text:
Should I report a PhD student’s incompetence to their supervisor?
I had a short exchange of mails with a PhD student who wished to ask some questions about a paper of mine, which was troublesome in many respects:

The student had a blatant lack of basic knowledge and techniques of their field. (Imagine a computer scientist not knowing what object-oriented programming is, a mathematician not knowing what fields are, etc. They did not change fields for the PhD.)
The student should be able to answer some of the questions with very little work.
Initially, the student did not give me even remotely the information I needed to answer their question in a useful manner.
The student seemed utterly overwhelmed with their project.
There were strong hints of a “do my work for me” attitude.

For the purpose of this question, assume that I am very likely correct in my assessment.
Going into details about why I arrived at this conclusion would be beyond the scope of this question and be disclosing too much.
I also wish to make clear that I am not annoyed by the questions or similar, I am just worried about the situation.
I am now wondering whether I should write a mail to their supervisor (whom I don’t know and who is not at my institution) informing them about this incident.
My considerations so far are:

I am pretty confident that this student will not finish their degree (by honest means).
As long as they continue with this, they waste time and resources: their own, their supervisor’s, and other researchers’ whom they are emailing.
If I were this student’s supervisor, this is something I would like to know since it can prevent me from wasting my time and resources on them.
On the other hand, I hope that I would quickly notice these qualities in a PhD student.
This problem will likely escalate soon anyway.
It’s the supervisor’s job to talk to the student and give them the possibility to clarify in case I misjudged them.
However, if I am wrong, my information may wrongfully harm the student if the supervisor overreacts or the student’s image is tainted subconsciously.
Depending on the situation, such a communication as mine may allow the supervisor to smoothly get rid of the student – which is good if I am right and the student is incompetent, but bad if I am wrong.

My question is this: Is there anything else I should take into account when making this decision? (I know that, at the end of the day, I have to weigh the arguments myself.) Note that I already sent a mail to the student but am skeptical whether they got the message.

Comments are not for extended discussion; this conversation has been moved to chat.

"(Imagine a computer scientist not knowing what object-oriented programming is.)" Yeah, FWIW, I like to think I know a thing or two about my trade, but I probably couldn't give you an academic-level definition of OOP (nor do I really care to, if I'm being honest)

Next time, you may want to consider including their advisor in the cc. list when you reply. IMHO, this is a good practice that the student should really get used to have in the first place. The advisor can still ignore conversations when he is not interested about them or if he trusts the student enough. At the same time, he can keep an eye on how the student approaches his own colleagues, the scientific problem at hand and how he works in general - if needed.

Perhaps a warning that certain aspects of the student’s communication could result in “someone else” making an unfavorable report to the advisor?  In other words, you’re being “nice” by being silent, but someone else might not be as kind.

I would ask a lawyer before proceeding.  An adversely affected student could sue for libel/slander.  Truth is a defense, but the court - not you - gets to decide what is true.

Unless I missed something, one piece of vital information is missing: How long has this individual been a PhD student? If they've been at it for a good year or two, then the student should know better by now, and perhaps these red flags and alarm bells are rightfully being waved and sounded. However, if this student has only been at it for a couple months, this could be a good opportunity for some informal mentoring.

@J.R.: I do not know, but I do not think it matters that much. You do not start a PhD at zero after all, but after you already acquired a certain knowledge about your field and scientific working. What they are lacking is (mostly) not something they should be acquiring at their stage of the career, but something they should have acquired earlier or much earlier.

Wrzlprmft, even though in principle I agree with you, I think you definitely overestimate the average PhD student.

Never make bad-looking anybody in the eyes of others, even if you think it is justified!

Why wouldn't you e-mail the student, not the supervisor? PhD students are adults and we should be able to appreciate feedback, even very harsh. I'd e-mail the student explaining why you think there's a problem and suggest discussing it with their supervisor (or offering your assistance, if you're willing to help). Saying that they'll most likely not graduate isn't fair, in my opinion. Personally I started a PhD in a field I had no idea about (not even basics), so it took me a while to stop asking ridiculous questions. PhD is a process!

@Paula: Saying that they'll most likely not graduate isn't fair, in my opinion. – How can you claim this without knowing the precise situationo? Is there no level of incompetence that will ever make you arrive at such a judgement? (Also see my edit.)

@Wrzlprmft "How can you claim this without knowing the precise situationo? Is there no level of incompetence that will ever make you arrive at such a judgement?" - do you know the precise situation? It seems like you cannot. But, you appear to have already made up your mind. I don't think what you have described is really enough justification for you to know that the student is incompetent, especially given that PhD students sometimes and even often do lack knowledge but later turn out fine. I'm not sure any good can come out of this to yourself, the student, or their advisor.

When I started my PhD in computational fluid dynamics, I couldn't solve a differential equation to save my life. But things seems to be turning reasonably well.

@ThomasKing: you appear to have already made up your mind – No, I haven’t; hence the most likely. All statements about such questions are inevitably uncertain, but that doesn’t mean we should not make them or that they are automatically unfair. However, I do have more information than you do (and I cannot share it). I also do not claim that the information I gave is sufficient for you to share my judgement (and I wrote that explicitly).

@Wrzlprmft "However, I do have more information than you do (and I cannot share it)." - so then it seems like you cannot provide enough information for you to trust an answer that suggests you do nothing, which seems to amount to having made up your mind.

@ThomasKing: I do not seek for advice on my assessment (and I hope to have made that clear in the question); I seek advice on how to act upon it.

@Wrzlprmft but how to act upon it (in your interests, the student's and their supervisor's) depends on the reliability of your assessment! We've all seen students start from a poor intellectual state and then become good researchers, and also the reverse, and some of us have been one of those students too. We do not know what we're not seeing, for example some of the factors that can make someone look incompetent are not intellectual.

This problem will likely escalate soon anyway. Why are you asking the question then? It sounds like you're eager to snitch, surely got better things to do?

I witnessed such an occurrence (a totally incompetent phd student). There was a mix of specific situations that lead to that. He was working hard and the supervisor went through with no doubts. He finally got his title and was even enrolled in another country for a post doc. Than he disappeared from academia. It is hard to say what could have been the best for him, or solve the dilemma if such a person should own a phd. The supervisor compare himself to a gardener "I try to get the best out of every plant, flowers or potatoes".

"This problem will likely escalate soon anyway." - then let it run its course...

On a not so funny note, if the student is really incompetent, maybe it’s not a bad idea to report the supervisor to it’s supervisor for selecting such an incompetent student. I know people tend to disagree with medling in other’s businesses but we are talking about a PhD, which needs to mean something. Afterall, that student may have took the spot of a more competent one.

@Ivella, "couldn't solve a differential equation to save my life" that's why we have Lattice Boltzmann.

Not going to offer an answer, you have plenty, and most say "don't," which is probably for the best, but if you choose to do so anyway, do it neutrally, e.g., "Professor X-- See attached discussion with student Y. Some of it gave me slight cause for concern, but I may be misunderstanding the situation, so I will let you come to your own conclusions. --Regards, Professor Z"

Some people find it best to 'play dumb' when encountering something which is outside their field of expertise - it's a very effective way to avoid assumptions. It's also very possible that the questioner and the OP are in different fields, or are in very different specialties of the same field. There may also be a language barrier, or even a terminology barrier within the same language. It is a very hard thing to evaluate.

"...allow the supervisor to smoothly get rid of the student...": you seem so eager to dismiss this student. After a "short exchange of emails" you shouldn't say that you know the student better than the supervisor. Once a PhD project started, there are plenty of ways of finshing respectably, with proper guidance. The problem with some PhD subjects is that there's no flexibility. The project should adapt to the student, and the original objective should be broad enough to allow enough intermediate results for a young researcher. Otherwise, it's just a bad subject to give to a PhD candidate.

Reporting your concerns to the student directly is probably your best option, for reasons beyond those already mentioned.

Contacting the supervisor escalates the situation dramatically. Dealing with any ensuing situation might take away time and energy you need for your job. Do not borrow trouble.
The student asked for your help and will likely benefit from your feedback.

From the question, we do not know whether the student has already reached candidacy.

If not, they would benefit from hearing directly that they need to improve their skills.
If so, they may need to review the material, be less lazy, or have a reality check about their path. They can recognize which option is relevant for them, more than you can, or even than their supervisor can.
Either way, an appropriate response might include: "The questions you're asking can be answered by applying core knowledge from [key subfield--especially if this is a subfield that usually has its own qualifying or comprehensive exam]. Please check with your supervisor about these points, and she can contact me with further technical questions."

A central problem seems to be professional communication, and you could address that directly with the student.

You could write: "I recommend that you review these tips on professional academic communication from Dr. Matt Might and these guidelines on asking questions of experts."

The question did not specify that the student and supervisor are at the author's institution.

