasonable.

@user253751 About v>c for DM should be made the central theme of this paper example mentioned and not rather a "footnote" I would agree if it was not established today that what is already unambiguous clear that DM is new physics. As such I find no reason why DM  must obey SR since all our attempts to detect it using SR have failed and see no any reason to make a connection of SR with DM. Also, v>c does not necessarily refer always to translational speed but could be in-place vibration speed.

@Markoul11: Indeed, I would find the notion that laws of physics as presently understood would perfectly match the behavior of everything, including things that are not understood, less plausible than the idea that there would be some deviations.  The likelihood of stumbling upon a deviation would be rather remote, but it would hardly be implausible that some parameter might be a physical constant for all forms of matter for which it has been measured, but different for some other form of matter which has not been measured, especially if such difference would make measurement difficult.

@supercat Could be even worst by DM being just a gravitational side-effect of Dark Energy DE.

@Markoul11: If DM isn't really matter, the fact that its behavior is inconsistent with the laws related to matter wouldn't in any way contradict such laws.  But there may very well be situations where the laws as presently understood don't hold.  If someone e.g. analyzing spectra from supernovae were to derive a theory which fit the data better than any existing theories, but would be inconsistent with relativity, I wouldn't find it astonishing if particles are affected by some effect whose magnitude is e.g. a really tiny number times the sum of the squares of the magnitudes of...

...all gravitational forces acting on an object, in a way which would be below measurement noise anywhere other than in the heart of a Supernova, but caused statistically significant deviations from what normal theories would predict in such contexts.  Supernovae involve densities, temperatures, and energy levels so far beyond anything that occurs anywhere else that it shouldn't be at all astonishing if rules of physics that are accurate everywhere else fail to accurately to describe them.

@supercat I agree. Failing to explain over the last 70 years unknown physics exclusively with known established physics and the limits they impose maybe is not the best way to go. We should create new physics matching our observations even if these are effective theories but accurately describe the results. If superluminocity accurately explains such results I see no reason to discard such theories and make it an open possibility. That is actually what Maxwell did. "Once you have eliminated the impossible, whatever remains, however improbable, must be the truth."

@Peter-ReinstateMonica I get what you are saying... But please remember that before the 1800s it was the scientific consensus that disease was caused by bad smells.
Theories should be rejected/accepted based on their validity. It is unscientific to reject something just because it disagrees with the scientific community.

@Peter-ReinstateMonica: I find it to be less disturbing. My first hypothesis was in fact they really have the dark temperature energy per unit mass and just used the value that got spat out of orbital dynamics or something, not realizing they needed a relativistic version of the theory. But even if not, dark matter exceeding c is less problematic than visible matter. There is no proof in GR that such speeds are unattainable but rather such speeds are rejected because the consequences are considered bad.

If they don't address the issue in the paper or claim this suggests an issue with their assumptions that should be addressed in future work then I'd say it’s fair. In addition, this is an extraordinary claim and so there should be some extraordinary evidence/reason to accept it, though experimental work might have a lower threshold than theory to avoid a desk reject.
Don't forget, to not give a desk reject to a paper that has clear issues with it is unfair to the reviewers since they will probably just come back with the same issue you had initially, but they have now wasted time reading the article.
It’s also possible they had a typo in the paper such as the number really being v^2 or got the exponent wrong where it should have been 10^4 rather than 10^14. If you are worried about the desk reject being unfair you could perhaps ask the authors for clarification or make the reason very clear during the reject so that if there was a mistake then it’s very easy for the authors to contest the desk reject in an appeal (depending on your journal's policies).

I think the LHC-faster than light-neutrino paper ( https://link.springer.com/article/10.1007/JHEP10(2012)093 ) is a good example on how to do it properly. They had a result that broke relativity and they focused the whole thing on ruling out mistakes in the measurements. (Or at least, all mistakes except for a loose cable)

The LHC-faster than light neutrino paper was actually the reason I said experimental papers should be given more leyway than theory papers. If reality violates our theories, no matter how nice/beautiful our theories are, then we need new theories. Though if an experiment says I timed a pulse of light reflecting off the moon with a stopwatch and got the speed of light is 6*10^8 m/s then the experiment is probably wrong.

@NAMcMahon (1) Look up distance to moon. (2) Double it to get round-trip distance. (3) Time a pulse reflecting off the moon. (4) Forget about step 2. (5) Halve the time to get one-way time. (6) Divide these numbers, yielding c = 6×10⁸.

My suggested approach can be summed up with Sagan's standard:

Extraordinary claims require extraordinary evidence.

Or rather, Laplace's original formulation:

The weight of evidence for an extraordinary claim must be proportioned to its strangeness.

Note that the maxim is not "Extraordinary claims should be desk rejected to save everyone time, because they're almost certainly wrong". They are almost certainly wrong, but there's a non-zero chance they might be true, which is why they need the proportionate amount of evidence to back them up.
If an author is submitting a paper which makes an extraordinary claim (e.g. that relativity is broken), it's incumbent on that author to provide the extraordinary evidence that it's not a whole lot of hogwash -- that is, to provide sufficient evidence that the claim is valid. It's your job as an editor to provide the first-pass assessment of the provided evidence, and see if the extraordinary-ness of the evidence is proportioned to the extraordinary-ness of the claim.
How much evidence is needed is somewhat topic and field specific. As mentioned, the needed evidence is proportional to the strangeness of the claim. The more fringe and unexpected the claim, the more effort the author needs to go through to prove that the claim is valid.
Generally speaking, a decent scientist who knows the claim is extraordinary (and who ascribes to Sagan's standard) will put in the effort to back up the claim. It will be rather obvious that they're doing their best to back things up. (They will make some effort in that regard.) On the other hand, if someone is not aware that the claim is extraordinary, or if they don't care about the standard (e.g. if they're a crank), the paper will be tissue-paper thin on the point. In these latter cases you can safely desk reject with a note they'll need to spend effort backing up their claim.

While I agree in principle, note that Relativity itself had little actual evidence when first presented. After 100 years we still find confirming evidence. It was largely a thought experiment (i.e. a theory) to Einstein. It comported with (didn't contradict) the then available evidence, which was slight at the time. "What if I can still see myself in a mirror if we are traveling at the speed of light? Yes! Yes I can."

Einstein, himself, spoke tentatively about proof of the theory.

@Buffy If you want to make an analogy here, it would be better to bring up the Michelson-Morley experiment, whose result is also something that at the time must have seemed almost certainly wrong. Relativity came later to resolve that issue. In that sense, the required evidence was already there, Einstein didn't need to provide it.

It's worth mentioning that at the time of Einstein there was a lot of opposition to GR and it wasn't accepted for a long time.  The Nobel Prize citation for Einstein does not mention GR, if I remember rightly.

I really do not understand this obsession to say that superluminal theories about DE especially have anything to do with "breaking Einstein's relativity"?  Since by definition it is unambiguously clear today that DE and possible also DM do not obey our known theories. If they did I think 200 yeas is more than plenty of time these to be explained using our known and established theories? For me this argument is irrelevant.  Why all energy in the cosmos should exclusively come in the form of matter and light and not also a third type yet to be discovered or theorized type of energy? Dogma?

Uhlenbeck and Goudsmit introduced the electron spin in a model that would require the electron to spin faster than the speed of light.  Yet they were right although their model of a classically spinning electron was wrong.
So: breaking relativity should not lead to an immediate reject.
The story of this “error” can be found in

Electron Spin and Its History, Eugene D. Commins,
Annual Review of Nuclear and Particle Science 2012 62:1, 133-157

and available here. The salient part (with relevant actors properly identified) is as follows:

Discouraged by Lorentz's criticism, Uhlenbeck and Goudsmit told Ehrenfest that they wished to withdraw their paper. However, Ehrenfest had already sent it to the publisher. He told Uhlenbeck and Goudsmit not to worry: They were young enough to be forgiven for their stupidity! The short paper was published in Naturwissenschaften in November 1925 (13), and an even shorter version, in English, appeared in Nature in February 1926 (14).

Why wasn't it rejected? Did they have extraordinary evidence?

I was unfamiliar with the story, but Duck Duck Went for a bit. https://www.lorentz.leidenuniv.nl/history/spin/goudsmit.html looks like an approachable introduction. (@ZeroTheHero maybe check the spelling of Goudsmit?)

@tripleee thanks for spotting the typo.

