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How to ask dumb questions
I am having trouble asking questions in seminars, conferences, and public talks.
As a graduate math student I often fail to keep up with speaker and more mature members of the audience at events like seminars and conferences. It is very frustrating to lose track of the talk simply because I am not familiar with one key theorem/definition referenced by the speaker.
On the one hand, many people encourage me to raise my hand whenever something is unclear, motivating it by "no such thing as a stupid question" quote.
On the other, very often I observe people being annoyed by a student asking an "obvious/elementary" question and wasting everyone else's time.
I have seen several related posts on this website, most of which are asked on behalf of the speaker.
However, my question comes for the opposite side of the barricades.
I want to know how to find balance between not annoying the speaker and the audience too much on the one hand, and catching up with the talk on the other hand.
More formally, my questions are:

Are there any (semi-, non-) official recommendations on professional etiquette for mathematicians?
I understand that every situation is unique and highly subjective. However, I would be glad if someone gave me advice on when to ask, how to ask, and what to ask at research talks.
How can I tell if my question is "dumb" (i.e. the answer is well-known or searchable), or if it addresses legitimate ambiguity?

In particular, how can I quickly determine whether a definition/theorem/lemma mentioned by speaker is a part of common knowledge?

All relevant links or examples are appreciated.

Having reviewed over 2000 first posts on this site, I must say this is one of the best questions I ever reviewed. I can imagine many graduate students would have the same problem. One sugegestion, start your question by saying "I am a graduate student." so that everyone in the room will understand.

The answer to the question may depend on how tight the time slot for the talk is. If if is very short, then the speaker may simply not have the time to give a relevant definition that was not originally planned to be given. Just to let you know.

A related question, which will probably be worth reading the answers to is http://academia.stackexchange.com/questions/34187/what-can-i-do-to-be-able-to-understand-conference-presentations/ While it does not answer this question, it is related because the more you understand, the better you can judge whether a question is "stupid".

In particular, one of the things mentioned there is to prepare by reading some abstracts (when available), which will allow you to have looked up some of the definitions beforehand. Also note that an important factor in whether you should ask the question is whether there is a good chance that getting an answer to just that one question will really help you understand the rest of the talk.

Posting as comment as I'm not sure if this answers your question: I generally avoid asking questions unless I'm sure of the value of the question. Now, if I find myself in a domain that I am both very interested in and am very unfamiliar with, then I take notes diligently. I practically copy the slides and transcribe the speaker's words. I may then go through the notes and discern the finer points at my leisure. If required, I may then correspond with the author and/or other experts.

I do wish people would stop saying "there's no such thing as a stupid question". It's simply untrue. In any given context, there are stupid questions. And given almost any specific answerable question, there exist contexts where it is a stupid question.

One thing you should consider is what your question will achieve in the context of the given forum. Questions that are appropriate for a presentation at a research group meeting may be considered inappropriate for a short formal conference talk, and vice versa.

It would be great if you could find a seminar buddy.  You sit next to each other, you write a brief question on your pad of paper, tilt it toward your buddy, and your buddy scribbles a quick answer, or gives you a nonverbal signal -- shoulder shrug, shaky of head, thumbs up.

Remember that sometimes there are other people in the audience who wanted to ask the very same question, but were afraid to do so. But in other cases the question might indeed be so basic, that asking it would show that you don't have a chance of understanding the talk anyway. If you only need to ask a single question during a talk, and if that question can be answered in less than 10 words, and if that answer makes the difference between you understanding the rest of the talk and you learning nothing from the talk, then you should not feel stupid for having asked it.

@EnergyNumbers Yes, in theory there are such things as ``dumb questions''. In practice, I have seen many talks where the speaker lost all of the audience, and no one dared asking a question, but I have not yet seen a single talk that was derailed because of someone asking too many dumb questions. It seems that the real dumb questions aren't a problem yet.

Ask a math professor teaching complex analysis whether changing from 'i' to 'j' will matter. Ask him if 'i' stood for current or not just to be safe - Those are some real questions I have heard coming from other students

@IllegalImmigrant I can expect the math professor teaching complex analysis to be disappointed that their students are so confused, but that does not mean that the questions are not appropriate! A good professor is happy to get such a question. It will save them from wasting time lecturing students on, say, the Riemann mapping theorem when students do not know what 'i' is.

Do speakers care whether the audience understands the talk? I'm not sure the answer is always "Yes".

People might feel like you are wasting their time if you ask a question to which the answer is obvious to them, but you're wasting your time and money if you don't ask, and there is every possibility that there are others present who have the same questions as you, but are afraid to ask them because they seem "dumb".

As a postdoc in math doing interdisciplinary work, my collaborators and I have adopted the phrase "every question is dumb because we are all dumb, so ask your dumb question." Ask your questions.  Another option is to do a quick google search on your smartphone/laptop in real time and then ask the question.

@BorisBukh stupid questions are a real problem already. I've seen conferences, seminars and workshops lose valuable time to stupid questions - in extreme cases, even the whole Q&A session. And while there is some utility in having fools flagged up early so that they can be bargepoled, we'd still all be better off without the stupid questions.

Conferences are not for learning. They are events. If you want to learn, read the paper, or book if there is one.

@qthrowaway I disagree. A conference is where you can learn things that are not in books or papers yet. Okay it's a different kind of 'learning' to picking up a text book but I would not categorize it as "not learning". Aside from networking I'm not sure what else a conference offers.

It may be better to attend these events as a group and when clarity is needed, to elect someone from the group to raise the question. This way you have the support of the group along with actually asking a question which would benefit the group even if it is "wasting" the time of the other attendees.

@IllegalImmigrant: actually changing 'i' to 'j' matters a lot in the scientific literature, because physicists and engineers use opposite conventions for phasors, that is, i=-j, and this is a source of confusion between the two groups.

Not an answer, but a great article about The Importance of Stupidity for graduate students (and, really, everyone)

@EnergyNumbers : what happens if you are that "fool"? What is the consequences and what can be done about it to modify it? Moreover, if there are "stupid" questions, I suspect this is not a precisely-defined term, so if you need clarity, why not determine it by experiment? "Breach" a little. If the problem is wasting time then just limit the number of Qs you ask. As with the other answers here, if you find yourself having to ask too many, you probably won't understand the talk anyways and should just leave. Then nobody's time is "wasted".

@The_Sympathizer oh, there most definitely are stupid questions, given a specific context. Given almost any context, then there exists some stupid questions for it. Given almost any question, there exists a context within which it is stupid. Stupid people ask questions that are almost always stupid. Q&A time is precious, and not to be squandered by fools and their experiments. As with swimming, so with a Q&A session: one should know how deep the water is that one is swimming in; if one is completely out of one's depth, one should go somewhere else.