If they are not co-located, this would complicate the idea of informally chatting with the supervisor and would likely force the conversation to be through email.

Under U.S. law, this email becomes an "education record" for the student, which would be turned up in the (hopefully unlikely) event that the student is acrimoniously parted from the program and has a competent lawyer.
Other jurisdictions may not have the same law, but emails can be forwarded and may still drag you into a mess or reflect poorly on you out of context.

Feedback from an outsider may have a larger impact on the supervisor's judgment of the student than the author anticipates.

More prominent academics are often more distant supervisors, and this supervisor may have had very little contact with the student to compare this with. By Bayesian updating, your assessment would loom large in the supervisor's mind.
If the author and supervisor are in different countries or at institutions of different status, the supervisor may feel ashamed that the student attracted your negative attention.

Finally, if you communicate about this with anyone, do not say anything remotely like: "I am pretty confident that this student will not finish their degree (by honest means)." (You stated this as an assumption, but it is not clear to me whether this would be part of your intended message.)

If I received a message saying this about a student I supervised, it would sound like you suspect academic dishonesty, an extremely serious charge.

If you have such a suspicion, it is worth approaching an ethical advisor (an ombudsperson?) at your institution and/or the student's institution, beginning with hypothetical questions.
If you are merely worried about the student's competence, do not appear to impugn their ethics.

Whether a student will finish is a very difficult judgment to make accurately.

Since your contact with the student appears to be limited to this unflattering correspondence, you may not see the student's countervailing strengths.
There are many different doctoral student trajectories. Traits like taking initiative (as demonstrated by cold-emailing the author of a paper) and perseverance help students who are behind academically make up their deficits and finish.
Let's assume you could accurately peg the student's odds of graduating at 25%. Stating your conclusion that the student is unlikely to graduate and is a poor target for resources (even if phrased as a "worry" or otherwise softened) will draw everyone's attention far more than the specific details you have to offer. Whomever you communicate with about this, recognize that the base of your evidence is limited and refrain from extrapolating.

I agree and I also think the op should consider how unflattering it would be for the OP should his concerns be unfounded.

This is unethical and unprofessional. It is simply none of your business. You are not in a position to evaluate the student. Writing to his supervisor will make you look bad. It is an insult to his supervisor.
The Ph.D. degree can not be awarded to someone incompetent. If this happens, it is going to be shown sooner or later. Let future employers and assigned assessors who are in a position to evaluate him say this, but not you. Judging a Ph.D. student requires external assessors. Let him take his time. If he is incompetent, it will be revealed sooner or later. But every student has the right to take his own time and attempts. Even the Ph.D. examination process allow several attempts before reaching such a conclusion, simply because such conclusion destroys a human's future.
If he is annoying you, you can refuse answering him. He might be wasting your time but how did you judge he is wasting other people's time? You can tell him that his questions are not reflecting the basic required knowledge and he should first build solid foundation in XYZ then come ask you. Otherwise you can not afford helping him.
Finally, we are all learning all the time. You are knowledgable in this topic he is asking but in someone's eyes ignorant in that someone's topic. You also have been ignorant about your paper's topic until you learned gradually. The world would be more peaceful if we remembered that we have not been born scientists and we learned through other noble people who gave us a hand to help us rather than a hand to destroy us, even when we have not been asked about our opinion.

"Phd. degree can not be awarded to someone incompetent"- Empirically speaking, I think that the reproducibility crisis has conclusively demonstrated that PhD's are awarded to incompetent researchers.

Phd. degree can not be awarded to someone incompetent. — Of course they can! The only true prerequisite for a PhD is to convince four faculty to sign a piece of paper saying that you deserve a PhD (and perhaps a little theater).  You're not assuming that all faculty are competent, ethical, and well rested, are you?

Not only is it possible to do so, it's also possible for an incompetent Ph.D. student to finish the project and get their Ph.D. See Milena Penkowa for an example.

I did not say it is impossible. I said if it happened it is going to be shown sooner or later. Any competent place will assess this graduate capability. At the end, he can not get more than what he deserves. As they say: You cant fool all the people all the time.

Phd. degree can not be awarded to someone incompetent. OMG. Thanks for the laugh!

@None I reallly wish you were right, but I've seen too many incompetent PhD holders, some of whom are now faculty and (horror of horrors) supervising PhD students. Sadly, neither intelligence nor competence are always prerequisites for getting a PhD degree. Sometimes, all it takes is persistence.

@EricDuminil lol. yep that's todays laugh.

@EricDuminil Perhaps it was not intentional on your part, but I feel that mocking someone's answer by saying that it is laughable always comes across as unnecessary.

@Carl-Fredrik: I genuinely laughed out loud while reading it. What can I do? There are a few sentences here which are quite simply and objectively 100% wrong.

@Eric Duminil @Carl-Fredrik Complete the sentence please! If this happens, it is going to be shown sooner or later. Let future employers and assigned assessors who are in a position to evaluate him say this, but not you. It is true that if he passed unqualified, he can not fool everybody after that. He will not get the job that a competent person will. What is really funny is that an external person who does not know the student nor the supervisor in person nor worked with them, from short exchange of mails can make judgement that a phd student is incompetent and take actions.

@Nat I strongly disagree that the reproducibility crisis is caused by researchers being incompetent. Rather, the problem is that the system is broken/suboptimal in a fundamental way (read Ioannidis to understand why).

@EricDuminil "What can I do?" For an excellent example of what you can do, without any hint of mockery, look at terdon's comment below your own.

@None Now you are assuming the people who make those decisions in general would be any more competent than the ones they are hiring ;)

It will however still show sooner or later as you say, but not to the ones who hired them, but instead to the future students who have to put up with them when they reach the higher levels.

NOTE: this answer was based on the assumption that OP works at the same institution and on the same campus as the supervisor in question. OP has since clarified that is not the case. I will leave the answer to stand in case it is helpful to those in a similar situation - see meta discussion.
At the risk of sounding like an old man: does no one talk to their colleagues any more? This sounds like the ideal situation for an informal chat with the supervisor. Even if you don't know them, surely there's a chance to start up a conversation after a departmental seminar or something.

Hey, your student X has been contacting me recently about their work. Sounds like an interesting project...

And play it by ear from there. It should become obvious whether the supervisor has a high opinion of their student or not. If they seem receptive, drop in that you think the student seemed to be struggling in this area. Don't charge straight in accusing student of being incompetent. If the discussion opens up then great, you can give more detail. If not, you tried, you can't force the issue.

Do really people use Hey in conversations? I find it plain irritating.

@RuiFRibeiro Do you prefer more formal salutations or simply to omit 'em?

In this context in written form, I would prefer omitting them than seeing them probably...

This seems to me the most reasonable approach. There is no reason to force anything on the supervisor, but if they open up and hint at their student struggling with the project, an honest conversation could help.

@user37208 That is an excellent point and one that I had not considered. I have added a note regarding my own assumption.

@RuiFRibeiro: Yes.

+1 This is the best answer in the general case if you're concerned about the competence of a Ph.D. student at your institution. If they're not at your institution, the other answers are correct that it's simply none of your business.

I'll propose a variant of the answer written by @user88253.
Email the student, with a cc to his/her advisor, encouraging him/her to work on the topic(s) with his/her advisor. Invite the student to let you know how it goes, and invite the professor to draw on you as a resource.  Starter text (which you can edit and make your own of course), where A is the student and X is the advisor.

Dear A and X,
A - I'm glad you're interested in (name of paper) which covers (such-and-so) topic.  It's a worthwhile area to delve into, especially since it provides a fruitful opportunity to apply a number of basic concepts which will stand you in good stead in your study of (field).  I would encourage you to work with your advisor to review the basic techniques that provide the underpinning of this paper.
X - Let me know if I can be of assistance.

Advantages of this variant over the original:

The "quite busy" excuse, which could be hurtful to read, has been eliminated.

The "Basic questions" phrase, which could come across as offensive, is avoided.

This variant tries to be upbeat and encouraging.  Still, it politely sets up a boundary (without getting annoyed, frustrated or angry), and hands off to the person who should be helping the student.

Similar to the original, bridges are not burned.
Please note, there is no need to embarrass the student by including the chain of previous emails.

I like this suggestion but would omit the word 'basic' in the phrase 'a fruitful opportunity to apply a number of basic concepts'

+1, this is the way to go OP. It puts the burden of the (perceived) problem on the shoulders of the person who should have them - the student's supervisor. If they don't do anything after some time, it's more likely that either you misjudged the student, or the university needs to seriously re-evaluate their admission standards and the standards to which supervisors hold their students

No, and you're very possibly misjudging the student
I'd like to buttress @user37208's answer and counter some of your factual claims, OP.