@PeterMortensen Uhlenbeck and Goudsmit sent their work for publication before this was realized.  Pauli or Eherenfest spotted the contradiction pretty quickly and said something to Uhlenbeck and Goudsmit to the effect that "You are young, you can make mistakes" (can't find the source but it's something close to this).

@PeterMortensen I think it was initially rejected.

@Tom it seems not as per my edit above.

@PeterMortensen see my edit above.

That is a completely different situation.  Relativity was not a firmly established theory in 1926.

@AnonymousPhysicist relativity had 20 years of history and Einstein had already received the Nobel (putatively for something different).  Nobody doubted the validity of special relativity, as witness by the concerns of Pauli and Lorentz on the results of Uhlenbeck and Goudsmit.

@ZeroTheHero I was already aware of that.  There is a very large difference between the experimental evidence available then and now.  I do not care what theorists believed or doubted.

One should be very clear about what it exactly means that a theory violates Special Relativity (SR), because there exist at least three classes of situations where "violations" of SR are harmless.

In relativistic quantum mechanics, the magnitude <x| \exp{-iHt} |x=0> of a free particle is positive definite everywhere. So the particle can be located, with some probability, outside its forward light cone. Does this violate SR mathematically? Yes, big time. Does it violate SR in actual physical experiments? No way. Because it is impossible to localise a particle to volumes less than the Compton wavelength λ. (Such an attempt would boost the energy uncertainty and generate a multi-particle state.) In short, a single-particle theory is inconsistent with the SR causality. Thus, if a violation of SR happens within a less-than-Compton volume, it may be ignored. For it will not enable you to send superluminal signals.

Similar is the situation with quantum entanglement. Does the entanglement violate SR mathematically? Sure. Does it violate SR in real life? Never. Because quantum entanglement does not enable superluminal signal transmission.

In cosmology, inflation permits for patches of the manifold to recede from one another faster than light. This, however, does not imply superluminal signal exchange.

To conclude, we should be tolerant to the violations of SR that are purely mathematical.
If I were an editor, I would ask the authors of SR-violating theories the same question: does the "violation" imply the possibility of superluminal exchange over longer-than-Compton scales? If no, then the theory can be considered for publication. If yes -- then the author must, at the very least, discuss the extent of the violation, and its consequences.

As mentioned in comments, phase velocities exceeding c are perfectly normal in relativistic wave theories. Group velocities can also exceed c when there is anomalous dispersion. Recessional velocities in cosmology can also exceed c. Without context, it's not clear whether the claim in the paper is related to any of those, but I hope that before rejecting the paper you'll consider the possibility that the claim isn't as extraordinary as you think.
Of course, the authors should make that clear. But if they mention that the velocity is a phase velocity, or mention anomalous dispersion, I wouldn't expect them also to say that it doesn't violate causality, since the intended audience would understand that. I think the paper wouldn't be improved by adding an explicit disclaimer in those cases.

@Allure Well, I'm not too surprised. I suppose my answer may still apply to these situations sometimes in principle.

I don't do physics, but if I, as editor, come across results that look implausible, my first impulse is to see how good the rest of the paper looks; in most (or even all, see below) such cases I easily find other issues and then I have a good basis to desk reject. One sentence alone shouldn't be enough if the thing looks well done otherwise. If I remember correctly, I never needed to reject a paper based on one implausible result alone; there was always more to be found.

No, this isn't a situation in which a desk reject is appropriate unless the editor can see a fatal flaw in the paper themself. It is actually, in that case, a situation in which you bring in the "big guns" to look at the arguments.
There is a misconception by many, possibly including some editors, that the "real world" somehow conforms to or obeys a set of theories. In fact the real world is what it is and our theories, including relativity, are just our (feeble) attempts to understand and explain that within a theoretical/computable/mathematical context.
The fact that "all available evidence" (if true) supports relativity doesn't mean that there isn't some better, more inclusive theory in which it sometimes fails. Don't suppose that "all available evidence" is the same as "all evidence". We haven't, certainly been able to examine all possible phenomena in such a complex world. It is hubris to assume otherwise.
In particular we have lots of limits on what is visible to us, both at micro and macroscopic scale. Likewise on long and short term time scales.
Of course, an editor would/should be skeptical of claims made in such papers and errors can be missed by the best researchers, but there is still a lot to be examined and understood about the real world. Not all possible phenomena are apparent to us. That is one reason that it took Einstein about ten years to come up with (even) special relativity and he did so in the face of opposition of the scientific establishment of the time, who were pretty generally committed to the theory of the aether. And his results weren't IIRC immediately endorsed and accepted by all.
Big world.
Quoting from Wikipedia

Galileo's championing of Copernican heliocentrism (Earth rotating daily and revolving around the Sun) was met with opposition from within the Catholic Church and from some astronomers. The matter was investigated by the Roman Inquisition in 1615, which concluded that heliocentrism was foolish, absurd, and heretical since it contradicted Holy Scripture.

Don't do that.

Yes, I do realize that the Inquisition of 1615 were, indeed, the big guns of the day. Oh well.

I think a key point is if the author doesn't comment on it and how it violates whats been seen in the past then the desk reject is warranted, if there is a discussion about how locally we preserve speed of light, but we have constructed an FTL drive, then this isn't a good reason to desk reject. Another example would be what if I said I had an effective algorithm for solving 3-SAT but didn't talk about P=NP then that suggests I don't understand the field and a desk reject may be appropriate (or a response saying that you'd like a section on how this relates to P=NP problem before review).

Just because there are occasional cases where going against current orthodoxy leads to substantial scientific advances does not mean that any paper that violates the current orthodoxy ought to be afforded "review by the big guns". 99.99% of those papers will still be wrong.

@WolfgangBangerth, then the special relativity paper would have been rejected immediately. It clearly went against "current orthodoxy" which was the theory of aether as proposed by Poincaré, the leading scientific authority of the day. If the editor doesn't see a flaw and the claims are extraordinary then people with extraordinary skills should get a look.

@Buffy by the time Einstein came around the theory of aether was having trouble due to the inability to detect it and the fact you needed to jump through a number of hoops to modify the theory so that non-detection of the aether was predicted. So a paper which proposed there was no aether wasn't going against the "current orthodoxy" since the "current orthodoxy" had already been beginning to break down.

@NAMcMahon, I think you are misremembering Poincaré and his circle. And, FWIW, relativity is also showing a few cracks.

I'm not taking a position on Einstein's paper, but would like to restate part of my comment with added emphasis: "...does not mean that any paper that violates the current orthodoxy..." Others have made the point elsewhere: Extraordinary claims require extraordinary evidence. Einstein's paper did. Those 99.99% of crank papers don't.

@WolfgangBangerth, see: https://daily.jstor.org/why-no-one-believed-einstein/

I am not sure there is a one-size-fits-all answer to your question. There are different journals with different policies, some journals have more appetite to speculative ideas than others. I doubt one should think about desk rejections in terms of fair/unfair. Desk rejections (or maybe any editorial decisions) always carry the risk of an error, but many (or maybe most) journals cannot do without desk rejections. There are a lot of journals nowadays, and an author should not be too thin-skinned and give up after a rejection.
Let me refer you to Gell-Mann's views in (a very short) https://www.youtube.com/watch?v=GSLY2ojYZHg , where he criticizes the policies of Phys. Rev. and Phys. Rev. Lett. Apparently, his letters to these journals were not always met enthusiastically :-), so he tended to publish elsewhere. Was Phys. Rev. unfair to Gell-Mann? I don't think so, they just had policies that he disliked (and I am not trying to say that Phys. Rev. was right and Gell-Mann was wrong, or vice versa).

Is it fair to desk-reject the above paper based solely on this one
sentence?