@EnergyNumbers So what hapepns if you are the asker of those questions and you are the fool? How do you "deserve" to be treated? What happens if you are not the fool, but decide to not treat the other "fool" with what they "deserve"? And what I said was not that there were/weren't, but that the way to find out is to ask it anyways, but mind the time. Ask 1-2 questions at best, no matter the content, if that's not enough, then you should just not bother trying to understand it at this time and it was probably too deep.

There is no such thing as a dumb question is a good adage for the classroom, where our mission is to teach students, and we have a number of weeks to accomplish the learning objectives. We use this maxim to encourage students to ask questions rather than fall behind.
However, there is such a thing as an annoying question can be an equally true corollary, particularly in a conference setting where someone is trying to cram months worth of research into a 45-minute talk in front of presumed experts in the field. In such cases, it might be preferable to not derail the speaker's presentation.

How can I tell if my question is "dumb" (i.e. the answer is well-known or searchable), or if it addresses legitimate ambiguity?

Really, there's no way to tell for sure during the talk. However, if you venture to ask your question, you can preface it with something like:
"I'm sorry if I'm asking something obvious here, but..."
So far, my answer doesn't really differ from some of the other advice you've gotten in other answers. However, I want to address the professional etiquette part of your question. While you are asking your question, and in the immediate wake of getting the question out of your mouth, pay careful attention to the body language of the audience at large. If you see several heads nodding affirmatively, that might be a good indication that you were brave enough to ask something that was nagging in the minds of everyone else, and the speaker has made some erroneous assumptions about what was presumed to be fundamental or obvious. However, if you notice some sideways glances accompanied by grimaces or eye-rolls, then maybe you've touched on something that would be better left until the end of the session, or until the next break, in which case you can quickly add:
"If you'd rather discuss that with me off-line, that's okay."
In summary, be aware of your environment:

What is the purpose of the talk?
Who is the intended audience?
What are the speaker's time constraints?

Moreover, be cognizant of non-verbal audience reaction to your initial question, and use that as a barometer before venturing to ask follow-on questions. In my experience, people are rather forgiving the first time a presentation is interrupted by an elementary question, but they begin feeling exasperated when that one question transforms into a hijacking of the presentation as a whole.

I really like the "read the audience" part of this answer. Nodding heads or exasperated looks are easy enough to distinguish.

Wouldn't perhaps the better method not be to screen for content of the questions, but rather for how much time of the presenter is taken up by asking them? I.e. if you're asking too many questions, to the point that it's being bogged down, then that's too many. But if you only ever ask one or two questions, it shouldn't matter how "stupid" it is or isn't (And moreover, if those one/two questions take the whole thing to answer and the lecturer actually chooses to go that far, chances are that means they are NOT "stupid" questions!).

Moreover, the term "stupid question" does not admit a well-defined meaning, and thus one should/can only (it is only possible to) learn from experience and don't be afraid to make mistakes and yes, do "bad" things - that's what being human is about. Just ASK the damned things, but limit the number.

In theory, trying to read the audience while asking a question is great advice. Personally I find that very hard to do though--depending on the number of people in the room as well as their seniority, my heart will be racing as soon as I raise a hand and then a lot of focus will be required basically to not stammer..

Keep asking the dumb questions! It is better to look like a fool, than to be one.
You worry that many speakers are annoyed at the `elementary' questions. Some speakers do it because they are stressed about public talking, and any question upsets them. For some, communication of mathematics is not the aim of the talk; they give it because it is a condition of travelling to the venue, or simply because everyone else gives the talks. Instead of feeling joy at an opportunity to clear the confusion, they might get annoyed at having to do the extra work of explaining some of the background. The annoyance has no long-term effect --- nobody holds grudges for asking dumb questions. To give up actually understanding math for such a petty reason is just not worth it.
There is only one situation in which you should refrain from asking a question. That is when you are representative neither of the actual audience nor of the intended audience. So, if you are graduate student at your department's colloquium, it is OK to ask anything. If you are a graduate student at a seminar in your field, it is OK to ask anything. If you are at a seminar in another field, and there are several other students in your field in the room, again it is OK to ask anything. Only if you a lone outsider at a seminar or a conference that is not in your field, there is a reason not to ask questions.

This is good advice.  I will add that "when you are not the audience" works the other way too.  For the first year or two of my tenure track faculty job, I attended the Graduate Student Seminar.  There were a lot of talks in different areas of mathematics, which I enjoyed, and I thought "Well, if these talks are meant to be understood by students, then I should really be able to understand them.  So a few clarifying questions have to  be okay..." It took me way too long to realize that my questions often pulled the talk away from its intended audience and actually had an intimidating effect.

Well, I can think of other cases where asking (more) questions is not a good idea. For instance, if a talk is really hard to follow and/or ridden with mistakes, and any answers you get only exacerbate the confusion, it may be a better idea to just let it go instead of prolonging the agony. But yeah, that's good advice IMO.

There has been a rare occasion where someone who wasn't paying attention will ask a question that was just covered. I find that incredibly annoying, both as the speaker and as a listener. Please don't ask unless you're actually paying attention.

@DoubleDouble: I somewhat agree, but sometimes when you are paying attention, just lose focus for a second (because a colleague asks you something, or because you focused on a different part of the presentation), you still miss something, sometimes even realising that something was just covered and you missed it. In this case, I don't think you should refrain from asking just because of that.

"You worry that many speakers are annoyed at the `elementary' questions." - it's indeed rather the audience than the speaker that I'd worry about. As a speaker, I am usually relieved when some of the questions I get are actually easy to answer for me. It gives me an opportunity to explain something that may have been mentioned before in different words. And quite possibly, I'm too mentally busy to think about whether it was a really sharp question, anyway. In a way, I actually prefer somewhat basic questions about my topic to some other types of questions I regularly get - questions about ...

... possible future extensions or shifts of focus, which are so far away from my current contribution that they have to be considered a different topic, and also certain questions about the premise of my work, which "hijack" the Q&A session on my paper to become a Q&A session about another paper (written by someone else) that I was merely building upon. Both of these latter question types leave me as the speaker with little insight into the thoughts about my concrete contribution. To the audience, of course, these two latter question types might be more interesting.

Before asking a question, ask yourself the following:

If I get a nice detailed and understandable answer to this question, will I be able to understand a significant part of the rest of the talk?

If the answer is "no", then you should probably not ask the question even if there really is some ambiguity that could be cleared up, because chances are that asking the question will not benefit anyone (i.e. that those who are able to understand the talk would also be able to answer the question themselves).
If the answer is "yes", then that is a good start. In that case you should probably ask the question unless it is something you really ought to know (if you don't know that it is something you should know, then that is usually good enough). Of course you don't want to become known as "that annoying grad student who keeps asking trivial questions", but you also don't want to miss out on learning something from the talk because you are missing some small detail, and usually the above will limit the number of questions you ask sufficiently that people should not be annoyed.