The student had a blatant lack of basic knowledge and techniques of their field. (Imagine a computer scientist not knowing what object-oriented programming is.)

I know more than a couple of computer scientists who don't know what OOP is, or at least - "know" that it's "programing with objects", but have barely ever programmed anything, if at all, and not with objects. At least one of them is a very esteemed professor.

The student should be able to answer some of the questions with very little work.

Maybe he misunderstood the question, or misunderstood your assumption that he was supposed to do work?

Initially, the student did not give me even remotely the information I needed to answer their question in a useful manner.

Do you know how many times this happens to me with people who ask me for things? If I thought those people were incompetent I'd think everyone is incompetent. Now, I suppose you could make an argument that this is the case, but then you don't have anything to complain about...

The student seemed utterly overwhelmed with their project.

Being overwhelmed with something can make one not bring one's competence to bear.

There were strong hints of a “do my work for me” attitude.

Ah, now this is an ethical failing. Unfortunately, sometimes it's exactly those people who manipulate others into doing work for them that get ahead in academia, i.e. some people would regard this scrupulousness is a merit. I wouldn't. But - that only weakly correlates with incompetence.
Now, it could very well be that I'd get the exact same impression as you if I'd actually read the email exchange. But if that's what you were to "accuse" the student of - your case is pretty weak.

Maybe relevant to your first point: I'm a math PhD who has written code off and on for many years (Basic, Pascal, and Fortran years ago; C++ and Perl more recently) and I don't really know what object-oriented programming is, in any more than the very vaguest sense.
Now maybe a computer scientist should know what object-oriented programming is. I don't call myself a computer scientist. But it's conceivable that someone in an area adjacent to computer science might try to have a meaningful conversation with a computer scientist about a specific algorithm, while not knowing what OOP is.

I think your logic is problematic. Just because that some of my individual observations can be explained by exceptional circumstances (which I do not dispute the slightest), doesn’t mean that my overall conclusion is likely wrong. Also when I said “not even remotely”, this was not rhetorical emphasis. As an SE regular I am familiar with the all the difficulties of asking good questions, but the question in case could be compared to “My code isn’t working; what am I doing wrong?” (without any additional information).

@Wrzlprmft: And is he a PhD candidate in the equivalent of software engineering?

@einpoklum does that really matter? Remember what PhD stands for, he's supposed to love thinking; the expectation he has to fulfill is trying to find ways to solve his problems that do not rely on others doing his work. So I argue that the actual background doesn't matter, as long as it's vaguely technical or scientific.

@MarcusMüller: Yes, it really does matter, because you might have a perception bias of his behavior (plus - very small sample). Maybe he's a flunkie, but - you're not in a position to make that argument to his advisor IMHO.

I actually believe that experience might objectively give someone the ability to assess the abilities of an individual based on the way they ask, and what, for one's own specific area of expertise! Op is not judging someone's skills as brain surgeon, op is assessing a student's abilities on the very field he's an expert on. Sure, being an expert gives you a bias regarding simplicity of concepts that might be hard to grok for someone new to the field, but there objectively are basics to every field that even a new student would have to bring. I'm an EE, so my example comes from that world:

Assume you're an expert in the design of microscopic antennas for the detection of terahertz radiation. You obviously think antennas are easy, so that would skew your assessment when you meet a new PhD student that doesn't know much about antennas, though he should have already learned that in his master's. But now you've got this student that keeps emailing you about concepts in your bleeding edge research, but his questions betray a total lack of basic understanding of even basic antenna principles. Like, he doesn't even know what an electric field is. You really can't work with him,

Because he really can't even understand the most basic explanations that you could give. Instead of taking your hints of maybe reading a basics electricity theory textbook, he continues to ask about concepts that with his background are impossible to understand. He starts demanding you do his work, as he really stands no chance of doing it himself, and he also doesn't stand a chance of learning the basics "on the job", because he tries to still work with your very involved stuff, trying to find a shortcut around the basics. This is the situation I think op describes!

Remember, that the whole object-oriented-programming thing is just an analogy to illustrate the level of lack of knowledge. You can also imagine a physicist not knowing what a Hamiltonian is, a mathematician not knowing what a group is, or what @MarcusMüller is describing, etc. And it’s not my field they are lacking expertise (though I am under the illusion to have some expertise in this field). Finally, I never talked about making an argument to their advisor. I am at most considering informing them (and letting them act upon this information however they seem fit).

In general, I do not provide unsolicited negative opinions.  So I would wait until the student's supervisor asks me for my opinion.  I anticipate that the supervisor will never ask me - and I am OK with that.
On the other hand, it is good practice to praise generously and publicly.  If and when you are impressed with a student then make sure his/her supervisor knows about it.

I agree with all the answers that say not to contact the supervisor.
I'd make one exception to that: If the student's questions indicated that they may be putting animals or people in danger, I'd find some way to flag this to the supervisor. I'd do this as tactfully as possible, but if a student is unintentionally causing unnecessary animal or human suffering I'd definitely try to intervene.
Otherwise, I'd go under the assumption that the supervisor is more familiar with the student that I am and already is aware of any weaknesses I perceive.

I would approach the situation differently. If you are a faculty member in the same department as the student with the same rank as, or higher than, the supervisor, you can swing by the supervisor's office and have a closed door conversation about how unimpressed you are with the student. Under no circumstances would I want a written/email record of the conversation. I would try and avoid doing it over the phone because face-to-face is more personal.
If you are a faculty member with a lower rank, depending on departmental culture, you might need to tread more carefully. If you are a post-doc or not in the department, instead of telling the supervisor that in your opinion the student is incompetent, I would provide the evidence that you have and allow the supervisor to come to that conclusion on their own.
I would send an email to both the student and supervisor that includes the entire email conversation. You could provide some textbook suggestions or very simple answers to the easy questions and suggest that the do it for me stuff would require a collaborative effort (if you want to work with the supervisor you can offer yourself up, or just say you do not have time to collaborate).

Personally, I don't see what you're suggesting as being as bad as other commentators are making out (and I note from your explanation that this is not a matter of annoyance, but concern).  I note from your profile that you are a post-doc, which essentially makes you a member of the academic class, and so you are effectively a fellow colleague to the supervisor (albeit a lower-ranking colleague), not a fellow student to the student in question.
Rather than communicating your concern to the supervisor explicitly, perhaps you could obtain the same effect in an entirely different manner.  You could send the supervisor an email advising that you have been providing some assistance to the student, and for his/her information, you are including the email chain in question below.  Make no negative comment on the capacities of the student, and leave it to the supervisor to review the correspondence that caused you concern, and make his/her own professional assessment.  That way, you are doing nothing more that being helpful by sending an email query from the student to the supervisor.  If this still seems a big presumptuous/rude, you could even lighten it further, by seeking the supervisor's guidance for how to provide clearer help.
Here is an example of what I mean:

Dear Prof. [Name]
I am a post-doc in [area] at your university.  I just thought I'd write to let you know that I have been attempting to provide some assistance to your PhD student [Student name].  This relates to an initial query about [subject] and we have been corresponding on the matter to try to figure out the best way to proceed (see email chain below).  I'm not sure if I'm doing the best job explaining this stuff, so perhaps you could give me some guidance for how to provide clearer help.
I hope I am not stepping on any toes by giving outside assistance.  If you would prefer to assist the student directly, please let me know.
Thanks,
[Your Name]

Without knowing the department culture, I would never suggest that a post-doc assume they are an equal colleague to a faculty member. In ideal departments that is the case, but in reality, that can get you in a lot of trouble in a lot of places.

@None outside the classroom, there is not a sense of privacy. I wouldn't think twice about forwarding on a research related email to another colleague unless the the original email suggested something was confidential.

@StrongBad: I have edited to make clear that I am not suggesting that a postdoc is an equal colleague to a prof, but still a colleague.

The basic approach could work as an alternative to consider in case OP isn't able to visit the advisor in person.  But I would cut down the prologue a whole lot.  Specifically, I'd just remove the whole first paragraph.  It's not needed.

Hmmm, I think what you suggest is illegal. Forwarding email without consent

@SSimon That would be insane. Think about it for a moment.

I like this in general. In a first exposure to a colleague, don't deprecate your explanation skills ("I'm not sure if I'm doing the best job explaining this stuff"), especially to cover for the student. At the same time, as aparente001 said in another answer, it may not be necessary to include the past exchange.

Downvoted inadvertently. (Fat fingers...) So sorry. My abject apologies. Please apply a small edit, so that I can reverse the downvote.

@Mico: No worries.  I always assume any down-votes on my posts are accidental.  ; )

Rather than doing something like that behind student's back,1 I would send a blunt and frank email to the student. Like, judging by your questions, I don't see any effort on your part, and don't understand how you can be competent in this field. Please don't waste my time. Etc.
If I was confident in my assessment, I wouldn't even spare student's feelings that much. Sometimes it can be deserved. I find it creates a healthier culture than being nice upfront but making decisions behind the back. While being a student, I would certainly prefer it that way.