My claim: Under no circumstances, is it fair to reject heretic claims based on their contrarianism alone, without going through the reasoning. It may still have to be done due to other constraints of time and effort, but it is never fair.
May I request you to have a look at three case studies that I have compiled to beg reconsideration of the antithetical expression established knowledge. The point is that contrarian claims should never be discarded solely on the basis of their contrarianism alone. These historical examples emphasize that the presumption of established knowledge can be unfair to contrarian thinkers.
The following three counter-examples are a case in point as to why desk rejection mainly because of heresy may not be a good idea. It only intends to provide a proof by counterexamples; to complement other answers which are well written and cover all of the important points.
Evidence
Case 1: Pulsars
Thomas Gold (1920 - 2004) was a brilliant contrarian scientist and a professor emeritus in the Department of Astronomy at Cornell University.
As a consultant to NASA in 1955, Gold conjectured that "the Moon is covered by a layer of dust resulting from the ceaseless bombardment of its surface by Solar System debris" (Bondi 2004). He was opposed by many of his colleagues, but was eventually proven right when the crew of Apollo 11 returned with samples of lunar rock and regolith in 1969.
Astronomer Jocelyn Bell Burnell along with her doctoral adviser Antony Hewish (1924 - 2021) discovered a regularly pulsating source of radio waves in the sky with a pulse width of 0.3 seconds and a period of 1.337 seconds, later called as a pulsar (Hewish 1968). Thomas Gold hypothesized that pulsars are vigorously spinning neutron stars emitting radio waves from their magnetic poles (Gold 1969). This was considered so outrageous that he, a Cornell University astronomer and a NASA consultant, was not even permitted to present it at the first international conference on pulsars (Dermott 2004). This incident begs the following question: Is any idea ever outrageous enough to exceed the outrage of not allowing the idea to be talked about in the first place?
The experience of Thomas Gold is not uncommon. Contrarian thinkers challenging the consensus narrative have always faced unfair, sharp, and inconsiderate rejection from the mainstream thinker community as well as the public at large.
Case 2: Rogue Waves
For centuries, sailors had reported witnessing extremely large, sudden waves, over 30 meters high, that seemed to come out of nowhere. Such waves became known as rogue waves or monster waves. Scientists almost universally believed the rogue waves to be a myth for hundreds of years.
When the renowned and respected explorer Jules Dumont d’Urville (1790 - 1842) confirmed experiencing such waves in his expeditions, which was corroborated by his co-travelers, it was ridiculed even by the then French Prime Minister François Arago (1786 - 1853) who was also a well respected scientist and mathematician (Levine 2018).
In 1994, the Draupner E oil drilling platform was constructed in the North Sea, 160 kilometers off the coast of Norway. It was designed to withstand waves of 20 meters height, estimated at that time to occur once every 10,000 years (Levine 2018). On 1st January 1995, the platform was hit by a 26 meter high wave in a relatively calm sea with no storm in the vicinity, thus conclusively establishing the existence of rogue waves. It is now believed that rogue waves are much more common than previously thought, with many of the strange unexplained maritime incidents in history now being hypothesized as having been caused due to rogue waves (Groch 2022).
Case 3: Quasicrystals
Dan Shechtman (born 1941), a professor of materials science at the Israel Institute of Technology, was studying rapidly solidified alloys of aluminum with the transition metals, a major class of metals based on their chemical properties. He discovered quasicrystals in 1984 while on a sabbatical at the John Hopkins University.
It was believed at that time that the solid state of matter can show only two fundamentally distinct states of arrangements of the molecules: crystalline and amorphous. In the amorphous solids, like glass for example, the arrangement of the molecules is aperiodic and disordered. Aperiodic means that the pattern does not repeat itself over space; disordered means that there are no simple rules governing the pattern, meaning that the unseen portions of the pattern cannot be predicted over a significant range. In the crystalline solids, the arrangement of molecules is periodic and ordered, meaning that it is repetitive and predictable.
The quasicrystalline state of arrangement of molecules, discovered by Shechtman, is characterized by patterns that are ordered, but aperiodic. In other words, the arrangement of molecules is predictable, but not repetitive. It was an unexpected discovery, and Shechtman faced enormous backlash from the scientific community.
The paper describing his research findings was repeatedly rejected, getting published only after more than two years. After the publication, the head of his research group at the John Hopkins University called him a disgrace to the group and asked him to leave (Jha 2013). Linus Pauling (1901 – 1994), a strong contender for the title of the greatest chemist ever, one of the founders of quantum chemistry and molecular biology, strongly rejected Shechtman’s findings, saying, "There are no quasi-crystals, just quasi-scientists" (Jha 2013). Shechtman was eventually vindicated and even won the Nobel Prize for Chemistry in 2011.

As far as I can tell, your answer is being downvoted on account of barely addressing the question, which is about how to make a decision in such a case and not about detailed historical examples of non-mainstream hypotheses prevailing.

@Wrzlprmft Barely addressing the question? The question is not about the theory of relativity. It is about whether contrarian ideas need to be desk-rejected primarily because they are contrarian? How are counter-examples not a good way to argue that this is not a good idea?

@Wrzlprmft I understand that it is not a complete answer. It was not intended to be. Because others have covered the important points. But does it not add valuable examples to other answers?

It is about whether contrarian ideas need to be desk-rejected primarily because they are contrarian – No, it is not. If you read the question, it becomes pretty clear that the asker is aware that going against the scientific mainstream may be necessary for progress. Also have a look at the other answers and how much they focus on such examples – which is barely at all as this is simply not the point.

@Wrzlprmft Quoting from the question: "Is it fair to desk-reject the above paper based solely on this one sentence?" My counter-examples clearly underline that the presumption of established knowledge can be unfair to contrarian thinkers.

@Wrzlprmft Others are saying exactly this. In fact, the highest upvoted answer says, "That procedure would remove the ability to overturn incorrect mainstream results." I only provided supportive examples for this statement.

Exactly: Your answer is providing excessive evidence for what boils down to a single sentence (of many) in the highest-voted answer and is something that the asker is apparently aware of, as evidenced by statements like: “It's conceivable that whatever theory eventually supplants these two theories could allow velocities greater than the speed of light.”

@Wrzlprmft I understand your point. Would you agree that it is unfair to downvote based on a subjective opinion of excessiveness of evidence in support of the most important point in the other answers? While one counter-example may be enough for some, others may need multiple examples to feel convinced that it is not a good idea. As a matter of fact, the example of Dan is mentioned in multiple highly upvoted comments. I wanted to add the complete story of Dan, while also adding 2 more examples. How is that so bad?

@RiteshSingh Community members can pretty much vote however they want, as long as they vote on content rather than for or against individuals. If someone thinks a post spends too much time on a side track that's a valid reason to downvote.

@BryanKrause Thank you for your reply. I understand. But the answer is just an extension of the highest upvoted comment. And it is common in SE to post too long comments as an answer.

@RiteshSingh: This is common because many highly upvoted comments should actually have been answers in the first place. This does not apply here. If you want to post a comment that is too long for a comment, your first instinct should be to not post it at all.

@Wrzlprmft Thank you, I understand. Will take care in future.


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If I don't get a job but am told I'm the second-choice candidate, is it reasonable to think I am indeed the second choice candidate?
Inspired by Dave L Renfro's answer here and Interpreting the statement "We received an extremely strong pool of applications this year." in US grad school decision letters
If I apply for a postdoc (or tenure-track, etc.) position, fail to get the position, and am told I was the 2nd-choice candidate, can I reasonably assume that I am indeed the second choice, or is that something departments do to soften the blow of getting declined?

Most of us learned at an early age that lying is bad. Lying to a candidate in a formal hiring process that’s fraught with legal liability seems an even worse idea. So, basically: no, that’s not something any person with common sense would do.

@DanRomik that's one wants to think, but see this comment: https://academia.stackexchange.com/questions/188578/how-much-does-being-denied-tenure-damage-your-career#comment508571_188578

FYI, in my particular case I knew I was in the top 3 because I was among the 3 who were selected for the final round of (telephone) interviews. My experience, and from what I've heard over the years, is that ranking after the final stage for an academic position is almost never provided, but in my case the person I would have worked with (the post doc was for anyone in the department, not for a specific person) told me a few months later that it was basically a toss-up between me and another candidate (the 3rd candidate being clearly 3rd), and the other candidate wound up being selected.

I think it may occasionally happen (due to the personality of the chair and/or specifics of the situation, e.g., the applicant is already in the department) that someone is told they were the next choice when they may not have been.

@Allure I’m not saying it never happens. But lying to a candidate about how they ranked relative to other candidates would be a seriously stupid thing for a department chair or hiring committee to do, and isn’t something they would do if they were competent at their jobs.

Why the downvotes? This seems a legit on-topic question to me.

The ethics of lying aside, it would be very strange if the hiring committee was so heavily invested in flattering the candidates and protecting their egos that they would actually make up a specific false statement rather than just throwing a vague encouraging phrase in their direction. I'm guessing virtually no sane person cares that much about making candidates feel good about themselves.

-1 for being way too broad. ‘Do departments do X….’ - maybe some do, maybe some don’t style questions usually end up with people guessing answers and giving anecdotes which aren’t that useful.

@AdamPřenosil no sane person cares that much about making candidates feel good about themselves Then why do people care about making graduate students feel good about themselves?

@user438383 Ok, I will rephrase.

