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I have noticed from time to time that some PhD students at good, sometimes top, universities (who must have shown promise at some point) start to act as if they're gone a bit crankish. Say, they speak cryptically about how they are working on [insert famous research problem here, like quantum gravity or the Riemann Hypothesis].

As an advisor, how would you deal with such a student? Clearly there is a risk for the advisor's reputation if this gets out of hand (e.g. the student suddenly posts a patently wrong preprint purporting to solve that famous problem). But how long to wait, and how much efforts to do, before thinking seriously about terminating their contract?

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    Can you clarify the extent of the evidence that the student has “gone crank”? Just “speaking cryptically” about working on the Riemann Hypothesis etc does not convince me that the student is incapable of doing genuine, serious work (and if they are successfully doing that then I don’t care what they’re thinking about the rest of the time). So I feel like answering your question would require understanding better the severity of the problem and how it would affect my ability to work with the student.
    – Dan Romik
    Sep 6, 2019 at 7:26
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    Related: academia.stackexchange.com/q/56220/40589
    – Dan Romik
    Sep 6, 2019 at 7:30
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    @Dan Romik : I implied to mean that the student has not published anything else yet, and asks questions which show a lack of the type of knowlege one would expect from someone serious (on technically easy steps, or is unaware of key references). I hope this clarifies the type of situation I was thinking about.
    – Archie
    Sep 7, 2019 at 17:20

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In my opinion, every smart theoretician should take a crack at the major open problems in their field, at least for fun. Often the key to "solving" such problems is to not actually bang your head through the same wall everyone else has been stuck by, but to change the perspective or cheat the problem somehow. Even if you aren't smarter than everyone else, perhaps you can think of an outside-the-box idea others haven't. And as John Conway put it, we aren't as smart as we think we are. If I remember correctly, he said this when interviewed about Fermat's last theorem, which was unsolved at the time.

Of course if someone actually is "smart" enough to continue in research, they will be able to tell if they have failed. And they will be able to make a rational assessment of risk versus reward in pursuing such a path where the likelihood of funding and career success would be low. Hence pursuing such problems isn't all that different from pursuing other very-high-risk ideas. Just a little farther out on the spectrum.

One situation where this risk-reward assessment breaks down, of course, is mental illness. A manic person might think they are superhuman, for example. And there are some professors that I really wonder about. But mental illness is beyond the scope of sites like this, and I'd note that people with diagnosed illnesses can still function normally (or better) and be very successful in research.

The short answer is if you can't prove (rigorousy) they will fail, you can't prove they are fools for trying.

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I assume that you mean that they turn into a "cranK" prior to graduation. There are some famous cases of this happening later, though the seeds may have been there before graduation. One famous case is The Unabomber who holds a doctorate in mathematics from University of Michigan. Had his "career" as a crank been started earlier, the proper response would be to contact the police.

But in the more common situation, people can just drift away, not needing action from the advisor. Large universities usually have some sort of counseling available and people drifting into abnormal behavior can often be referred to them, though privacy laws might prevent that from happening. It is available if the person her/himself notices it.

And, don't forget that some "crankish" graduate student might just succeed in solving some long standing important problem someday, though the likelihood in individual cases is small.

Also being crankish takes many forms. Some are benign. Everyone is different. Your crank may be just my brilliant, but socially awkward, sister.

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    I'm not sure I understand your Ted Kaczynski example. As far as I'm aware, he never became a crank in the (mathematical) sense the OP is referring to. I've never read any of his mathematical work, but it all seems quite respectable. (His dissertation was impressive enough to win the Sumner Myers award at UM for instance). My impression is that the OP is using the word crank, not to speak about people who've "lost it" in any general sense, but rather to refer to say, the type of people that post 2 page 'proofs' of famous conjectures onto the general mathematics category of the arXiv.
    – user109454
    Sep 5, 2019 at 20:58
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    @BenLinowitz, his math was fine. But "being crankish takes many forms". And in the sense that you and perhaps the OP intend, people just sort of disappear and don't finish, despite some advice and/or counseling. But being disjoint from reality can be a symptom of many things.
    – Buffy
    Sep 5, 2019 at 21:19
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    @BenLinowitz 's comment is implicitly pointing to an important distinction here. We have procedures if students have mental health issues or if they have become a danger to self or others (discussion of how effective those policies are or whether they strike the right balance is secondary). Often cranks don't have any standard mental health problem, except maybe some topic-specific narcissism which isn't something in the DSM. How one should respond to a grad student trying to prove the Riemann Hypothesis seems substantially different than responding to one with mental health issues.
    – JoshuaZ
    Sep 5, 2019 at 23:53
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    @JoshuaZ "a grad student trying to prove the Riemann Hypothesis" doesn't seem too concerning as long as it doesn't become obsessive. A crank usually is convinced they have proven it or that some well established and accepted proof is wrong. Also, cranks typically focus on something outside their advanced education. (Several years ago we had to deal with a crank convinced that photosynthesis doesn't actually use CO2 as a substrate. They were not a biologist or chemist.)
    – user9482
    Sep 6, 2019 at 7:31
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    @Roland, I think that part of the question is implying that it is becoming obsessive, or at least to the detriment of their other, more likely to be successful work. I agree that crankish behavior is much more common in people who aren't experts in a given field. When it is someone in the field, it is sometimes more difficult to tell where to draw the line as genuinely crankish behavior.
    – JoshuaZ
    Sep 6, 2019 at 10:56

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