1 Note I don't imply, like some, that this is inherently unethical. Situations differ. There are fields where turning a blind eye creates an even more unethical (or even outright dangerous) situation overall.

I actually already sent such an email to them (being as frank, but less blunt); I am skeptical whether they got the message though.

Perhaps it will help if you will e-mail the student and tell him that you are quite busy and do not have time to answer his basic questions. However, you suggest the student to e-mail his supervisor with same questions and add you as cc (and request his supervisor to send you also his answers), and if a more advanced question will arise, you will try to briefly explain what you know on the subject.

I suggest you read this "....." or do research on these techniques "...." then you should be able to answer questions 2,3, and 8 on your own.  Also, it should help you ask better questions which include all the information I need to answer them. I don't have enough information to answer the other questions, I need the following information. "........".
me
The student will then have to do the research.

I understand the intellectual and moral challenge, and plenty of thoughtful answers have been given here.
I would suggest first do no harm as first guiding principle, and then hard on facts, soft on people as a second-line guideline.
In my perception, beside the do-nothing option, the safest strategy is to offer to the student your availability to establish a connection with his/her supervisor. What happens then becomes a matter for other posts.
A risk not to oversee is that the supervisor is no better than the student. The proof of the pudding is in the eating.

I am now wondering whether I should write a mail to their supervisor
  (whom I don’t know and who is not at my institution) informing them
  about this incident.

No. Selecting and evaluating PHD students is the sole responsibility of the supervisor and their institution. It's none of your business if another institution suffers from selecting bad people.

Write a mail to the PHD student that you don't wish to continue the exchange on the topic.
If you want to be super-nice and interested in a collaboration, then write a mail to the supervisor where you ask for a brief alignment on the way of collaborating, and then make clear that you don not desire to communicate with the PHD student directly.


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How much does the ranking of your Master's program/school matter when applying to PhD programs?
I'm applying for research-based Masters in Oceanography. So far, I've lined up advisors at University of Massachusetts (an intercampus program), South Florida, Georgia, and New Hampshire.  According to here,they are in top 150 globally for oceanography. Moreover, my current work supervisor says 3/4 are well-known.
I've also been talking to a UCSD prof. He told me he is unsure if he will be taking on students but is more inclined to accept a PhD student (since they'd be with him longer). I would really like to go to UCSD for my PhD.
Will going to a lower ranked MS have a big effect on PhD applications?
What can I do to distinguish myself in PhD application 1.5 or 2 years from now should the MS be low rank?

Generally, rankings matter very little. Having good recommendation letters matters much more.

Don't know the answer for your field but took a hacksaw to the lengthy question.

Is there any reason you can't apply to the PhD program at UCSD now? Unless they require an MS, it might be your best option.

Don't get hung up on the ranking thing because it doesn't really matter when it comes to a PhD admission.  Top things that make your application stronger are

Skills you gained during MS and prior education/work experience
Letters of recommendation (high impact)
Writing sample (usually your statement of purpose/intent)
Willingness to learn new things
Your vision for a future career

That being said, try to find a program (either MS or PhD) that is best suited to your interests and that has the faculty expertise for you to learn from.

The ranking of the institution does not matter. The ranking of the institution in the subfield in which you work may have a small impact, but the main factor of importance is the prestige of your reference letter writers. If the writers are unknown, then their recommendations do not carry much weight.
Also, smaller departments with one or two famous people have the problem that if these people leave, then the rank/influence of the department drops a lot. To hedge your bets, a department with a greater number of well-known academics is better.

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Should I complete a master's thesis if I intend to move to a different graduate school?
I'm currently a first year graduate student in mathematics in the United States. After having completed my requirements for a master's degree, I'm considering moving to a different program, based on reasons I outlined in a previous question.
My question should one necessarily work on a master's report/thesis to boost one's application to enter a different PhD program?
I can probably get a master's degree without having to write a report/thesis as I have already passed some of the qualifying exams at my school.
So, to what extent would having completed a master's thesis help me in switching programs? My interests are a bit broad, and interdisciplinary, so I'm, as of yet, broadly interested in a couple of areas, and not a single, specific area of research. So, I would like to apply to schools that have research groups in at least 1-2 fields of my interest.
Any suggestions on how graduate committees in math or STEM fields view the issue of masters thesis or reports.

Perhaps it does not directly address your question of how math admissions committees view master's theses, but I'd suggest that doing a master's thesis would let you judge how much you actually like doing research in at least one of the areas you're interested in pursuing as a graduate student.

@virmaior I am sorry, but I didn't quite follow. What do you mean when you said broke it apart?

@virmaior Oh, right. Makes sense. I didn't see you had edited the question.

Since the drop-out rate in PhD programs is quite high, it's a good idea to secure the master's degree so that you'll certainly have something to show for your efforts.

Since the doctorate is about research, having some solid evidence of doing that successfully is a plus. You can change your main focus to another subfield after writing a master's thesis, so don't worry that it will limit you.
But my main question back to you is what you will be doing in the meanwhile if you don't do the thesis. Will it be at least as positive an influence on admissions committees and potential advisors? It seems doubtful, but possible, depending on your situation.
An MS isn't really required for admission to the PhD in the US, but a positive record that predicts success certainly is. Do whatever you can to build that record.

I was thinking of using the time to take more coursework, prepare for the subject test GRE in math and shore up on my math background in general. My interests are interdisciplinary (probability, PDE, and it’s application to machine learning and physics, mostly quantum mechanics) so I suppose my main goal of getting a master’s was to shore up on advanced mathematics background. Writing master’s thesis may bog me down in a specific topic area, and I suppose I should use this time to learn more geometry, topology etc. to build my math background and then branch out. So, I’m delibereatly thinking

Of doing a master’s degree with a coursework intensive focus. I try an self study books outside of class, and I could always ask faculty to advise me on my readings.

Definitely get the "en passant" masters if possible (masters that you just submit a form for, based on your coursework).  At least you have something.  The same would apply if you are in same spot for a degree program.  But it's even more important if you are shifting schools (hard to go back for the certificate if you moved on already).
I think most places will look at you based on your old record.  If it is competitive, the master's thesis is not so important.  For example if you are moving towns because your spouse got a job, etc. If instead, you are trying to "move up", some record of work would be helpful.  Ideally publications (even small LPUs, but peer reviewed somethings).  But a thesis is next best.
Obviously the amount of effort involved affects choice to write.  If you have a topic and think you can jam out the document quickly, I would be more likely to spend the effort.

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Is it possible to be admitted to graduate programs in subjects one minored in in undergraduate?
I'm currently studying electrical engineering and considering on getting both a physics and math minor. I'm wondering though, is it possible to get a Ph.D. or a Master's degree in Physics or Math when I only have the minor and not a real degree?
Is it likely at all to be admitted into a graduate study with only a minor? I'm wondering because I don't know whether or not the minor is actually worth it if I won't be able to pursue a graduate degree later on with my credentials.
For reference, a math minor at my school typically means 3 extra upper-level math courses (300-400 level), and only 2 more physics courses (one in waves + light + thermodynamics; the other which samples different fields of modern physics, all required for the degree program)

It is indeed possible with the right alignment of the stars.

In the UK, as long as you can demonstrate that you have the necessary interest and skills, you can.

Why not just major in physics or math?

@Azor-Ahai at this point, I'm more interested and likely to do eecs, but I just want to know if it's possible, just in case I get interested some time soon.

@101010111100 I'm just wondering in that case, if a minor demonstrates the skills?

@A_Happy_Student I am not too familiar with the US system, but it may, especially if supplemented with private study.

In the U.S., for graduate work in mathematics, even a minimal major in mathematics is not really adequate preparation for a good graduate program in mathematics. Not enough coursework, which means insufficient exposure to higher-level ideas (assuming, as is apparently mostly the case, there's not much interaction with mathematics outside coursework). Even though various engineering programs require quite a bit of mathematics, in my observation the topics exactly do not get to fourth-year mathematics, so that exactly rigorous analysis, abstract algebra, point-set topology are not seen. The latter are a very minimal preparation for graduate work in mathematics of almost any sort. Still, if there's room for those courses in a "minor", then some minimum thresh-hold would be reached for appearing to be prepared for math grad school.

Are 2-3 extra courses reaching the threshold?

If those "extra" courses are rigorous analysis (not just "advanced calculus", at most places), abstract algebra, and maybe a semester of point-set topology, then at least you won't be immediately disqualified. That does not mean that your application will be competitive, though, since math grad programs would prefer (all other things the same), not to have students spend too long acquiring or mastering undergrad material.