@Allure it’s reasonable to try to make graduate students, or job candidates, feel good (or less bad) about themselves, without lying to them. The lying is the important difference between your hypothetical scenario and the one discussed in the other question you linked to.

@Allure I really don't think that people are so invested in making other people's grad students feel better that they would make up a specific lie just for the purpose of making them feel better. It's not "other things being equal, make the other person feel better", it's "make up a specific lie which goes beyond mere politeness with the sole purpose of making someone feel better". People generally don't do that unless they have either a significant emotional investment in that person or some sort of psychological problem with saying something even mildly unpleasant.

@DanRomik are you saying that when graduate schools say "We received an extremely strong pool of applications this year", they really did receive an extremely strong pool of applications?

@Allure yes, but that’s almost always true for highly ranked graduate programs, so it’s true in a vacuous sort of way. That’s why they say “extremely strong” and not “stronger than in any of the last 20 years” or some other formulation that actually carries useful information… (See the answer I just posted for additional thoughts.)

@Allure I think you're comparing apples and oranges here. The scenario you're asking is not at all like throwing a vague encouraging phrase in the direction of a student or applicant. The scenario you are asking about is more like telling a specific student in your class in private that they had the second highest score in that class, when in fact they didn't. Ethics aside, that would just be a really, really weird thing to do, don't you think?

Competent people involved in faculty hiring in some official capacity (say as department chair or a member of the hiring committee) will never outright lie to a job candidate, even for well-intentioned reasons. If they want to hide certain awkward truths they will instead use vague or diplomatic language. It is then up to job candidates to interpret what the vague language means (there are many questions on this site giving examples of this). But the general rule is that such vague statements usually do not contain more information than their literal meaning - they are crafted precisely to have that quality. So yes, if you are told that you were the second-ranked candidate, I’d say there is a high chance that that factual statement is in fact true.
Two caveats to keep in mind are:

It’s still not a guarantee that the statement is correct. There are incompetent people who deviate from normally advisable ways of doing things, so it’s certainly possible that a chair will lie to a candidate, and probably happens every once in a while. (This would probably be correlated with other weird or alarming things the candidate might have noticed about the chair’s behavior.)

If the candidate heard that they were the second-ranked candidate not through a formal channel but through some informal gossip, say through a friend on the faculty who isn’t an administrator or on tbe hiring committee, the possibility for lying (especially a “white lie” told out of a misguided desire to protect someone’s feelings) seems more likely. Telling such a lie is still a stupid thing to do in my opinion, and still creates many possibilities for real harm as well as legal liability for the university, but professors do not always have common sense about such things, especially when they are not themselves administrators and are not sensitive to some of the legal issues related to hiring.

As other people have said, they should definitely not lie about something like this.
However, as far as I know it is not normal to tell someone they were second choice in the first place. So this is an unusual situation and I wouldn't be too confident that the usual norms apply.
The exception would be if there is a good reason for giving out this information. For example, I have heard of people being told "you are the second choice, and the first choice has not yet decided whether to accept". This seems like helpful information to give (if true), and there is no good reason to lie about it, so I would be inclined to take it at face value. The probability of subsequently being offered the position is perhaps not high, but I have heard of it happening.

For a tenure track candidate for a position at a large university that uses typical recruiting practices, if you haven't been invited for an interview, you're probably not the second choice, even if you were told you were.  This probably includes distance interviews these days.
Beyond that, it can be really tough to say.

Lying to a candidate is in no way helpful to them, so telling them "to soften the blow" is totally misguided.
You tell them they came second if they actually did, and if you had been willing to hire them if the first candidate hadn't been there, to keep a possibility open to hire them if (a) the first candidate rejects the offer, or (b) you suddenly get more budget and can hire two people, or (c) you want to pass their name on to someone else who is hiring.
And you tell them they were good enough to get the job, but one (or two) people were better, if that is actually the case.

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Is a lecturer job early in my career a bad thing?
I know some of you may say "it depends", but in the US is a lecturer position inferior to Asst Professor? I am in the job market and get lots of calls for adjunct or lecturer. In the interviews, it is emphasized that "you will be teaching large classes and your job will be just teaching".
My question is that if I get into a lecturer position at the start of my career - will that be a bad thing? I do hope to keep applying for tenure track jobs, so will my lecturer experience be considered?

Generally, yes, in the U.S. "lecturer" is a lower-tier, non-tenure-track job, while "assistant professor" is tenure-track. "Inferior" in that sense, yes, although it does happen that assistant professors fail to get tenure and must leave, while lecturers go on and on... admittedly, often term-by-term.
I think that once-upon-a-time any job other than the deluxe ones would have been a sign that you'd already not "made the cut", but nowadays things are so much tighter that it is widely acknowledged that not managing a "research post-doc" or similar is not a disgrace, but as much just bad luck.
It is true that the same factors that push one to a certain threshold so that bad luck becomes the dominant factor will most likely still be in play, so hedging one's bets should take that into account.
Your experience in doing serious teaching will be taken into account in the positive sense that not everyone is able to cope with this, but it is very much the bread-and-butter of academic mathematics, no matter what anyone says. Not glamorous or heroic, no, but...
So when you ask "will my lecturer experience be considered", well, yes, that's legitimate work, and not everyone manages it. But/and if you're applying for somewhat-upscale tenure-track jobs (=assistant prof), those lecturer jobs are approximately neutral. The question will be about what papers you've written/published... given that you apparently pass the minimum threshold of not-awful/intolerable teaching chops.
The most awkward part of such situations is that, often, doing sufficient adjunct/lecturer teaching to make a living leaves insufficient time/energy for scholarship/research/thinking, and this is not usually taken into account formally...

"The most awkward part of such situations is that, often, doing sufficient adjunct/lecturer teaching to make a living leaves insufficient time/energy for scholarship/research/thinking, and this is not usually taken into account formally..."    Your research productivity will be evaluated as if you had spent those years in a research post-doc rather than teaching a heavy load.  Thus someone with two years of experience as a lecturer or visiting assistant prof with no publications will look unproductive compared to a new PhD or a research post-doc.

Note that if you end up applying for a permanent tenure track position at a teaching oriented institution, then your lecturing experience may well be a positive experience.

As @BrianBorchers notes, your experience may help you toward teaching-oriented schools, but I've had PhD students of mine not taken seriously at "little teaching colleges" because ... supposedly... they were not "sufficiently research-oriented". Whah? This after my recommending that they demonstrate that they could reach undergrads... That is, some little colleges are violently delusional. So one should try to test the waters.

I'm not sure about the U.S., but if your aim is to be a professor that do research, if you accept a position where you are not required to do research (e.g. lecturer), it will certainly not help.  To get a professor position where you do research, you need to get some good publication record. If you are a lecturer and do not have enough time to publish and do research, then, yes, it could have a negative impact on your chances to get such position.
By being a lecturer, you would get teaching experience, which is worth somethng but for position where you need to lead a research team, your publication record is generally MUCH more important than teaching experience. In fact, for an assistant professor position, I have often seen universities hiring persons without prior teaching exprience (in Canada).
So in my opinion, it is  better to do a post-doc than to be a lecturer if you want to get a professor position where you lead a research team. Being a post-doc will let you continue building your publication record. If you are a lecturer, what might happen is that you may teach too many courses and not have time to publish papers, and thus your publication record may even look worse because you may not publish for a few years, which may leave a huge gap in your CV, and as a result you may never get an assistant professor position.

A variant on this scheme would be to teach a part-time load, as an "adjunct", while continuing to build up your publication list.

@aparente001 -- great advice.  I've had several friends/jr. colleagues go this route with great success.  It says that you can teach and are collegial while still allowing time for the research/publishing that will make or break a professorial position. (I'd add that if teaching is your main love, there are amazing lecturer positions that can be just as fulfilling as professorial positions).

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Is pursuing a second PhD in the same or related field a viable means of returning to academia after a decade in industry?
I have a PhD from a reasonably good program in pure math, but I had no support from my department or adviser and so went into industry afterward. I'm unhappy with the situation, and I'd love to return to academia. The situation is a bit tricky, though, and I was wondering if a second doctorate (whether again in math or in another related area like mathematical physics) would help at all. Further complicating the issue is the fact that I've been in industry for the last decade, so my CV has hardly gotten any better than it was when I applied, or even left, grad school the first time. (The situation may be different in other areas, but there's very little you can do in industry that looks appealing to an admissions committee in pure math.) So, would applying to PhD #2 be a waste of time, or is this a weird but possible means of clawing my way back into academia?