It seems hard to predict in advance whether you'll be able to "understand a significant part of the rest of the talk"--if it's anything like the talks that I've had trouble following, you're not always lucky enough to know whether a reference to some concept you haven't heard of is a sign that the rest of the talk depends on that concept.

@MiloPrice Yes it can be hard to predict. But a metric which you can only apply retroactively is still better than no metric at all. Additionally usually there is some sort of understanding between speaker and attendees about what the prerequisites for understanding the talk are.

@MiloPrice I think the best you can hope for along those lines is to use inductive reasoning to make a reasonable guess as to whether that definition is a key to "[understanding] a significant part of the rest of the talk."  It won't always be right, but you'll learn more about yourself by trying... and that's worth a lot, in and of itself!

@MiloPrice Indeed, it is not a perfect method, and if in doubt, I would usually err on the side of asking the question.

@MiloPrice Well, there are some clear cut cases where the answer is unequivocally no.  I still ask those questions.

@AaronHall The answer does not at all depend on the comments mentioned, so I will just remove the first part when I get to a computet.

Fixed it for you.

I have recently finished a PhD in Particle Physics. Over my time as a student, particularly early on in my studies I frequently encountered this problem.
I would start by pointing out that, generally, you won't be the only student in the room and there will almost certainly be others thinking of the same 'stupid question' but not asking. Many times these questions arise due to the incompetence of the speaker and not the audience. Secondly, it is an academic environment, questions should be encouraged. Those academics who don't encourage bright young students are the problem, not you.
Saying that, the timing of the question could be considered. From my experience I found that in a seminar it was best to ask questions from the audience which were relevant to the topic, perhaps not so much for your personal understanding but to encourage discussion. More often than not there will be ample opportunity to ask the speaker personally in a coffee break or at dinner, maybe take these opportunities to understand the details you are missing. Often this approach will allow you to spend more time gaining an understanding and benefit you.

Thank you! The last paragraph sounds extremely reasonable to me.

First, it is important to identify whether you are the intended audience for the talk or not.  If you're not the intended audience (you're going to a seminar well outside your subfield, you're a second-year graduate student at a conference that's mostly not graduate students), then you should be careful not to annoy the audience.  However, if you are the intended audience, then you're doing everyone a favor by asking more questions.
Second, if you're generally good at reading social cues, then you should trust your instinct and feel free to ask questions until the speaker says something to shut you down ("we can talk about this afterwards", "that's standard material", etc.) or you notice the audience being annoyed.  If you're not as good at reading social cues, then be a bit more careful and try to get honest feedback from a friend about whether your questions were out of line (e.g. "I'm not so great at reading social cues, so I was wondering whether you could honestly tell me whether some of my questions went too far.").

I'm not in your field, so I'm not sure if this'll be helpful or not.
When I was a grad student, I convinced myself that if there was a talk I didn't understand, then it wasn't my fault, but the speaker's, and that there would be other people in the room with the same questions and confusion as me. It might have been overconfident of me and I might have been wrong, but I nevertheless acted according to that belief.
So I was always "that guy" in the audience who asked at least one question after every talk I attended, and many of my questions were just asking for clarifications of things I didn't understand, and if the answer didn't help, I wasn't shy to say "I still don't understand".
And sure, I often wondered what all those big shot professors in the audience were thinking when they listened to me asking all these questions. I still don't know. But after a session at an annual meeting of our society, in which I had again asked a lot of questions, a Berkeley grad student came up to me and thanked me for daring to ask all my questions, which she said helped her a lot in understanding what the presenters were trying to tell us. And she was a pretty smart student to begin with.
It's possible that you don't understand things you should understand, but you should have a reasonably good estimate by now whether you are a normal, brilliant, or terrible grad student. If you're in the two former groups (which I'm sure you are), I can pretty much guarantee you that there will be others in the audience (professors and grad students alike) who will appreciate your questions.
As to what to ask: If you don't understand the presenter's talk, then there must've been a certain point where you got confused. Ask a question about that very thing that made you lose track. It may or may not help you in understanding the rest of the presentation, but ask anyway, because others will have been confused by the same thing.
And for the record, I'm a professor now, but I don't understand more of talks than I did as a grad student ...

Unfortunately most attendees at most pure maths seminars do not understand most of the talk. A good talk aims to make the first third comprehensible to phd students, second third to faculty, last third to experts. Most talks do not achieve this.
You are generally expected not to ask questions unless they are to the point which is very hard for a phd student.
I try just to get a few ideas across in mine recognizing that no one will follow details. However, I sometimes get sarcastic comments that my talks are too "philosophical." I also sometimes get great compliments.

While a given person is not usually expected to ask questions, they are not typically discouraged.

my experience was that asking basic questions about the material was discouraged.

Asking basic questions about things introduced in the talk is fine in most seminars I've been to.  E.g., speaker defines moufang loops, gives an infinite example and starts to move on.  Q: Can you give an example of a finite moufang loop which is not a group?

yes but if he says a moufang group is a Lie group and you ask what a Lie group is, you'll irritate the senior academics present who expect you sit at the back and shut up.

I always heard "1/3 understood by everyone, 1/3 understood by specialists, 1/3 understood by no one."

The balance should be struck in terms of usefulness. Let us assume that the question is genuinly dumb, it might still be useful, first to you if the answer will really allow you to make a significant breakthrough in your own understanding of the field. It might also be useful to other persons like you in the audience, who likewise lack some information to follow the talk properly; it might also be useful to the speaker, who has failed to realize a fraction of is audience is not aware of such and such premises (s)he relies on. In other words, any question is going to cost at least one minute of the speaker's time, but the gain might outweigh the penalty even if the question is dumb.
Typically, even a dumb question may enlighten someone in the audience because it gives them the opportunity to realize that the matter discussed  opens itself to some level of misunderstanding when perceived from a different perspective than that intended by the speaker. In the end, exposing such different views is nowadays one of the main reason to attend talks in person.
Then there is always the possibility that the question is not dumb :-)
So the bring-home message is "Bite your tongue, and ask yourself, how useful will my question be?".

For the case you mention of being unsure of a definition or a named theorem, your phone is your friend. Look it up. It should be possible to gather whether the concept is well-known, for example if it is mentioned in a course, a survey, or on Wikipedia. If it is, and the mention is in the research seminar's area, that suggests the question would not be welcome, as the audience is reasonably expected to be familiar with it.

This sounds  to me like a good advice in principle but not in practice. If you start concentrating on Wikipedia you are likely to loose track of the talk? It does answer the question asked by the OP though!