What is your definition of "advanced calculus"? The class for analysis at my school is called "advanced calculus"http://www.math.lsa.umich.edu/courses/undergrad.html#451

@A_Happy_Student: Yeah, terminology does vary.  I think you would want 451, 490 (but better 590), and 412, and hopefully 452 also.  Again this is a bare minimum to not have your application thrown in the garbage and most of your competition would have much more preparation.  By "advanced calculus" Paul means something like 351, which would not be suitable.

@A_Happy_Student: But you will get much better and more specific advice by talking to someone in your own math department.  They will have a clear sense of what's in their courses, and how it fits into a course of study to prepare one for grad school.

@NateEldredge yeah I should see what the advisors think, thanks

You should talk to the student advisor of the corresponding graduate program.
In general it should be possible to change your major, but there might be some rules to that, e.g. you not only need to take the 3 courses you already take but a few more to fulfill the prerequisites for the program.
I don't think that only undergraduate students from the same university and the same major are admitted to the graduate program, there surely are also externals, internationals,... and yes, also people changing their major might happen. Thus, just talk to people and find out, the rules might be different at each university.
Also, depending on how far you already are into your studies, you might consider changing your major already now. If you are only just a few months in, that might be easier than to do it later.
Switching major and minor, you might even be able to keep all the courses you already did in your record. Once again: Talk to the responsible persons at your university, the rules are different everywhere.

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Will past medschool failure history affect Graduate admission in Literature/Linguistic major?
14 years ago, I attended med school. After 6 years and chronic depression issue, I decided to withdraw from that program. This means I have an official transcript, but of course did not earn a degree. (In my country, you can start med school right after graduating high school, and I enrolled when I was 17 years old)
I am currently taking a bachelor's degree in Literature, and so far I have gotten straight As, and my GPA is 4.00 (The maximum grade I can get in my country). Since I feel very passionate about my current major, I plan to do a graduate degree (MA&PhD) in Literature/Linguistics in Europe or the US. After seeing the application requirements, I have noticed that most institutions require all college level transcripts to be submitted. That would mean, I have to submit my med school transcript as well.
My poor performance in med school experience and the transcript are haunting me. The med school grades certainly look terrible compared to my Literature grades.
Will this affect my chance to be admitted to good graduate school in Literature/Linguistics? I start to worry that they will see my past failure as inability to work in PhD research.

Literature and linguistics are two unrelated fields.

@AzorAhai Not necessarily in the United States.  The general norm here is that anyone who studies one is required to study some of the other, similar to how those in studio art are required to take some number of art history classes, and those in art history are required to take some number of studio art classes.

@guifa That's funny, I wasn't required to study any literature to get my linguistics degree, because knowledge of literature doesn't inform linguistics as a science.

@AzorAhai Were you general linguistics (from a self-standing Department of Linguistics) or were you specific to a particular language / language family (from a Department of Foreign Languages)?  What I have always seen in, for example, Spanish departments is a literature track or a linguistics track, but each includes a handful of courses from the other.   Given the OP's repeated mention of Literature/Linguistics, my guess is that is the type of program he is looking at.

@guifa yes, I studied in a department of linguistics.

Your past career as a medical student was so long ago that it might as well have been someone else. I doubt that a grad school in your current field will still hold you liable for problems you experienced much earlier in life, especially given your accomplishments since then.
If you are concerned enough to mention it as part of your application, then just make sure to comment about why you believe it’s not relevant to who you are today and why schools shouldn’t worry about a repeat episode. Of course, you can point to your current record as proof of the change!

Thank you for your reply, @aeismail . It's relieving to read your helpful comment, and I agree, I will mention it and comment about it. Hopefully the grad schools will not dwell too much on it, and if one of them does, then it's not the right school for me. Thank you again.

Of course you can't determine how others would judge you. It may well be that some reviewers will see your past as an indicator. That is true for anyone and for any sort of failure, academic or otherwise.
You can't, and shouldn't, hide your past, and you shouldn't be overly fretful of it either. It is what it is. I wouldn't try to over-explain the past either. A short acknowledgement of your past problems is enough.
However, since the number of applicants to your desired program probably exceeds the number of available slots, you need to give the reviewers something positive - a reason to want you in the program. It seems to me like you are doing that now, so the other answers here seem good to me if a bit sanguine on how you might be viewed.
But since this is Literature you want to pursue, do something interesting in Literature that you can point to, even if it is in local publications. What can you add to Literature and its analysis that says to a reviewer "This is the candidate we want." The interesting thing you do, however, needs to have some visibility, though, as an undergraduate, not excessively so.
Note that everyone makes mistakes. Most people realize that. Most people also realize that people can change. Most people, in fact, do change over a lifetime.

Most language programs that I'm familiar with care little about your performance in non-language courses, so long as the grades aren't awful.
 As long as you have decent performance in them they will only care about the difference between an A or a C in, e.g., a math course if they're looking to distinguish between two near candidates.
If you want to pursue a particular subfield then they'd really just be concerned about mediocre grades in related courses (for example consistently getting Bs or Cs in computer science courses if you're looking to study digital humanities, or in statistics classes for linguistics).
The fact that you continued in the medical program for 6 years before dropping out, and currently have a perfect GPA tells me you would little problem being accepted into a program, especially if you explain why you dropped out.  In fact, if you were interested in a translation/interpreting program, you'd probably have a solid advantage with the medical background you have.


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Quality and effectiveness of remote/online vs traditional teaching?
Currently when a lot of traditionally taught courses are converted to online and remote education - what does science say about the effectiveness of that compared to the traditional classroom situation?
I am taking a language course and just had my first online (MS Teams) session and thought it was horrible (much more distractions sitting at home, no live interaction with the person next to me and so on).
I assume there has been a lot of research on this subject made before (and that even more will be done after corona) and am looking for "rule of thumb"-answers, just to get an idea in which direction the results are pointing

I would specify clearly that these rule-of-thumb answers should still be science-based, otherwise the answers will be full of unsourced personal takes.

Yes, of course. I just meant that the effectiveness might vary a lot between subjects, context, type of course and so on. As an example of a rule of thumb answer I am looing for, organic farming requires approximately 100 % more area for the same yield but certain crops only need 20 % more are while others need 300 %. 100 % is a good rule of thumb if you look at the average persons consumption of vegetables and fruits.

Yes, there are many published works assessing the effectiveness of e-Learning. My personal opinion is that physical attendance is always the best and e-Learning didn't come to existence in order to replace traditional learning but to be an alternative solution when traditional learning cannot (or difficult to) be reached.

From what I've read, it's the consensus that high-quality online teaching is possible, if rethought and redesigned from the ground up. The existing research will presumably discuss that. The current situation -- where in-person courses have been hastily converted into online courses, by faculty members with little background or interest in online teaching, and who in many cases are coping with adverse circumstances -- is unprecedented, and I think it's inevitable that in many cases it will go poorly.

Have you tried a basic search of the literature? e.g. https://scholar.google.com/scholar?hl=en&as_sdt=0%2C50&q=online+vs+traditional+education&btnG=&oq=online+vs+

@RichardErickson Yes, but I am not sure what I am looking for.

I don't believe there is much research done on this, though some research definitely has "concluded" that the MOOCs courses were unsuccessful.
My own take on the matter is this, and it is backed by some personal experience and student feedback:

When done with sufficient care, online learning is way more effective, pedagogically, than traditional lecture-centered face-to-face learning. Students can go back and forth in their own preferred pace to understand thoroughly what the lecturer says.

It is a fact that many (20%?) students (and people) simply cannot grasp notions and arguments from live lectures. While they can grasp it if they have control over the speed and can freeze the lecture.

Online lectures tend to be better structured, because the focus is on mere educational communication, and so teachers must work harder on the content, and cannot rely on spontaneous interaction with students.

Interaction with students is in fact better many times: you have a chat, and students can ask questions in a clear way, and not live in front of the whole cohort. This gives many students much better interaction with the lecturer.

The only BIG disadvantage of online lectures I see, is not the lectures themselves, pedagogically, but the psychology of sitting at home isolated from spontaneous interaction with peers and staff. This psychosocial deficiency detracts severely from the teaching experience, and students motivation and hence students engagement, and is probably the reason why online learning only programs are not sufficient and cannot replace entirely face-to-face teaching.