See Is doing two PHDs a good path?, Is a second PhD ever necessary?, When does one go for a double doctorate?

I did see those, in fact, but they were mostly in the context of whether it's useful to have doctorates in separate fields or whether they're worth pursuing toward the beginning of one's career. I'm interested in whether this is a possible way to return to academia. (Also, I didn't realize there was a 'second-degree' tag. Spiffy.)

Is your job anyhow related to Math? I don't know Math, but in Chemistry, Physics, etc there are plenty of people who can come back easily academia to tenure track. Have you tried to apply for jobs, postdoc or tenure track?

@Greg: For hiring (postdoc or tenure track) in pure math, they would look at your recent research publications.  If you have none, then you will need another plan.

Another PhD in the same discipline is quite rare in my experience, although I have seen a few cases where students first earned a PhD in some foreign country and then immigrated to the US and completed a second PhD.

@BrianBorchers: Right, I've run across that too in a small number of cases. That's not the case here, though; my PhD is from a decent enough university in the US, and I don't think the reverse tactic of emigrating to a foreign country and getting a second degree there would be conducive to an academic career (at least, one in the US).

In order to return to academia, you will need to satisfy certain requirements.

Most academic jobs require holding a Ph.D. in the relevant field. You already have one. A second Ph.D. won't help you here.
Teaching jobs require demonstrated teaching experience. A second Ph.D. won't help you here. You could certainly try to build a track record of successful teaching by getting in touch with a local community college and offering to teach a few courses - possibly even while holding your industry day job. If you are serious about returning to academia, this would be an investment, not a source of income, so you could offer to teach for little or no remuneration.
Research (and teaching) jobs require publications. A second Ph.D. won't help you here. Instead, get up to speed on current research, find a good open problem, work on it and publish some papers.
Finally, contacts will be invaluable. A second Ph.D. won't help you here. Instead, attend conferences, give presentations, get to know people, leave a good impression and hint that you would like to return to academia.

Note the common thread going through all these points: a second Ph.D. in the same field won't help you. Instead, invest your time in building the kind of portfolio (teaching, publications, contacts) that you do need.
This will likely be an uphill battle, but if you are motivated enough, then go for it. Good luck!

This is a more concise, less judgmental version of my answer: +1.

Seems like this is a waste of time, then. Oh well, thanks for your answer.

Unfortunately, it's very difficult to "get up to speed on current research, find a good open problem, work on it and publish some papers" without some kind of community.  Do you have any former collaborators who think highly of you?  You might be able to get a visitor office at their university for a year to try to get back into doing research.  Odds of success are still low, but higher than trying it on your own.

I think it will be very difficult -- perhaps prohibitively difficult -- to get a PhD in mathematics (or mathematical physics, which is all but mathematics) in the US given that you already have a PhD in mathematics from the US.  It came up within recent memory in the math PhD program at my university that we had an applicant (from outside the US) who already had a PhD.  We went so far as inquiring to the graduate school whether it would be against the rules to admit such a student.  It wasn't.  Nevertheless, suffice it to say that the student was not admitted to the program, and if this happened again I highly doubt the outcome would be different.
Maybe it will help to look at it from our perspective.  We are trying to train future academics.  (It is true that some PhDs leave academia, but for graduates of my department this is a minority: probably less than 1/3.  Moreover, no matter what they go on to do, it is only fair to say that we are training students for academic jobs.)  We want to spend our limited resources on people that want to say in academia and are capable of doing so.  Someone who wants to get another math PhD ten years later is getting the horns of that dilemma: if you are actually capable of continuing on in academic mathematical career, then one math PhD ought to be sufficient.  A US math PhD is not meant to be merely an apprenticeship in a certain specialized subfield of mathematics: it is meant to train you to do teach yourself new mathematics and do research independently.  By attempting to enroll in a second PhD program, you are signalling that you didn't acquire these skills the first time around....or maybe you did but chose not to exercise them for so long that they atrophied.  Either way makes you a poor candidate.
What can you do?  I would have to say that getting a PhD and spending a decade away from mathematical research and teaching is closer to a "lockout situation" than most others I can think of.  If I'm honest, I don't completely understand the story as you frame it: you write

I have a PhD from a reasonably good program in pure math, but I had no support from my department or adviser and so went into industry afterward.

You lost me: awarding someone a PhD is a highly non-negligible level of support.  If you have a PhD from a reasonably good program in pure math, then you can get some kind of academic job: maybe an adjunct position, but some kind of job.  You chose not to and went with that choice for ten years.  If you really loved academia, wouldn't you have done some academic activities in that time?  I'm afraid it sounds more like: you don't like either academia or industry.  This is not a contradiction: there are more things you can do than that!
But let me give you the benefit of the doubt that for whatever reason you spent ten years away from the thing that you really want to do.  What can you do?  Well, what I said before about having a PhD getting you some kind of academic job is still true: in fact, at certain institutions, "even" in pure math your industry experience will be a big positive.  A community college teacher may be a "pure mathematician" in her own private time, but those skills are very unlikely to be drawn upon in her work.  Someone with industry connections of any sort would be preferable.
So I think you claw your way back into academia by taking a job at a community or regional four-year college: quite possibly a temporary job.  If you play it right, you might be able to do this for a time while still keeping your industry job: I certainly recommend this.  After you build up a teaching portfolio, the ten-year gap in your CV will recede into the background, and -- assuming you do well, work hard, and so forth -- you should eventually be able to land a permanent teaching position somewhere.
If you want to break back into mathematical research: go ahead and do it, you don't need to leave your job for that.  Go to some conferences, read some papers, start contacting people in your intended field...just do it.  It will be slow going at first, but because you already have a PhD in mathematics you do have the capacity to learn and acquire mathematics.  So learn and acquire mathematics.  Have fun...

I would have loved to have had an academic position during that time, but I needed a job on short notice after finishing grad school (I only found out I was finishing a month or two before I did), and neither my adviser nor my department in general were helpful in trying to find any sort of position afterward. It wasn't any deliberate choice of mine; I would have loved to have any sort of academic work. What I'd really want would be to start over in a different, more useful math program. It sounds like that's not an option, unfortunately.

Also: Declaring that I could have had some sort of academic position if I had just wanted it enough if is unwarranted and unhelpful. While I'm not going to go into my own personal story here for reasons of privacy and space, telling me that what I really want is some alternative to both industry and academia is not a good answer. While I appreciate your taking the time to write up this answer, I can't say it was helpful.

@anomaly: I don't know you and your story, but you wrote up your question as though you had explained why you couldn't get an academic job.  But I didn't understand why based on what you said.  Did you actually apply for all kinds of academic jobs as soon as you knew you were graduating and at regular intervals thereafter?  If so, I read the situation wrong and you might consider including that information in your question.

Also, I did not say what you wanted.  What I said was that based on what you said, it sounds more like you don't like either industry or academia.  To be honest, even that alone ought to be helpful information: namely that your language gave some generic math professor that impression.   Also, there was more to my answer than that: a part where I took you at your word and gave you concrete advice for how to "claw your way back into academia".

Let me also say that what you are asking about is way too weighty for you to take too seriously what a few random academics who have seen only a small snapshot of your situation tell you.  Here's one last piece of advice that I guarantee you will be helpful if followed: find a mentor.  Some math professor who knows you and feels positively about you will do fine.  You would very much benefit from guidance which is both expert and personal.

Given the very short notice on graduating (in fact, I only found out secondhand), I put out only a few, almost desultory applications to academic posts at that point. I continued to do so thereafter, but nothing came of it. Even in the current situation, I suppose I could try to, say, teach calculus at a low-tier university (I've gotten this suggestion before, and some specific places were mentioned). Still, I'm not convinced that's really much of an improvement over a math-related industry job.

You did give more concrete advice, and I appreciate that--- thanks! As for the mentor, that's tricky to arrange. No one does pure math outside of academia, and professors have their own students to deal with; a former student or, worse, some random person from industry calling on them is pretty low-priority. (After all, this is a weird situation.) I've put out some feelers and sat down with a couple of professors to chat, but I haven't found a long-term mentor. I suppose there's no harm in keeping at it, though.

While considering a more recent question by the same OP, I came across an alternative solution that seems more on-topic for this question.
I checked my memory of the graduate student admission requirements for the school where I got my PhD, UC San Diego. Unfortunately, my memory is correct and there is no option for duplicating a PhD in the same subject.
However, it also mentioned "Nondegree Study", coursework only study for a year, with a possible extension for a second year. You could look whether any schools in your area with good math departments offer that option. If you could get into a suitable department, taking pure math graduate courses, for a year or two you could aim to do several things:

Refresh your pure math skills.
Demonstrate aptitude for pure math.
Reconnect to current research.
Find a mentor.
Find people to write letters of recommendation for a postdoc.