Usually, the in-depth questions come from people who are familiar with the topic. So, if your intent is to play an active part, you can read up on the work of the speakers who will give the most interesting talks. I would not worry too much about the speaker getting annoyed by questions. From the speaker's perspective, what is more annoying are all those famous professors who the speaker wanted to make an impression on, who are sleeping.

There are a couple of factors I use:

Have you asked a few questions already?
And if so were they well received?
If you have, and they were not, you get away with a lot less, and rightly so. Most of the etiquette is not hogging the speaker. You can avoid this by either asking questions that help everybody or by not using too much time. Either way you are safe.
Did you give it long enough to know you are roughly at the right level?
If you followed the majority of the talk thus far, and that was a non-trivial fraction, then it's reasonable to assume the bit you're stuck on, others might be too. Even if this is not the case, other's will recognise that you have restrained yourself at least a little.

If you are 'clear' on both points, I think it would be hard to object to your questions.

The literary technique known as Lampshade Hanging may be useful here: specifically call attention to it yourself, acknowledge it, and then just move on.

"So in the interests of making sure everyone's on the same page here, let's ask the obvious question first: are we talking about African or European swallows?"

This looks intriguing but I don't completely understand what you're suggesting in the seminar context.  Could you expand on this a bit, please?

@aparente001: By pointing out yourself that you're asking (what could be seen as) an obvious question, you beat any potential critics to the punch. It basically helps to deflect criticism preemptively, because now if someone says "that's a really obvious question!" it makes them look bad because you just said that.

I see.  Thank you.  Well, I will suggest a slight modification: "In the interests of making sure I'm on the same page here, are we talking about African or European swallows?"  (Reason for modification -- the OP will probably want to avoid seeming to be claiming to speak for others.)

One simple heuristic I use is considering how long it will take to answer the question. For example, if I don't understand the notational conventions or assumptions in a talk and they seem important, I might ask -- even if it's an obvious question, it can be answered in a sentence and won't waste too much of everyone's time. If it's a deeper or more substantive question, then I might save it for the end of the talk or take it offline.

A good way to ask a dumb question is as follows:

Sorry, this is probably a dumb question, but [ask question here]

for example

Sorry, this is probably a dumb question, but isn't a differentiable function continuous?

This signals to the audience that you are just having a momentary lapse.
Another tactic amongst all the people who wish to appear intelligent is to use the word "surely", eg.

Surely that theorem is not true if you don't have any regularity?

Please do not strive to appear intelligent. Freely admit your ignorance and stupidity --- nothing is more intimidating to a student than a bunch of professors who want to appear intelligent. Do not use other pretense tricks either --- if you have a memory lapse, admit it, but if you don't, then don't fake it!

@BorisBukh I am not saying that one should do it. I am just saying that this is what happens in real life, and my post tells you how to do it if you wish to emulate it.

Your answer starts with "A good way ... is". I felt that I had to make a strong warning since the answer did not come with one.

Actually, I think the "Sorry, this is probably a dumb question, but ..." part is an ok way to do it, if you're feeling too insecure to ask your 'dumb' question without such a preamble. In most cases, the speaker will answer that it's not at all dumb, thus encouraging more people who feel like they have a dumb question to ask it. BUT I won't upvote this answer because the other suggestion "Surely, ..." is just horribly impolite towards the speaker, because it assumes that the speaker got something wrong instead of leaving the option open to explain why he did it that way.

Yes, the theorem is true even without regularity, and don't call me Shirley.

@Sumyrda If you spent any time in the UK, you would know that the second suggestion is used by all the pseudo-intellectuals. as I said I am not recommending this course of action. But I think downvoting my answer is a bit unfair since I have given practical advice that the OP requested!!


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PhD. in the US directly after European BSc., or is it better to go for a MSc. at first?
I'm currently in the final year of my BSc. in physics and the second year of my BSc. in mathematics at a university in german-speaking Europe. I'll have my degrees finished in summer 2018.
Eventually, I'd like to end up in theoretical or mathematical physics research (most probably gravitational physics) and would love to experience some time at a foreign university. Firstly, since my university is stronger in the experimental field and has a relatively small theory group and secondly for the sake of broadening my horizon. The US (or generally Northern America) would be particularly interesting, because of the somewhat "more open" research culture, in comparison to a highly bureaucratic and kind of inert (and often poorly financed) apparatus in german-speaking part of Europe (and probably in the whole EU - at least that's my naive perception of it, or am I wrong in this?).
I read that the PhD. in the US is a bit different to the one in Europe, e.g.  that a MSc. is kind of integrated in the coursework-heavy first one or two years of a US PhD. Nevertheless, there are some interesting MSc. programs in Europe, e.g. in Edinburgh or at the ETH in Zurich.
So, my questions would be (from a rational/scientific-career point of view):

Is it reasonable to go for a PhD. directly after a BSc. in the US? Or would it be more sensible to pursue a MSc. in Europe first and only then head for a PhD. in the US?
Small side question for physicists since I'm just researching into this: Which universities in Northern America have a strong gravitational physics department (aside from Stanford and Princeton)?

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Should I answer email from a student who based her project on one of my publications?
I published a paper that received a bit of attention several years ago. A few months ago, I received an email from a student who expressed interest in my work. I don't know her, she is not in the same institution as I am, and she appears to not be located in the same country. I thought I was corresponding with a peer.
Later, I found out she is an undergraduate student who is doing her own research project with my model. She has been sending me emails asking very elementary stuff which she should have figured out herself and sometimes she tries to urge me to reply by implying what I did in my paper was wrong.
I answered a few of her emails just to clarify that what I did in the paper is correct. Moreover, from time to time, she said that she has severe depression and the deadlines were close.
What is the professional and ethical way to deal with this student? I don't want to waste time on this student, to whom I have no obligation to train, but I also don't want to increase her depression.
Currently, I plan to stop answering her emails.
Edit 1: After reading @Buffy's answer, I realized I forgot to mention an important point. At first, I thought the student was somewhat experienced in the field. After a while, I found out she did not know various basic concepts in the filed, so I pointed her to various introductory materials. However, the next time she asked a bunch of elementary questions that were covered by the introductory materials, as if she had never read the materials. Several days ago, she said she just wanted to finish thesis on time, suggesting she would just throw all things away after her thesis was finished. I felt very discouraged at the moment. At first, I thought I was communicating with a potential collaborator. After a while, I thought I could inspire a curious mind. Now I found out that she have never been interested in the things I discussed with her.

The notion of her (alleged) depression is also possibly a (maybe subconscious) emotional blackmailing technique which some people use in many ways all their lives.  "Something bad will happen me if you don't fix my problem for me".  This is the way some people avoid taking personal responsibility, sometimes consciously sometime not.  Big red flag that you need to walk away from this person.