It seems that the consensus over the last two years is that online teaching is inferior to face to face, for many students. But the change has changed the balance a bit between different groups of students. For me, the change would have been highly negative (were I still a student) as I like to ask a lot of questions. That is harder to do remotely. But some students would rather study on their own and the provided materials are adequate for their need.
Some things are clearly harder. Exam proctoring, for example, has become difficult to impossible. Changing the nature of student evaluation is probably necessary, but the time frame has been too short for that to be developed and tested.
Giving individual feedback to students has become harder also. I used to be able to write short notes on student's paper assignments. But exchanging paper has become difficult to impossible and electronic equivalents are, at best, awkward.
But the biggest factor is probably that it is new for everyone. It isn't that we have two highly developed "modes" of course delivery that everyone understands so that we just flip a switch to go from one to the other. Everyone is struggling, instructors included.
But, we are unlikely to go back to the original system as economic factors start to intrude. If it is cheaper to deliver online, since a lot of the cost gets pushed to students, then those responsible for institutional costs will start to find that advantageous.
At the moment (early 2022) we are still struggling. A number of news stories recently suggest that students are unhappy, both with online teaching and with being forced to meet in person in uncertain times.

I like to ask a lot of questions. That is harder to do remotely: Actually, in my experience in the last two pandemic years, students tend to ask more questions remotely rather than less, because the chat lowers the psychological barrier. Giving individual feedback to students has become harder also: Not really. You don't exchange papers, you exchange PDFs, which can be easily annotated with a tablet and a PDF editor. I'll in fact continue to use this method also this year as soon as I'll restart classes in person. Other things, yes, are difficult (e.g. experimental labs).

Pass rates for MOOCs (high enrollment online courses) are usually less than 10%.  This was well known before the pandemic.  The best MOOCs have lower pass rates than mediocre in-person classes.
Online courses have objectively inferior effectiveness.  There is no reason to think that cannot be changed.

Another reference, one of many: https://www.erudit.org/en/journals/irrodl/1900-v1-n1-irrodl04927/1065273ar.pdf


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Can a casual comparison of phenomena and assumptions based on it be presented in a journal/conference paper?
In comparison to the size of the universe, humans on earth are like bacteria on a speck of dirt. Much like in the animated movie "Horton hears a who".
When writing a journal/conference paper, if one would like to present a fact:

Universe being huge and Earth being negligible in comparison.

and compare it to similar phenomena and also present assumptions:

There might be a civilization of creatures living on specks of dust
  that we see everyday.

... is it considered acceptable to present such assumptions? Normally, scientific papers expect experiments and tabulated results, but can one present something purely philosophical as long as it seems logically sound (but there's no evidence to support it)? If yes, then what is the appropriate format to present the text? Can it be written as shown below?:

Observable Phenomenon: The universe is 92 billion light years in diameter. In comparison, the Earth is a mere 12742km in diameter, and
  the average human height is 1.7m and viruses can be just 20 nanometer
  in diameter.
Our hypothesis: If the Earth could be assumed as a universe, it may very well be possible that there may be an entire civilization of
  creatures living on atoms or electrons.

Would mentioning something like this make the reviewers reject the paper immediately or make them criticise it for being too casual or lacking in evidence? Or is a researcher allowed to philosophize such thoughts and present unproven theories/assumptions? If yes, then how should it be presented?
ps: This is not for the field of humanities or pure philosophy. This is for a journal in the field of computer science. The question is not exactly about the universe and civilizations on specks of dust. The question is about how to present the author's assumptions that don't have proof.

Your specific example, makes little sense, actually. "Civilization" has a meaning. But some things are just metaphors for other things. No one worries too much about that as long as you don't push the analogy too far. But for civilization you would need some evidence.

Perhaps you could cite Two New Worlds. I. The Infra-World. II. The Supra-World by Edmund Edward Fournier d'Albe (1907) or He who shrank by Henry Louis Hasse (1936). :)

It seems like what you are describing is closer to a blog post than a research article. Some journals will publish such musings when they are made by someone particularly influential, but these fit more in the category of editorial rather than scientific paper.

I think this question is about the content of your research.

@DaveLRenfro : Both the sources you've pointed to are absolutely fascinating!!! Thank you for sharing it.

In comparison to the size of the universe, humans on earth are like bacteria on a speck of dirt. Much like in the animated movie "Horton hears a who"... There might be a civilization of creatures living on specks of dust that we see everyday.

I think the example you have chosen here is obscuring your real question. It's not clear that this example makes any scientific sense, so if I saw this in a computer science paper I would find it very strange and probably suggest that the authors delete it. But your real question is about some hypothetical opinion, not about the (scientifically indefensible) suggestion that there are entire worlds on specks of dirt.
So, to the question I think you wanted to ask: it is OK to write casual comparisons if the comparison is useful and illustrative. Not all arguments have to proceed with a hypothesis test or linear regression or something. Your job is to convince the reader of your hypothesis; the argument for that can be heuristic, philosophical, or on a totally meta level. If it's a compelling point of view and the reviewers buy that it is plausible, they will likely be happy with your paper.

Observable Phenomenon: The universe is 92 billion light years in diameter. In comparison, the Earth is a mere 12742km in diameter, and the average human height is 1.7m and viruses can be just 20 nanometer in diameter.
Our hypothesis: If the Earth could be assumed as a universe, it may very well be possible that there may be an entire civilization of creatures living on atoms or electrons.

Assuming that the "Observable Phenomenon" and "Our hypothesis" are replaced by something less ridiculous, this seems entirely fine. You are starting with some quantitative observations, and asking the reader to draw a conclusion about some hypothetically possible scenario.


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By what main criteria does the mathematical community judge whether an open problem is important?
By what main criteria does the mathematical community judge whether an open problem is important among many others open problems in mathematics? For example if we say that the Riemann Hypothesis is an important problem because it is related to the prime distributions, are all open problems related to prime distributions important?

Terry Tao wrote an article What is good mathematics? that might be insightful for you.

@ KimballThat is an excellent article, indeed, though half of the mathematical community I know was rolling on the floor laughing because of the fact that the main example of "good mathematics" there was the one Terry worked on for the last few years and the other half was "somewhat perplexed" by it. I just discard the fact because it is natural to draw the examples from what is currently on your mind and because everyone can get carried away a bit when writing (that is certainly true as far as I am concerned).

@fedja, for example why RH is important then Goldback conjecture however the 2 topic are about primes ?

@user51189 1) "Both being about primes" is not any more meaningful comparison than "both being about numbers" 2) It is not RH, which is important, but the validity of RH (a counterexample will result just in a spectacular collapse of a huge body of conditional statements) and 3) Have I ever said in public or in private that Goldbach conjecture is less important than RH?

@fedja, i don't said you that you said in public what YOU wrote , but just asked you a question related to :http://academia.stackexchange.com/q/34800/24870 ,why RH is important for any mathematical open problem ?

@user51189 why RH is important for any mathematical open problem I hope the meaning of any is some here.  Then that part is easy: if we can prove it, we can answer quite a few other natural open question about primes, etc. The RH is related to the rest of the "prime number theory" via Riemann's explicit formula (well, as explicit as it can be, at least) for the number of primes up to $n$ and if you know the latter, you know quite a lot.

@user51189 My #3 was just a way to attract your attention to the fact that if I had asked you "Why are flowers more important than crocodiles?", you would be in the same position as I was when looking at your question as posed. How am I supposed to answer a why question about the statement that I have trouble even assigning a clear meaning to, not mentioning agreeing with? :-)

Asking "Why does the mathematical community think RH is important?" is somewhat like asking "Why does the film community think Citizen Kane is a great picture?"

An open problem is important if its resolution affects many, diverse areas (indeed, the farther flung the impact, the more significant the problem).
In the case of the Riemann hypothesis, there are many theorems that are conditional on it (see Consequences of Riemann hypothesis), though they are largely in the broader realm of number theory.
Other problems, like the P vs NP problem could have consequences beyond automata theory that include cryptography (and, by extension, our entire
e-commerce industry), computational biology, manufacturing and optimization, etc.

Why roll back those edits? At least keep the ones that are unambiguous fixes.

Sorry ... one of the edits produced a sentence fragment, and there was no obvious way to just accept some of the edits.

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Discouraged by advisor to take courses, is there a way to navigate through this situation?
Graduate schools in North America usually have a required number of courses that you need to pass. Afterwards, the school is usually pretty flexible at the student taking more courses (and it is often free of charge), but it is another story with advisor/PI/supervisor.
After I passed my courses, I have been either explicitly or implicitly reminded by my advisor that I should not take more courses. None of any other graduate students are taking courses either in some sort of tacit acknowledgement. Taking courses takes away time from research and performing badly could cause serious problems, so it is understandable.
However, I find that after a year or two of not taking any courses, I feel less mentally "sharp". And sometimes I say to myself "I wish I had taken information theory/blackhole physics/statistical methods!" because I think they could open up more research ideas and plus some subjects are just hard when studied on your own. Finally, taking courses open up more employment opportunities, either through TAship or working in industry. I get the feeling that some/many advisors in academia (such as mine) are fairly indifferent about transitioning from graduate school to industry and do not appreciate the importance of courses/technical skills to these future opportunities.
Has anyone been able to successfully navigate through this dilemma? There are pro and con to each side of the argument and I could't make a move in fear of jeopardizing my relationship with my advisor.