It is unusual but it is more likely to fit university regulations than the second PhD idea. You don't need any more degrees. You do need a pure math academic network.
You would probably have to self-finance, either keeping a day job or first saving enough money for a year or two of tuition and living expenses. You might be able to get paid work as an adjunct professor, teaching the material you know from your industry career.

The reason why I mentioned a second PhD is that my time in grad school was quite atypical, and I feel like I was cheated out of the experience that most other students (even in my department) get. The coursework itself is not inherently very useful, aside from how it might like on an application; rather, I want the connection to current research, the mentor, and the sources of recommendation letters you mentioned. Given my experiences, I think that I would have been able to continue in my academic career happily with a different department or even just advisor; that's what I want to try.

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Finding a suitable research topic from outside academia
I'd like to find a suitable research topic (specifically, in mathematics) to work on, where 'suitable' means that it's publishable and amenable to being completed in, say, 6-12 months. While there are plenty of topics I'm interested in and plenty of questions I'd like to answer, I'm not sure how to gauge the difficulty of a topic. I suppose this is normally part of learning to become a researcher in grad school; but while I have a (useless) PhD, my advisor handed me explicit problems to solve and summarily refused to allow me to change them when I'd asked. I'm also in industry now rather than academia (and not even in a real industry-research job), which means that I have little time--- certainly far less than a real researcher--- for useful work. Nevertheless, I'd still like to do whatever I can given my situation, and I can attend a few conferences, retain access to journals, etc.; I just don't have the time or resources that someone in academia would. So, how I can find a specific research topic?

A large part of research is simply discovering and clarifying problems to work on. Another part is simplifying them to the state they're still intellectually interesting, their solutions potentially add to the community, and ... are solvable. What's the 6-12 month timeline about? You have your whole life to explore your field. Also by "real researcher" do you mean "professional researcher"?

The 6-12 month timeline is an arbitrary scale that would hopefully filter out problems that are too easy or too hard. As for "real researcher," I simply mean one that who can contribute to the field in some nontrivial way, rather than forever bashing my head against problems that are completely untractable or restrict myself to recreational problems (which are fun, but it's the same difference between solving crossword puzzles and being a writer).

"I'd like to find a suitable research topic (specifically, in mathematics) to work on, where 'suitable' means that it's publishable and amenable to being completed in, say, 6-12 months."
Sorry to disappoint you, but that is not how mathematics works. If you are a professional, the possibility to produce a steady flow of results comes not from your ability to estimate the time needed for each particular project but from the general approach which is a combination of some pieces of folk wisdom like "cut the tree that matches your strength" and "don't put all your eggs into one basket". If you are an amateur, it comes from the sheer enthusiasm and the fresh point of view on things (also available to and important for a professional, BTW, but playing secondary role to a methodical approach there).
In both cases, the first question you asked is just meaningless (for that simple reason that no random person on the web can even approximately estimate your strength or tell which topics will carry you away and which won't, if you do not want to invoke higher level considerations). Either learn to live in the ever-shifting treacherous alternative reality based on the crazy concoction of high and low speeds, short euphoric moments and long depressive periods, persistent pressure and all-out spurts, rational calculations and pure luck, or consider another career path taking some "safe" job where the result is directly proportional to the product of the time and the effort spent.
As to your second question: "How to find a topic to research?", you don't find the topic; rather it finds you. Just go around and read papers you can understand and talk to mathematicians you know about what they are doing. Everybody is stuck with some "little thing" that does not require too much time to explain and if you can help someone to get unstuck in their project, you may easily get three things at once: a reputation, a joint paper, and a friend.

While I agree to some extent with fedja, I think it is possible to actively choose problems that you can finish in a relatively short time frame.  Here are general and specific suggestions.

Choose your problems carefully.  You can only make gauge how long a problem will roughly take you to make progress if (a) you understand what has already been done and (b) you have a fairly clear plan of how to make progress.  This means you may spend weeks or months going through potential ideas for problems before you come across one you like that you have a plan for solving.  Such plans could come from being something similar you've done before or following the strategy of a previous paper.
Ask other mathematicians.  This could be another professor you had or a maybe fellow grad student.  It's possible they have more problems than time to work on (I know I do), and some of them may be rather clear cut.
Read the (Amer. Math.) Monthly.  If you're not too focused on a certain area, the Monthly is a good source of ideas for problems which don't require too much background to understand.  I sometimes look at the there for ideas for problems for undergrads or high school students.   While a lot of articles in the Monthly aren't so "deep" (some are), many of them are very interesting, and can suggest similar projects (which can be publishable in normal journals).  Since you're not "in academia" anymore, I'm guessing you're wanting to do this for your own enjoyment, and you can do fun research that isn't necessarily deep.
Check out questions on MathOverflow and Math.SE.  (I'm just mentioning this since we're on SE now, but I think the Monthly is a better approach.) People ask interesting questions many of which are unanswered, and I've gotten ideas for little projects from these sites(none of which I've done, of course--but other people have).  (Unanswered doesn't mean the answer's not in the literature, but for a few questions it's pretty clear it's not.)  Of course, you will often need to do a serious literature search, because the right references may not be given to you as they are likely to with Monthly articles.

Thanks. I should mention, though, that my problem is not finding a problem I understand; it's finding one that's deep enough to be interesting but wouldn't involve wasting years I don't have looking for a solution. (There are lots of nasty open combinatorics problems, for example, that are easy to write down but quite difficult to solve.) In other circumstances, I'd prefer a broader research topic than looking for a specific, bite-size problem to solve; but, Fedja's condescending response notwithstanding, I'm stuck because of my situation with the latter.

@anomaly I understand, but I'm saying there are lots of Monthly articles where there are natural extensions of the work done there where at least a rough plan of attack is already there.  Then thinking about it for a few days should give you a sense of whether things are likely to work out.  If you start with a sufficiently interesting "bite-size" problem, it can easily lead into a whole research area for you.

While I am in a completely different field, what I would recommend you is participating local conferences, seminars in your area. Universities also regularly invite famous speakers, go learn, ask questions, talk to them.
This is a good way to see what kind of research topics other people do, and good way to find possible mentors or at least useful comments on possible research topics.
On the other parts I agree with fedja: there is no magic wand that tells you how long a project will be, especially if you are new at the field. These are not home-work assignments where people know beforehand the answer, the difficult and possible pitfalls. If it is a project that you can finish and publish in 6 month in spite you have no time to research, maybe it is not a particularly difficult project. But can lead you better ones on the long run, and lead to other projects.

I'm not new to the field, and I realize how research works. But I'm not in the situation now of being able to devote myself full-time to research, so I want to at least carve off a modest problem and make progress on it.

all_licenses: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
Is it possible for reviewers to ignore papers?
Sometimes, when I get the comments of the reviewers of a submitted paper (mine or some else paper), I feel like the reviewers just ignore the paper. More precisely, I feel that reviewers did not read the whole paper. Even more, I feel like they do not even complete reading the abstract (it is a conclusion from their comments of course).
Is it possible that reviewers ignore papers? Maybe because the authors are not known or maybe they did not write the paper well?
My only argument is that authors believe that their paper is a breakthrough paper and it has a very good contribution. Imagine reviewers start reading a paper in a high journal rank and they found (just examples, do not take them as serious ones): "This paper shows that 1+1=3" or "This paper demonstrates that if the determinant of a matrix is 0, the matrix still has an inverse."
Although, I do not think that in a high quality journals, reviewers can do some kind of ignorance. How can one be sure that their paper is well treated? How to write papers that make the reviewers do not throw the paper away.

Do reviewers sometimes do a bad job? Yes. Do reviewers sometimes focus on minutiae? Yes. Do authors sometimes think their papers are better than they actually are? Yes.

Related: Is it possible for reviewers to mistakenly reject a quality paper?

Also related-ish: Paper rejected. Should I appeal against biased reviews? and I believe I have solved a famous open problem. How do I convince people in the field that I am not a crank? (both examples where authors feel their paper was not reviewed properly)

Let me expand a bit on Thomas comment.
By its very nature, peer review is a guaranteed item of discussion in all academic discussions. Every author has stories to tell about unfair reviews, reviewers that missed the entire point of the article, reviewers that rejected a paper only to publish something similar themselves, and so on. What is much rarer talked about is that the vast, vast majority of rejected papers are simply not good enough! Indeed, this surely also includes a large percentage of those papers that the authors are sure were handled unfairly.
An important point is the following:

My only argument is that authors believe that their paper is a breakthrough paper and it has a very good contribution.