@StephenG-HelpUkraine this is not a helpful way to respond to allusions of sever depression. I agree that the students depression is not OPs responsibility, but none the less, it's 100x better that they are telling someone that something is wrong, even if it's not the right person right now. OP may be in a position to redirect that conversation over to the right person and potentially mitigate serious harm.

@Clumsycat The OP could make such a suggestion to this person, but that doesn't change my viewpoint.  My experience of depression (which is  from suffering it myself) is that people who are severely depressed do not email other people in this way , but in fact hide their depression.  People who ask for help with their own work (from a stranger no less), and link the request to their own claimed depression are far more likely to be manipulative people.  It's hard to believe that the email correspondent would be aware of their depression but unaware that this needs medical help.

@StephenG-HelpUkraine: Depression can express itself in a wide range of ways. Hiding/denial is common, but it’s far from universal, and using it for manipulation is also not uncommon.  From the OP’s description, the student’s descriptions of their depression sound certainly manipulative, but quite possibly also true.

I agree with Clumsy cat and PLL. One person's experience of depression being a certain way does not mean that all cases, or even most cases, are the same.

@StephenG-HelpUkraine No! People who say they are depressed are almost always genuinely depressed. Depressed people hear "you are just saying get attention" all the time. If you do not have the recources and the time to help them, that's fine, do what you can, but don t tell a depressed person they re making it up. That s the worst thing you can do!

@Nico did you notice the line which is from suffering it myself in one of StephenG's comments?

@Esther no, i missed it. thank you! i only read their first comment.

"Now I found out that she have never been interested in the things I discussed with her."
I am sorry to hear that.
I agree with @StephenG-HelpUkraine you were being played by someone manipulative When someone you do not know openly talks with you about their own sickness/mental problems/issues ... 99% of the times is a huge red flag.
And even for the genuine 1%, what can you do?
Now you see a bit of the background to my answer ... unfortunately it takes some burn to learn to deal with fire.

I guess I have a more expansive view of "duty" here. While you may not be paid for any interaction, IMO a professor has a "duty" to help people who ask them for professional help if done sincerely. It is part of what it means to be a "scholar".
But you can also make it less of a burden by pointing the student to some resource(s) that they can study to bring themselves up to date. You don't need to give them a course via email.
Imagine a different situation in which some eminent scholar from a related field sent similar emails asking for help? They don't have the background you have and don't understand some things. How would you treat them? Would you consider it a "waste of time"?
Personally, I'd be quite pleased if some undergraduate decided to follow up on something I'd done earlier. And, if they had serious misconceptions, then I'd be glad to set them on the right track. But not by giving them a university level course via email. I might suggest that they consult with a local professor to help them learn. Lots of "low impact" possibilities.

I wish I could upvote this more than once, especially the part about "pointing the student to some resources."  By doing that instead of answering elementary questions, you will help this student grow and learn to be independent.  That may, in fact, be more trouble than just answering the questions, but see Buffy's definition of "scholar."

Yes! Thank you! Students are also human beings, and should be treated as such. I was really afraid to read the awnsers here, but yours is really nice.

Yea i like this answer a lot. Just because someone is incompetent doesn't mean they are evil. If they mean well then telling them "you are making trivial mistakes but you have hope" and then pointing them to constructive resources would be the right thing to do. "Shutting down"/"ghosting" stupid people before offering them a path forward doesn't really benefit our society and exacerbates the image "academic elitism" for people not already in the circle/aware of how to interact with academia properly.

I do agree with you though I don't think it applies to my case. I replied to the first several emails with enthusiasm. Later, she started to ask elementary questions as if I was her supervisor. (Actually you don't want to ask questions so elementary to waste your precious time with your supervisor.) It's like my paper was about an efficient method to perform Monte-Carlo Markov Chain, she asked me to show her to code to plot the marginal priors (I posted all my code on github and specified all priors in the code!). How should I deal with it?

@wdg sounds like classic help vampire behaviour. If some random person started asking me to write code for them then I’d just politely end the discussion there and then. Once you help them once then they will keep coming back for more.

@Buffy, Regarding your third paragraph, a scholar would likely spend much more time and effort to try to understand the subject matter than this undergraduate student. A scholar likely would not complain to the OP that they have "a severe depression and the deadlines were close".

I plan to stop answering her emails.

Good plan. Put it in practice. Yesterday.

sending me emails asking very elementary stuff

When dealing with a paper from someone else, it is better to ask any doubt, also on the elementary stuff. Then, the answer may be "please go and find the answer in your notes from Introduction to inorganic chemistry" to a passive-aggressive "you should know that valency of carbon is 4"

she tries to urge me to reply by implying what I did in my paper was
wrong.

She is free to write a rebuttal paper or to go ahead proving your method wrong. You should welcome any effort in confuting your paper: it is the best peer review you can receive.
Back to the student: your duty with her were not different than with any researcher. Give her the keys of the publication, i.e. to give her all the tools to repeat the results you obtained in your paper.
If she cannot, it is not your duty to teach her how to get there.

@theonlygusti It is not, if you are discussing and offering it genuinely (I understand my writing is abrupt and it souns otherwise). See same concept in the most voted answer.

bringing full attention to the (as you are telling them it is) embarrassing fact that they are ignorant, instead of just kindly telling them the elementary knowledge they asked for?
There is a big misunderstanding here. Being ignorant must not be source of embarassment. Being ignorant is our natural condition.
There is no problem at all in being ignorant, nor in pointing privately to someone ignorance, as long as someone is truly motivated in learning and not in obtaining a free ride.

Essentially I agree with Buffy's answer, but I think there is a more important step again; you do not have a duty of confidentiality to this student, and they are displaying symptoms of serious mental distress, if you know their home institution you should raise this issue.
In the UK we have recently had two high profile cases involving student death due to suicide.
https://wearetogether.live/phoebe-grime-inquest/
https://www.bbc.com/news/uk-england-nottinghamshire-61534460
In the second case, the university was actually found liable for failing to take reasonable actions to protect a student who was known to be at risk.
If you know which institute this student does belong to, you should raise your concerns there. This will give the institute a more complete picture of the risk to that student, and their own institute is in a position to act on that risk. This could be anything from refering the student to appropriate support facilities, to starting a discussion with the student about a medical break from studies. Good friends of mine have done both these things, they do make a difference and they do matter.
But all you can do is make sure someone at their home intuition knows, so they can act on this information.

“Death due to suicide”? That's a bit of an odd formulation, as if suicide were something that happened to these students without their own involvement. Now – sure there's a point to be made about university/society piling a lot of burden on students that may push them to such an action, and that they should have received help from the responsible institution, and/or friends/family. But it really isn't something that a researcher from an entirely different institution should feel responsible for. Referring to appropriate support facilities can certainly make sense though.