I say to myself "I wish I had taken information theory/blackhole physics/statistical methods!" because I think they could open up more research ideas Study them alone. That can be brutally hard, but you'll learn more and become more mentally "sharp".

taking courses open up more employment opportunities, is there evidence to support that?

@user2768 Yes, but sometimes guided studies of condensed material could be beneficial.

I get the feeling that some/many advisors in academia (such as mine) are fairly indifferent about transitioning from graduate school to industry I suspect advisors are well aware that must students transition, they're just unaware how to transition or lack knowledge of industry, because it doesn't interest them

@user2768 Yes. The algorithm and machine learning courses are always filled to their maximum capacity. It is well known that these courses are populated by industry driven students.\

Right, but do they actually open up employment opportunities? Does taking a course make you more eligible for a position? I seriously doubt that, especially as it takes time away from research, which is surely the primary reason you'd be offered a position. As you've said, Taking courses takes away time from research.

Ultimately, I'd recommend finishing your PhD as fast as possible, that gets you to the next step in your career. (I'll stop commenting, because this is turning into a chat, for which comments are not designed.)

What are they going to do if you take a course? Fire you? Cut your funding? I would do it anyway and let them b1txh. I'm perfectly happy to quit my PhD program at any time though. One of the only reasons a PhD isn't a total economic loss is the free tuition, which means courses.

@FourierFlux - many moons ago, when I passed my qualifying exam, my tuition status was changed. Actually taking courses was no longer allowed (except for the 'thesis research' 'course'). Turns out under the accounting system, the department had to lay our real money to take courses and were happy to do it for the required courses, but would not do it for non-required courses.

«I wish I had taken information theory/blackhole physics/statistical methods!» you may be glad to recognize this lacuna as soon as possible; it shall be your trigger to acquire this knowledge.

If industry relevance is your concern, the problem might be smaller than you think, as you can safely remove information theory and black hole physics from your list.

How can you feel less sharp? I mean your research should be more challenging and teaching you more in depth than any module you could take. Are you afraid of putting in more work for research?

Maybe I am too cynical, but from what I have seen at my institution: "I get the feeling that some/many advisors in academia (such as mine) are fairly indifferent about transitioning from graduate school to industry and do not appreciate the importance of courses/technical skills to these future opportunities." is putting it mildly - there are definitely advisors that care only about you being useful for their research and producing papers to justify their grants. Sometimes it even goes further than that and they would actively go out of their way to sabotage other opportunities you may have.

When I finished course work I found that what I really missed about them was discussing the ideas from the courses with fellow graduate students. What filled in the gap for me was an informal logic and foundations seminar that another grad student had created. We would meet about once a week at lunch time, discussing articles, books, and occasionally each other's research. It was a nice way to avoid becoming narrowly focused on your own research topic, learn about things of broader interest, and keep an active sense of community. At the same time, it wasn't very time-consuming so it was never a major distraction from our actual research. Perhaps something like that could work for you. If there isn't already something like that in your department, you could always take the initiative and organize one.

This is a great answer. An advanced PhD student is somewhere between a student and a professor; students take courses, professors do not, but professors still (ideally) learn new things, so try doing this the way that professors do.

Audit the courses that you like, i.e., take them without receiving a grade/credit for it. You might not even need to formally audit courses, most professors will let you informally audit them. But as mentioned by others, research takes priority.

I did this a few times, and it must be said that few people auditing a course "for fun" will be there by the end of the course. Meetings with the advisor and research always seem to get in the way.

You should talk to your advisor about why you want to take more courses, and discuss which one-or-two would contribute to your development as a PhD student.
Your advisor is correct in the sense that it is easy to spend too much time taking courses for interest, and you need to focus on your research. Also, to some extent you should be able to study the material without a teacher now.
On the other hand, your advisor may have a different idea from you about what you are aiming to achieve by doing a PhD. It is certainly possible that they do not think enough about being in a position to transition to industry.
Things that are not of some reasonable use to your studies you should do in your own time. On the other hand, at least in the UK it is recognised that some time (about 10 days per year) should reasonably be spent on career-building CPD.

What does CPD stand for? Career/professional development?

Assuming you are getting a PhD:  Your advisor is correct.  Taking courses is not the purpose of a PhD.  A PhD is a research degree, and PhD students should do research.  If you want to take courses, you should be enrolled in a course-based degree.

In my time, going to academia was about education. I find the formulation "Taking courses is not the purpose of a PhD" a bit misleading. I think the learning process into breadth does not stop with entering a PhD programme. Perhaps better to say "Taking credit-relevant courses is not the purpose of a PhD". I agree with below answer that auditing may be the right way to approach this situation.

@CaptainEmacs Learning is certainly an important part of a PhD, but course based learning is not important to PhD programs.  Doing research is the primary vehicle for learning.  In many countries, PhD students take zero courses.

"In many countries" - such as? In Germany, some time ago, it was perfectly normal for PhD students to attend advanced courses and seminars. I have no idea how it changed after the transition to the BSc/MSc system (Bologna). Doing research is "learning in depth". Going to courses/seminars is "learning in breadth" and I believe education consists of both.

At some point in one's research career, you have to be able to teach yourself new things. And, the only way to learn how to do that is, well, by teaching yourself. And during your PhD is the time to do that because you have more time and fewer expectations that you will be up to speed by next week. So, I fully agree. Time to start learning how to learn on one's own.

@CaptainEmacs The French system formalizes training in your core domain toward the PhD thesis plus mandatory classes (e.g., research ethics, interdisciplinary courses across different working groups) and continuing seminaries that you are not only the co-worker of a PI, but member of an école doctorale / Doctoral school example Bordeaux.  And right in the first week I had to enroll in additional lectures and lab classes to learn about one of the techniques to be used as soon as possible, exam included.

@DanielHatton I started mine in 2009 also in the UK and opted to take a couple of graduate level courses and one masters level (as a refresher given a few years since my first degree).  Other postgrads at the same time had a fairly high required course load in the first year.  My current experience as a postdoc is that the picture is similar - DTC (doctoral training centre) students have to take courses, others are usually permitted or encouraged, at least up to a point (and I've even been welcomed onto one as a postdoc)

More to the point, if you're hoping to be awarded a PhD and you remain incapable of teaching yourself the material (ie: you still require a course and an instructor to assimilate new information) then it becomes difficult to justify awarding a degree that, effectively, certifies that the individual in question has demonstrated the ability to perform independent learning and research.

After I passed my qualifying exams and passed enough classes to get over that hurdle, I took additional courses pass/fail or credit/no-credit (with the consent of the instructor) where it was clear I could pass or get credit without it impacting my workload too much. I took a lot of courses this way and learned a lot. My advisor encouraged it. It also helped him out if the course he was teaching was shy a few people, he could ask me to register, too, if I hadn't taken it for credit already. I could have taken courses outside my area, but I mostly doubled down on topics that would reinforce my knowledge and understanding of my research: mathematical underpinnings, practical underpinnings, etc. It helped make sure that I came into campus every day and did work, went to meetings, and stopped by the office/lab.
I usually didn't take more than one course per semester this way in order to try to keep my workload sane and to be sure I was making progress on my research with the rest of my time.

If you want make an value argument that appeals to the research university's historical context and you have "enlightenment ideals", maybe appeal to Lehr und Lernfreiheit.
Probably someone could cite Weber / Foucault for evidence that enlightenment ideals are "problematic". Indeed if your advisor is hip they might also think that Lehr und Lernfreiheit is bullshit.
Regardless, I would like to see more Ph.D. students (in scientific fields in N. America who are being advised against taking courses in favor of doing research) ease up and just:

read along with courses they're into,

interrogate folks enrolled in courses they're into about the content of lectures they maybe had to miss (due to research obligations, administrivia, CS dept auditing policies, etc.);

just be present for courses they're into (usually oral consent from the lecturer will do, he/she might even be flattered, and won't even have to worry about assessing/grading your work!);

be honest about what they're doing (if it's safe) with their advisor (e.g., misbehave and own it) and their peers (invite them to audit with you);

go rogue (if it's safe) or just formally withdraw from the program to do what they want (e.g., doing research or building things or learning, but without any cognitive dissonance owing to the fact that "they are scared they should be ostracized from the group");

own the privilege of their choices and represent them to other Ph.D. students (e.g., if someone really needs statistical methods, they need statistical methods!).

I’m pretty sure you don’t mean “castigate” there. Maybe “interrogate”, but even that’s a bit harsh for the situation.