Clearly, all (ethical) authors believe their paper has merit. Otherwise, one would likely not submit it. It has to be said that the author's own opinion is not a good indicator of the actual qualities of a paper.

Is it possible that reviewers ignore papers? Maybe because the authors are not known or maybe they did not write the paper well?

Somewhere on this site, I have previously listed reasons why a paper can get rejected, but to be honest I think "the authors are not well-known enough" is very low on this list. The second reason ("the paper is not well-written") is more common, but it should be noted that rejecting a badly-written paper is not unfair. It is the responsibility of the authors to make sure that their paper is graspable. However, again, I suspect the majority of papers are well-understood by the reviewers, and just rejected because the reviewers were of the opinion that the contribution was not substantial enough.
So why do papers sometimes get very short, high-level reviews, even if the reviewers read and understood the papers? One possible explanation is that many reviewers follow the golden rule "spend your time on manuscripts that deserve your attention". In practice, a reviewer has finite time to invest into reviews. When receiving (say) 5 papers for review, two being borderline and three being clear-cut rejects, it is an understandable decision to spend more time scrutinising the papers that still have a chance, and only provide high-level feedback on the ones that the reviewer sees as clearly too weak.
Now on to your questions:

How can one be sure that their paper is well treated?

The philosophy I try to communicate to my students is to stop worrying about optimizing manuscripts for peer review and start doing the best research they can. I am convinced that the best way to ensure that your papers are treated well is to write good research papers.

How to write papers that make the reviewers do not throw the paper away.

Make sure that your papers (1) have a clear contribution, which is evident from the abstract and the introduction, (2) are written as clearly as you can, (3) are correct in terms of grammar and spelling (try to have a native-speaker spell-check your manuscripts), and (4) have a compelling and relevant contribution. If Your paper is technically sound but boring as watching grass grow, you are not entitled to complaining that the reviewers did not read the manuscript carefully.

The best way to ensure that your papers are treated well is to write good research papers. This. As a reviewer, I love reading (and usually recommending acceptance of) really good papers.

@ff524 Indeed. Unfortunately, it is a pleasure that I can enjoy way too rarely.

I doubt that it is common for peers to disregard the content of a manuscript under review to that extent. However, it is quite usual for authors to understand more on their paper than the reviewers do. The following simple things may help:

accurate title, that really absorbs the content of a paper
very well written abstract, that says precisely what is done in a paper
well written introduction
a clear cover letter, explaining what is done and why this is important and new;
recommendation of at least a couple of referees for your paper who should understand it.

I'd like to emphasize 5. In fact, if you can name many potential referees (say on the order of 10), it doesn't look like you are cherry picking and you are likely to get people directly from your list. If you only name 3, they are likely to take people who aren't on the list, for several reasons.

Your question is highlighting one of the major problems of the current peer-review  process.  Numerous studies have been conducted on various parts of peer review, and one of the most consistent findings across these studies is the absence of reliable reviews.  That is, reviewers often disagree with each other, and editors often disagree with reviewers.  At the same time, these findings are not at all surprising.
I do believe that there are very conscientious reviewers, and those numbers will vary considerably over time and by discipline.  I believe there are many persons who are simply unqualified to review for journals.  And, I believe that some reviews are not completely objective.  Some people may be particularly critical on a given day -- perhaps their grande mocha latte arrived with no-whip when in fact they wanted whip.  Or, they just had one of their own papers rejected, which makes them think that the system of reviews is becoming increasingly tighter, so the subsequent reviews are made under a more critical lens.
The bottom line is there are many places where there are faults in the peer review process, and this question certainly highlights one of them.  Proposals for improving the system regularly enter the debate, but the current peer review system is not likely to change much over time.
If the review is particularly egregious, you can certainly talk with the editor.  But, as a submitting author, you will always have less power and influence relative to the reviewer.

I am not convinced that the "state of peer review" is even close to as bad as people make it out to be.

@xLeitix Is this statement based on empirical findings of the peer review process, or your general appraisal?  Do you disagree with the empirical findings?

Anyway, I guess it depends largely on what you expect. If you expect a system with a number of false positives/negatives close to 0, peer review certainly does not deliver, but I am not sure if you will be able to find an alternative that works as you hope.

Numerous studies have been conducted on various parts of peer review - perhaps you could link to a couple of these and quote their specific findings. Otherwise those w/o database access have to just take your word for it.

@ff524  Everybody on this site has unrestricted access to Google Scholar.  Go to Scholar and search:  peer review process reliability

I am not sure if disagreement between reviews is necessarily a problem. If the paper is great or really bad, reviews usually agree but for anything in between disagreement is quite common. And, reviews are really educated opinions and as such not to be expected to agree, although some think reviews are definite non-negotiable truths to be followed.

Reviewer A:  "I loved the paper.  No revisions necessary.  Publish it." Reviewer B: "The paper has a serious error in the analysis.  More specifically ... And, therefore, this paper should not be published."


text:
How to reference a technical document (like an RFC) in APA style?
Like, this document, for instance:
https://www.rfc-editor.org/rfc/rfc791
It's an official Internet Engineering Taskforce document that defines things like network protocols, etc. Often times there's really no specific individual(s) credited as the author(s), and even the official title can seem ambiguous.
The information on the title page is:

INTERNET PROTOCOL
DARPA INTERNET PROGRAM
PROTOCOL SPECIFICATION
September 1981
prepared for
Defense Advanced Research Projects Agency
Information Processing Techniques
Office
1400 Wilson Boulevard
Arlington, Virginia 22209
by
Information Sciences Institute
University of Southern California
4676 Admiralty Way
Marina del Rey, California 90291

So what's the official title and who get's the credit; DARPA, IPTO, ISI, USC, or IETF? I was going to do something like below, but I thought I should check with you guys.

DARPA. (1981, September). Internet Protocol Specification. RFC 791. Retrieved from https://www.rfc-editor.org/rfc/rfc791

How do people in your field do it? I doubt that the Internet Protocol Specification has never been cited before

You should indeed check what people in your field do, but generally the late RFC editor Jon Postel gets the credit, as he edited this report. The RFC organization has prepared a list of references in their reference style. I'm not sure how this would be written in the APA style, so I'm leaving this as a comment.

The APA Style blog walks you through citations with missing info. There is also how to cite a book with no author or editor. You seem to be using DARPA as the author, but from the title page, it seems mote likely that ISI/USC is the institutional author. If you dig deeper, the linked document lists Postel as the editor. In that case you just do a book without an author, but with an editor: https://owl.purdue.edu/owl/research_and_citation/apa_style/apa_formatting_and_style_guide/reference_list_books.html

Edited Book, No Author
Duncan, G. J., & Brooks-Gunn, J. (Eds.). (1997). Consequences of growing up poor. New York, NY: Russell Sage Foundation.

First person to post a complete, appropriate citation for the document in question (RFC 791) wins the cigar.

These other answers are fine, but all pale in comparison to the real thing.

There is a post about how to cite RFCs using BibTeX which says:
@techreport{rfc4180,
  author = {Y. Shafranovich},
  :
  :
  url = {http://www.rfc-editor.org/rfc/rfc4180.txt}
}

Taking the first entry in the link posted by @Anyon:
RFC0001 |           | Crocker, S., "Host Software", RFC 1, DOI 10.17487/RFC0001, April 1969, <https://www.rfc-editor.org/info/rfc1>.
Then a BibTeX entry would look like:
@techreport{RFC0001,
  author = {S. Crocker},
  title = {Host Software},
  howpublished = {Internet Requests for Comments},
  type = {RFC},
  number = 1,
  year = {1969},
  month = {April},
  issn = {2070-1721},
  publisher = {RFC Editor},
  institution = {RFC Editor},
  doi = {10.17487/RFC0001},
  url = {http://www.rfc-editor.org/rfc/rfc1.txt}
}

I have inserted the DOI entry based on https://tex.stackexchange.com/a/3803/142724.
Then, in APA style, the reference would look something like:
Crocker, S. (1969). Host Software. RFC 1. doi:10.17487/RFC0001
The Institute for Electrical and Electronics Engineers (IEEE) and the Association for Computing Machinery (ACM) publish a number of journals and conference proceedings in which RFCs are cited, so finding an example there may also help.


text:
Is it financially worth it to teach a MOOC (e.g. Coursera)?
I am not sure what the requirements are to teach a MOOC. I guess it is a little different, depending on which platform one is going to teach. As far as I undestand, Coursera offers only 15% of the revenue to the teaching team. Somewhere else it might be more, but the number of participants is smaller. Does someone have experience whether it is financially worth to teach a MOOC?
For this question please leave aside other benefits like getting known or the satisfaction of teaching. I see those as benefits, but that is not the question for now.
Also I am only counting money that flows from the MOOC platform to the team/organisation/company, not compensation from a university to the lecturer for doing the course (like discussed in How are instructors compensated for teaching a Massive Open Online Course (MOOC)?).