@leftaroundabout the point of the wording it to emphasize that suicide is increasingly seen, legally and medically, as something that the sufferer may not have really chose. It's often a side effect of a medical condition, and the phrasing is ment to highlight that.

@leftaroundabout anyway, point is, at least one university has found itself culpable of neglect over this one, so OP would likely be doing everyone in the picture a favour if they reported it to someone who could intervene.

Maybe it would do everyone a favour, or maybe it would just kick off a big bureaucratic process that creates a lot of paperwork and takes everyone's time without actually helping anybody. The point is that it's hardly possible for the OP to assess this, and suggesting otherwise runs the risk of building up false feelings of responsibility and/or hope.

@leftaroundabout As we've seen, universities are perfectly capable of not following up on these things. The potential harm from reporting is not all that high.

"In the UK we have recently had two high profile cases involving student death due to suicide." So instead of removing the general causes of heavy distress, are you advocating for every one to focus on curing the manifestations (the suicides)?
It seems quite a diffuse and heavy burden for all the people involved...

@EarlGrey That's not what my answer says. My answer names the two interventions that are targeted at the underlying problem, and have been seen to be effective at improving quality of life. While I do talk about what can go wrong if nobody intervenes, it is in order to give case law as proof that I'm not the only person who has come to the conclusion that a student's university ought to be actively involved in mitigating mental health issues that arise in it's students. At no point do I say the depression is only a problem if there is also a suicide risk.

The situations in the two links seem to be quite different from the situation here. In one situation the student had apparently mentioned suicide plans, in the other situation assessment practices of the university were apparently seen as inappropriate given the knowledge the university had about the student's specific condition before. These do not imply in any way that a student mentioning severe depression in an email to a stranger is better reported. Of course it could make sense to ask the student whether they have professional support, but I don't think one has to go further than that.

I thought I was corresponding with a peer.

Later, I found out she is an undergraduate student who is doing her own research project.

I would go way out of my way to to provide appropriate assistance to this (potentially future) scholar. I have to go with @Buffy's:

...a professor has a "duty" to help people who ask them for professional help if done sincerely. It is part of what it means to be a "scholar".

Personally I think all human beings have this duty, but that's for other SE sites :-)

The issue of depression can be addressed completely separately from the help/don't help question.
These days most people (especially younger) are working hard to de-stigmatize mental health issues. And so you will see a lot more matter-of-fact discussion of depression.
There's nothing wrong with a short caveat like

I'm sorry to hear about the depression. On this topic I simply can't help you at all. I sure hope you are getting some help from qualified people.
But I can help you with some resources on topic X that might be most suitable for you and your project right now. A lot of the material out there is pretty high level, but the following...
I think it's great you are pursuing this project, and I wish you the very best of luck in your future!

The notion you should a priori suspect deceit and refuse help based on such preconception is ludicrous. You don't know and you don't need to. As long as the interaction helps someone to better understand your field of research, the potential benefit of helping someone in a rough spot outweighs the potential risk that they got help without matching the profile of the person you think "deserves" your help.

Several days ago, she said she just wanted to finish thesis on time, suggesting she would just throw all things away after her thesis was finished. I felt very discouraged at the moment. At first, I thought I was communicating with a potential collaborator. After a while, I thought I could inspire a curious mind. Now I found out that she have never been interested in the things I discussed with her.

This is very forthcoming of you and I applaud your willingness to be so open. This aspect might better be addressed with an additional question in Interpersonal SE, but I would say the best thing you can do is

Don't loose faith so quickly! They're in a challenging situation with pressure and simply may have slightly "overshared" how they were feeling at the moment.
What they ultimately do or don't do in the future should not be a gating factor for how much you do/don't do right now. The fact that someone helped may crop up later in their life, they may even choose to help someone else some day based on this memory.
That you care so much is wonderful, but we must always give without expectation of return or immediate gratitude. Life is a hard, funny and strange thing for each of us, and it seems you already got the notion that "this one is worth helping" based on correspondence. Disengage a little bit at the emotional level so your feelings aren't impacted, help where you can and then move on.

I think it depends, for example, if their questions are 'real' questions.
I would like sharing my experience, from the student side.
When I was doing my thesis project, I was studying many algorithms and I had to read their original papers and re-achieve their algorithms. I read their papers carefully, including their published codings. I find some errors in their codes, and some places stated in their papers are  not same as what they did in their codes. I also sent emails to the author about my questions. I sent some, but only one teacher replied me, and even he replied me three times. I invited him to my thesis defence, and even he would like to provide my recommendation letter for my phd application.
I appreciate it very much, I got lots of confidence because of him. I am not a confident student, cause english is very hard to me.
Maybe, next time, when you are not busy and you find their questions are meaningful, you could give them some time.

I don't think the idea that there is a (single?) "right thing" that one "should" do makes much sense here. There is a wide range of reactions that I'd consider as acceptable, and what I'd do would depend on things such as my personal time planning, deadlines etc.
Personally I think helping people who are interested in my work is worthwhile, also addressing concerns with my work. On the other hand I don't want to get myself into a situation in which I do other people's jobs. If the student lacks some basic knowledge required for understanding the paper, it is for sure not my duty to teach this to her, although depending on my personal time planning I may help a bit. At some point I'd tell the student that "XXX is basic knowledge, taught in course so-and-so and/or book so-and-so", and that I don't have the time to teach her these things myself. The central thing to tell the student is that my time is limited, and that they have to understand that my ability and willingness to respond is therefore also limited. This can be done without implying anything negative about the student, I believe.
Regarding the mental health issue, I think I'd just advise them to get professional help. So I can really only say what I'd do, without implying anything about what is "generally right".

If she is doing a research project, there must be a supervisor, and if they have agreed to base her work on your model, the supervisor most certainly has some knowledge of the model, too.
Kindly tell her to turn to her supervisor if she has any more questions. She obviously needs supervision, but that's not your job, but someone else's. This person is also responsible for teaching her the basics. If she contacts you again, tell her again to ask her supervisor.
In the end, it is her supervisor who has to be contented with her work, not you, so she has to communicate with them.
I would not talk to her or somebody else about her depression, that's too intrusive.

I would try to find out who is overseeing her work at her college/university, and make your concerns known about her ability, and her health.

Alhough good meant, this is very intrusive.

I tried, but the student did not use her institutional email address, and a google search of her name did not turn up anything useful.

@EarlGrey If I was teaching someone, I would definitely want to know this; if it wasn't evident.

@EarlGrey If the student is telling people they are depressed, it might well be because they need someone to know. OP is the wrong person to tell, but they don't owe the student confidentially, so letting the right person know would be a good thing.