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Dealing with repetitive introduction sections - copy-paste-edit approach
I often find myself writing summaries of the same sub-(sub)-branches of mathematics in the background sections of my papers. With time, this grows somewhat tedious, since there are only so many ways to state the same definition, and the exercise is not particularly intellectually stimulating. It would be convenient to cite a relevant "canonical" reference and follow the notation therein, but unfortunately in some cases there is no such reference - this is especially the case for relatively modern subjects, where the proper notation and set-up are something of a moving target.
Would it be considered self-plagiarism (or otherwise objectionable) to copy-paste the relevant section from a previous paper and edit it to add any details that are necessary for the current application (and remove the ones that are irrelevant)? It does not seem to me that this approach causes any loss to the scientific community, but I would also not be surprised if automated plagiarism checkers might notice the overlap and complain.

Do you hold copyright on the previous papers? If not, are there few publishers of your work, or many?

Are these math papers or science papers with a math section?  Standards are different.

I'm not a mathematician and don't publish in math, but I do read some math-adjacent and applied math papers and so I'm somewhat familiar with these issues and I can tell you what is useful to me as a reader.
For definitions, notation, etc, I would quote yourself (copy) and cite. Make yourself the canonical reference for your purposes if it's something you've already done. Be explicit about what exactly you are copying, for example you might refer separately to the source for the original mathematics and the source for the notation you are using. If others use a different notation you can mention that when you cite them if you think it is relevant.
I would not repeat long sections of a paper; take only the most necessary parts and quote briefly. Let your reader know where they would find additional detail.
For repetitive introduction beyond that, I would think hard about what is actually necessary. Each paper should not be a review of the field, and there should be enough difference between papers to emphasize what is actually necessary to begin to understand the paper.

Actually, plagiarism and self plagiarism are easier to deal with than copyright, assuming that you have given that up in publication. You can avoid all forms of plagiarism with citation of the earlier work. You are giving proper attribution (covering plagiarism) and providing complete context of your current work rooted in the earlier (self plagiarism).
Plagiarism checkers aren't your real issue here, provided that they don't lead to rejection by an automaton. An editor can simply let you know of the similarities to your earlier work. It isn't like a professor grading an essay.
If you don't hold copyright and don't have a rather permissive license from the copyright holder, then you are limited in how much you can copy-paste and even, to some extent, how much you can paraphrase the earlier work. A solution in such cases is to ask the earlier editor if they will give you a more permissive license. They might do that, especially if you are submitting to the same journal. It might be essential to do this if you are extending the earlier work.
But, note one thing more. For things that can be expressed in only one way, copyright is hard, maybe impossible, to enforce. If you need to restate a definition or formula from an earlier paper, even if it is by someone else, the "length limitations and quoting requirements" don't apply to the same extent as in prose. Axioms are hard to restate differently from the original as is the formula defining the derivative. While those are very important and may have been very creative, if you want to extend the ideas, as you are always permitted to do, the original formulations need to be kept or chaos will ensue. This is especially the case for a lot of mathematics. Editors will understand that. Copyright is about expression and reuse can be limited. But ideas, themselves, can't be copyrighted or owned. When idea merges completely into expression, things are different. You still need to cite as appropriate, of course.
So, cite earlier work as you should always do. Seek assistance from editors, especially if they note too much similarity, and try to avoid being too pedantic, which can easily occur when overusing copy-paste.

It is probably too strong to say that copyright can't be applied to things that can be stated in only one way, though IANAL.

As you pointed out, if you copy-paste with a little edit you probably will be detected by the automatic plagiarism checkers. Unfortunately, the real problem is not the ethical or the productive one but the similarity percentage your paper could trigger. Editorials often run plagiarism checks with no context of the sections of the actual text detected as "plagiarism". Commonly, the paragraphs or sections that are detected with the most overlap in a paper are the background or preliminaries section which mostly contain definitions of the theoretical background for the paper.
You have two options.

Write the definitions again. Yes, your original problem. Change them enough just to not obtain a 100% similarity score.
Cite yourself indicating which parts of the text are used in the new paper.

I understand why you think that self-copy-paste is not a big deal but consider the lack of context the editors, reviewers, and automatic tools have when determining something as plagiarism. The current copyright practices in scientific publications also don't help.

In math, your #1 is counterproductive and unnecessary for some things. It can lead to chaos. For example, restating definitions in different words can lead to different interpretations. The point is precision, for which the essence of the statement is its form. Moreover, in the broader, beyond math, context, paraphrasing doesn't guard against plagiarism, though your #2 does.


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How do I respond to a review of the wrong version of my paper?
I have just had the revised version of a paper rejected from a well respected journal.  However, one of the reviews is clearly a review of the original draft of the paper, rather than the revised version.  I pointed this out to the editor, and his response was

Please revise your paper again, and submit it as a new paper, by mentioning your old paper ID number, together with replies to reviewers.

I don't see how I can reply to that review.  How should I proceed?

As a side note: make sure that your hypothesis (the review is of the original manuscript) is not due to observational bias on your part. This happens more than we know, even if we try to account for it (Murphy’s law). Maybe you could ask a friend to read the review and give you an outsider’s opinion?

No, the evidence is objective and unequivocal (see my comment to your answer).

Concerning the "submit it as a new paper" part, I strongly recommand http://svpow.com/2012/10/03/dear-royal-society-please-stop-lying-to-us-about-publication-times/
This is a bad practice aimed at making turnaround times better. It seems widespread, but still very unhealthy.

Shocking example Benoit.  The first version of my paper was submitted Nov 2011, so if I do resubmit, at best the apparent duration of the review process will be wrong by almost a whole year.  What happens if someone else publishes the same idea during the "missing time"?

Consider the pragmatic approach to this issue: you have already revised the paper according the reviewer’s suggestions, so only the response needs to be worked on. (Apart from other comments orthogonal to this main issue.)
Given that you already called the attention of the editor on the problem and he did not follow up on it, you clearly need to weigh carefully your response (unless you want to pick up a fight). I would suggest that you write a very detailed response letter, in which you explain how the manuscript addresses the reviewer's comments and questions. Do it point by point, quoting all the parts of your manuscript that are relevant (and locating them: page and line numbers).
Just do so by being slightly evasive about the exact evolution of the manuscript. Where you would normally say “we have added a paragraph at the end of section B”, just say “the revised manuscript includes a paragraph addressing the reviewer's question at the end of section B”. It's not untrue, though it is not perfectly clear.

Also, consider going for another journal. If you believe the editor has been treating somewhat badly, you might want to just let it drop and submit somewhere else. It's a complex decision.

Or, if you think it's worth it: make a stand for it. Appeal the editor's decision to the editor in chief, on the basis that a factual error was made. You need to be both strong and diplomatic in your appeal, and back it up heavily with facts. It will be easier if the incriminated review actually quotes some text that has changed in your revised manuscript, in which case the error is evident.

It was the EiC that responded, the action editor is anonymous.  It is easy to show that the review was of the wrong manuscript as it points out an incorrect acronym, which I amended in the revised manuscript, and there is also a direct quote from the first draft that isn't in the second draft.  Essentially I have already tried the third approach (and was diplomatic and gave the evidence with page and line numbers).


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What are Industrial Case Studies and why are they required?
I searched on the internet about this but found no convincing answer. I have to submit a paper to a conference where they are also asking for an Industrial Case Study. I contacted the conference chair about this, they replied:

If there is a pragmatic solution to a problem or there is an innovative application developed then an Industrial Case Study is required. For more details you may search for case study submissions for similar conferences.

The domain is KDD (Knowledge Discovery in Databases). I did search but did not find any good stuff.
Could anyone help me out with this?

I basically want to know what are they ? Why are they needed ... are the different from a research paper ?

An industrial case study is when you take your solutions or your processes, apply it to some situation in industry, and then report on the results. You might use the Case Study Method research method if what you propose can be appropriately investigated using a qualitative case study. This is appropriate if you're in management sciences, human-computer interaction, and software engineering (which is what the Damian paper referenced above is about). Note that industrial case studies of this type are VERY difficult to do and take an extremely long time to write up and report on, so I actually doubt the conference is looking for this kind of case study.
Alternatively, in the field of Knowledge Discovery of Databases, an industrial case study may simply be applying your algorithm to a database that is used in industry to discover something novel or meaningful. This is more common for analytical fields in general and simply requires you to acquire a database or some other data source that was generated through industrial activity.
The reason for "industrial case studies" is because a lot of people create methods and techniques and test it only with "fake laboratory data" that they happen to create or acquire. This fake laboratory data doesn't reflect very accurately the real world and people want to see that techniques in fact work on, affect and influence the real world.

I contacted one the authors on the papers UV-D mentioned and according to the author:

"Industrial Case Study"  means the results presented have been validated in actual plant operation, they are not only based on theoretical considerations.

I even asked this question on Google+'s Machine Learning Community. You can have a look at the converstaion here.