Coursera works on a different model from Udemy. Basically, Udemy pays its instructors while Coursera does not. Coursera may split profits (or revenue) with the university, but as far as I can tell it does not pay the teaching team directly. Based on your constraints this means teaching for Coursera is not worth it. It is possible to make money teaching for Udemy. For example, in 2012 the top ten earners made 1.65 million [1] and a quarter made at least $10,000 [2].

Is Udemy the financially most profitable MOOC platform, do you know?


text:
University subscribed its entire community for a mailing list which is impossible to unsubscribe from
I realize that this is a very atypical question for this forum. To be frank, I'm not sure if this is the right place to ask. But it is part of my academic life, so I'm going to give it a try.
In the beginning of Fall 2020, our university (in the U.S.) created a mailing list and automatically subscribed the entire university community to it. It comes from a certain non-academic unit within the university, newly created to address certain social issues both within the university and in the wider community. Now, there's nothing abnormal about that. Of course, a university is a social organization, and there are multiple non-academic facets in its life. Our university has many such mailing lists on various academic and non-academic topics, which can be crucially important for everyone (payroll) or optional for interested persons only (university sports).
But one thing makes this new mailing list different: it seems to be impossible to unsubscribe from. I believe the university has a uniform mailing list platform for many of its newsletters, including this one. There's an unsubscribe link at the bottom of these emails. Clicking the link takes you to a subscription profile page. There's an option to unsubscribe from the publication at the bottom of the profile page. I checked the unsubscribe box months ago... and it had no effect whatsoever. The messages keep coming despite my unsubscribed status.
This was never an issue with any other mailing lists that used the same platform and that I was able to unsubscribe from (say, I'm not interested in sports, so I unsubscribed from that one). So my question is:

Are there any legal and/or ethical regulations and/or guidelines for non-commercial mailing lists within an organization, in particular within an academic organization?

I tried searching online. Clearly, CAN-SPAM doesn't apply here, because this is a non-commercial setting.
EDIT: In response to a comment below. Yes, I thought that it might be a bug. I emailed the university tech support about a month ago. They said that they don't know of an immediate source of the issue, and promised to escalate my request to a higher-level email team... and now it's been a month, and I haven't heard back from then again.

If there is an unsubscribe button present that does not work correctly, are you sure it is not simply a bug in their system? Perhaps you could check with the IT department / whoever manages it?

@GoodDeeds: Yes, I thought about that too. Let me add an edit to my post to address your comment. Thanks!

Most mail programs have a filter function. If you cannot unsubscribe, you can direct the mails to an archive or to autotrash (but be careful with the latter, perhaps better into the spam, so you can still review it for 30 days).

In response to your edit: it may be perhaps useful to follow up on your complaint first. The absence of a response and resolution does not necessarily imply that the behavior is intentional, which is the premise of this question.

Perhaps this new unit was implemented in response to an imposed legal requirement (settlement). I'd just go with sending it to a spam folder...

It's normal to not be able to unsubscribe from certain categories of emails from your school/employer. Though if it comes with a button to unsubscribe that doesn't function that sounds more like a bug to me.

This is more of a rant than a question. Do you want to sue the university because they want you to get emails that they think are important enough to deny you the ability to avoid them? I think that any set of regulations would (and should) have exceptions for such things. The people who most need to see some things are the ones who are most likely to avoid them.

For example "A man with a gun is prowling the campus" is probably a message that should go to everyone.

This seems like an IT, rather than an academic question

To address a few more of the comments. (1) Yes, it is a bit of a rant, and I realize that. But I still believe that it's not right to force something entirely optional on the university community. We're educated adults and can make an educated judgement on what we need to see. Unless it's required, but ... (2) But no, it's not anything legally required. It's not related to safety either. It's on a social subject -- an important and a worthy subject for ongoing discussions, for sure; but I can't see it as something that would warrant mandatory participation.

@Buffy: That's exactly the problem that I'm having with this: "They think [these messages] are important enough to deny you the ability to avoid them?" Who are they to DENY me of any abilities, unless required by law? And this newsletter is not required by law.

Simple, hit reply all and say "unsubscribe me please"

@Buffy Except for the distance learning students who may be many hundreds of miles/km away from the physical campus.

@shoover, that was an example. There are other things related to safety and the laws. Another example would be the rules you need to follow if you aren't regularly on campus but want to visit in the middle of a pandemic. I get those even years after retirement and even though I'm unlikely to visit. But they have given me my email address so I need to accept a few things that are irrelevant to me. Note that in the US, the president can send a message to every cell phone in the country. No exceptions. No opt-out. Amber Alerts are similar if more regional.

@zipirovich Does your email client have a feature that allows you to create folders and apply rules based on sender?

It looks like the common consensus is for me to create a filtering rule in my email to stop seeing these messages, since I can't stop them from coming. This doesn't answer my question that I asked, but it is a very practical solution. I guess I'll just go with that, for practical purposes. Thanks for good advice! But I'm still curious if there's an answer to my question...

You seem rather convinced that this is entirely optional.  Perhaps you are wrong about that.

@Libor: This is not about an employer/employee relationship. The university has subscribed all students too. Again, the same logic kinda applies to students too: the university has given your an email account, so the university can send whatever they want. But my point is that it's not about employment.

@shoover One university continued to send me emails about people prowling with guns after I moved to another continent.

Setting the leadership issues aside, there is an old saying in the IT: "If you are going to send mails to me with a cron (i.e., automatically), I am going to read them with a procmail (i.e., with an automatic spam filter)." Check your system configurations on how to do this. Whether you should filter those mails automatically, is a different question though.

Are there any legal and/or ethical regulations and/or guidelines for non-commercial mailing lists within an organization, in particular within an academic organization?

I’ll address the ethical angle. University administrators make many decisions every day that affect the lives of the thousands (for a typical large institution) of people belonging to the university community. One of those decisions involves deciding which information to send out as emails that go into everyone’s inboxes and that cannot be opted out of. And with any such decision that affects such a large number of people, it is impossible to make a perfect decision that pleases everyone and also helps achieve whatever other visionary, big-picture ideas the administrator in question sees it as their role to advance. There is always a trade-off involving the knowledge that an email will represent yet another small annoyance and nominal waste of time in some people’s workday.
The point is that making such a decision to prevent people from opting out of certain emails is generally ethical if it is done in good faith by an administrator who believes the email is doing enough good to justify the small negative cost it incurs. They don’t have to be objectively correct in that assessment in order for the decision to be ethical, and typically for any specific email some people will disagree and will think the topic of the email is not relevant or interesting to them. But unless the administrator is deliberately, knowingly sending out emails that they know are useless spam, there is no ethical problem here.
P.S. We all receive these sorts of emails regularly, so I feel your pain. If it’s any consolation, I think the situation is much worse in the corporate world...

It's normal for universities to require all staff and all students to be on a mailing list.  This is how university leadership attempts to communicate what it is doing to everyone.
It would be odd if alumni cannot unsubscribe.
I think you will find the university is actually sending emails to email accounts it owns.  There will be no regulations restricting that.

Although several comments alluded to this point, and actual laws (so not necessarily about ethics per se) surely vary from region to region, I have the impression (especially in my experience in the U.S.) that if the email account is "owned" by your employer, created fundamentally for your use as an employee, even if there are some privacy assurances, ... some emails will not be block-able.
I've had that experience in my univ, where a considerable amount of PR/spam/propaganda/informational [sic] email is sent from univ offices, and cannot be blocked by any of several devices.
Annoying, but... hard to be tooooo resentful if the univ is paying for the account, and wants to be certain that they can send certain messages. Probably they are not interested in my opinion of the value or interest of the messages. :)
So, no, not unethical. I'd wager legal everywhere. Annoying, yes. Like many things one's employer may impose, although hopefully one has benefits worth the cost.