@Clumsycat Their supervisor may not be the right person to inform.

@BryanKrause the person who is in a position to act on this may be their supervisor, and if it's not, their supervisor likely knows who that person is. At any rate, it's a non starter here, since OP doesn't have enough information to find their home institute.

@Clumsycat You don't know that; their supervisor may not be supportive in that way and might act against their student's interests with that knowledge. Remember that this is a student reaching out to someone other than their supervisor, they may have a reason.

@BryanKrause nobody can ever guarantee that talking about mental health problems will lead to better outcomes, and we should never make promises that might not hold. But it is a good general rule that trying something is better that expecting the situation to resolve itself, or indeed giving up on the situation improving. It is often true that the person suffering from a mental health crises tells no one, or the wrong person, for all sorts of reasons related to the nature of the beast. The supervisor has a duty to help the student, but it's quite possible they don't know there is a problem.

@Clumsycat There's a difference between "trying something" as a person suffering with mental health or reaching out to someone you know is suffering, and this situation which is someone who is basically a stranger (OP) contacting another stranger (the student's advisor) about a student's health. That's a big breach of the student's privacy and trust.

@BryanKrause the supervisor isn't a stranger to the student, the supervisor is a person with a (possibly formalised legal, but certainly social) duty to help the student.

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text:
How would one fail a master thesis defense?
So I have to defend my master thesis in computer science in two weeks. It's in Germany and the thesis is 30 credits. I'm wondering what could make someone fail his defense? In my thesis I mainly compare different approaches and algorithms and did experiments on them to answer some research questions. Would I fail if, for example, during the discussion a mistake in my comparison and evaluation approach is discovered which makes my results meaningless? Although I discussed my approach and everything I did with my advisor (postdoc not the prof), yet I still fear that I might have done something wrong and then everything collaps.
I still remember reading a question on this site where someone mentioned that he discovered a vital mistake in his PhD thesis after two years of the defense which made almost his entire thesis useless. Yet of course he didn't lose his PhD degree. So if something like this happens, but during the defense, what could happen?
So how would one fail his master thesis defense? Just to calm down a bit and feel safe and secure!
Edit: I passed with an excellent grade :D.

Your advisor should not let you defend if you are not ready.  Ergo, your advisor thinks you are ready.  Stop worrying and go for it!

Ideally, an advisor should not let you defend if you are not ready; unfortunately, some advisors still do let not-quite-there-yet advisees defend anyway.

Would I fail if, for example, during the discussion a mistake in my comparison and evaluation approach is discovered which makes my results meaningless? Although I discussed my approach and everything I did with my advisor (postdoc not the prof), yet I still fear that I might have done something wrong and then everything collaps.

This should not happen unless:

The mistake is blatantly obvious and something you should have really known better. And with blatantly obvious, I mean something that somebody who has not even studied computer science or a related field could easily spot. Even then you may have chances, if the rest of your work is appropriate – almost everybody brainfarts now and then.
You made not only one mistake but a lot of big mistakes.
It becomes obvious that you deliberately ignored that mistake, to avoid being stuck or to get “nicer” results.

(In the first two cases, your advisor is in big trouble, too.) The point of the master thesis is that you should demonstrate that you can investigate a scientific question under supervision (or something similar – check the regulations, if you wish to know). One mistake does not change this.
Moreover, in some examination regulations I am aware of, there is a procedure for the case that some important but localised flaw is detected in your thesis. For example, you could be given a month to amend your thesis.

How would one fail a master thesis defense?

I have not experienced or heard of such a case but from what I have gathered, you pass if you:

Give a talk about your thesis.
Be able to answer questions about it.
Are not detected to have been cheating.

Even if you suffer from a mental breakdown due to nervousness or similar, you probably can repeat the defense – at least with any reasonable examination regulation and examination committee. I am aware of one case where somebody had a nervous breakdown for understandable reasons¹ and the examination board let him repeat the exam. If you look into your examination regulations, there are probably some clauses that allow the examination board to do some things at their own discretion in exceptional cases.

Be aware though that being very difficult to fail does not make the defense unimportant. A bad defense may seriously (and in particular more than nominally) affect your thesis degree, which in turn has a huge impact on your total degree. In particular having passed a thesis with the lowest possible degree (or something close to it) is something you definetely do not want to have in your vita².

¹ It’s very complicated, but you might compare it to the following: In the middle of the defense, somebody who was the defender’s girlfriend until six months ago enters the room being obviously pregnant in her ninth month.
² Unless you are in one of those few disciplines where this is the norm.

You really calmed me down. A nervous breakdown means crying or what do "you" mean by it?

Why are you putting you in quotation marks? It’s not my story; if that’s what you were thinking. Nervous or mental breakdown here can mean all sorts of symptoms that make you psychologically incapable of holding your defense.

lol calm down. I didn't assume it's your story. I put that in quotation marks because someone else told me his definition of breakdown here and I wanted to know yours.

and what are those very few disciplines where it's the norm to have bad grades in the thesis?

@JackTwain: 1) I am perfectly calm and what you describe is not a good explanation to use quotation marks at all. 2) About 40 % of the final exams in law in Germany are passed with the lowest grade – though there is no thesis involved in this. Also, obtaining a doctor of law with the lowest degree (rite) is not unusual (though I have no numbers for this).

Great answer, however note that grading of the thesis or the thesis defense is no globally applicable.

The only person I've ever heard of to fail his thesis defense (Austria) in CS was a guy that started an answer to "What is object oriented programming?" with "Umm.. good question!" and everyone wondered ever after how the hell he cheated his way through all the exams leading up to the defense. It's pretty much impossible to fail your exam. I'd say it's more something to look forward to: Discussing a topic you're intimately familiar with some colleagues should be fun.

Evaluation criteria varies vastly, not only across different countries and cultures but even amongst different institutions within a country.
The only way I can imagine anyone failing a thesis defense here in Sweden is if you have a mental breakdown during presentation or questioning.
Alternatively if the defending student has plagiarised parts his/her work, that would also be a likely fail but otherwise, if you are allowed to defend, then you have practically passed already.

I believe the academia atmosphere in Sweden is similar to that in Germany. What does 'mental breakdown' mean? Gone mad?!

Mental breakdown would stand for a blackout where you transiently forget everything you know due to being overly nervous. Likewise, in Ireland, where I did my PhD, if your supervisor allows you to defend your thesis it would take unlikely catastrophic circumstances to fail the actual examination.

@Miguel thank you. That gave me some kind of relief and security.

@Miguel The situation with a PhD defence is slightly different, in that the PhD is an open-ended degree where your advisor shouldn't, as you say, let you defend the thesis until you're ready. But a Master's tends to be a fixed timescale, where you defend your thesis on date X whether you're ready or not.

@DavidRicherby my master thesis was an open-ended project. I had to formulate it from a to z. The contributions, the outline, the research questions and everything. I took me 6 months just to find out what I want to do. Then I presented it to my professor and he signed my thesis proposal. Then I started really implementing and working and writing. However as I said, I'm almost just concerned if I did something stupid which might turn my evaluation results useless and so almost my thesis.

It took me a complete year to finish my thesis.

Here you can always request your defense even if your reviews are negative. Your adviser can§t just not let you have it.

@JackTwain The possibility of being wrong is always present in research, and is a fundamental part of doing new stuff. You are exploring uncharted territory. Until your ideas have over the years been tested by others or applied to different problems you cannot be certain of the relevance or even correctness of your results. Only a tiny fraction of the original research literature enters the textbooks. You have done what you must: make sure that to the best of your knowledge and ability your thesis is a sincere record of your research.

Just to add: there is a grade associated with a Master's defense, so it is possible to get a low grade, if you don't present or answer questions well. But actually failing is very unlikely, as this answer explains.

@LuboAntonov even that varies, we have no grade associated with the defense or the thesis, you either pass or fail.

It's pretty rare for a student to fail the defense of a master thesis in the Swedish systems, but I have seen a few over the years. The three most common reasons include:

Did not show up to defend the thesis (AKA: Lose on walkover)
Doesn't know the material of the thesis (AKA: Didn't write it)
Unable to hold a discussion about the thesis (AKA: hid in a corner)

Note that unlike a PhD thesis the professor does not have lot of skin in the game. Having a PhD student fail a defense is extremely embarrassing and can end the career of a professor. Having a master student fail a defense isn't considered such a big deal.

"can end the career of a professor"? Haven't heard of that,  ever.

@Dirk good. Now you know so you can make sure that doesn't happen with your PhDs :D

Actually I still doubt that this really has ever  happened.

@Dirk would you fail a master student in his defense if turns out that he did a mistake and the results of the thesis are as a result useless because of this mistake? Suppose that this gets discovered only during the defense, and it was an honest mistake.

Dirk: I know of two cases, one in nursing and one in molecular biology over the last decades where a PhD student was recommended to defend and failed. In both cases this ended the research careers of several people involved.

@pehrs good to hear :p

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text:
Is copying equations plagiarism?
I defended my master thesis successfully yesterday. Everything in my master thesis was written by me and there is no problem. However I'm a bit skeptical about one thing. So in my thesis I have a part that explains a very well known model. To explain the model you basically need to calculate the gradient of some function, and calculating the gradient is basically lengthy and spans many subsections. So there are many papers and I believe even books that show how to calculate the gradient step by step. So I found one paper and I read it and understood it and understood how the gradient of the function is derived. Next I compared the calculations in the paper to other books and I found that they are very similar.
Since I'm introducing the model in my thesis I had to explain how to calculate the gradient. I explain everything in my own words. However when I wrote the equations they look almost like a copy from the paper!! Since I had the same understanding, so the calculations just almost look the same. So I was really confused what to do about this. Since I want to include the derivation of gradient in my thesis and it looks almost the same from the paper. So I was wondering if this is plagiarism! Since I was skeptical, I checked that the equations also look somehow almost the same from other papers. So to make sure that I don't be held for any kind of plagiarism and credit the authors, I mention at the beginning of the section that starts to explain the model is that my following explanations are based on the explanation given in the paper, where I learned about the model. Calculating the gradient is kinda lengthy and spans couple subsections, so in the subsections I assumed that since I credited the authors at the beginning then I didn't credit them again later for the other derivations of the gradient.
Now my question, is what I did correct?! I mean they were only the derivations of the gradient, so they were equations and I can't just "rephrase" them!
I remember once asking one postdoc that I'm using a lot of equations from one paper and he said that I just need to credit the authors that I'm using the same notation, which what I did. However, when I look at the formulas I feel somehow uncomfortable, because they look like a copy from that paper. Should I be worried?!
I really tried to credit the authors and I cited the papers in many places where I was explaining the models. But only the look of those large chunks of equations seems like copying.
As I said earlier I defended my thesis today, so I just hope that I didn't do anything wrong with that because I really worked hard for my thesis and every single word is mine.
P.S what I know is that copying common knowledge is not plagiarism. So this was my assumption behind the calculation of the gradient. Because the derivation of the gradient is not a contribution and the derivation is done in many papers. (like the gradient of the log-likelihood of logistic regression).
So in short, suppose that calculation of the gradient of the log-likelihood of logistic regression is very lengthy in general. And I just copied the equations from a paper that does the derivation. However when I motivated Logistic regression in my thesis I said that my explanations in the next subsections are based on the explanation in the paper where I copied the equations from. Is that okay?

If this is a standard calculation, why not just reference it? (and not include the equations)

@JoeManlove because the thesis, as far as i was told, should be self-contained. So I had to explain the details of the models that I address in my thesis.

I'd ask your advisor about this. Unless they add something to the exposition, redoing pages of calculations that are in the literature is not usually necessary. It's not a plagiarism issue if cited, but is a bit tedious to write (and read).

If one needs to cite Pythagoras upon every theorem use I'd say western civilization has one hell of a lawsuit on our hands.

If nervous, document where you got the formulas from. Which might be good practice anyway if they aren't well-known and/or obvious.

A resource that you should likely check with is the mathematics research librarian at your institutions library. They are there not only to answer questions such as these, but can also help you with your research, both developing questions and finding resources. They will also likely be aware of the expectations and idiosyncrasies of your Department, as they usually work closely with the professors in their subject area. It is a free service and they are usually more than happy to get and answer queries.

Reminds me of this: https://www.youtube.com/watch?v=kdAGXYmqbQ8&feature=youtu.be&t=19m18s

@JackTwain in mathematics "self-contained" does not mean "written in a vacuum." It means that the reader does not have to go look at other references to understand. Refer to the other article or book and then explain so that a competent mathematician could understand without having to read another article.

@TK In my experience, theses are expected to be written in a way to be understandable by peers. For example, if you study Computer Science and your thesis is about neural networks, you have to explain what they are in the background chapter so that a CS student would understand the main concepts.

Mathematical equations are not normally considered subject to plagiarism, since there are only so many ways they can be written, and because they represent ideas rather than material that can be "stolen" without proper attribution.
So long as you cite the material you are drawing from, and explain the equations in your own words, you should be fine.

Moreover, in the mathematical sciences it is very well understood that two people who are doing a standard type of calculation or technique, and both doing it correctly, have a good chance of doing it in a very similar way, even up to using similar or identical notation.  After all, we very much take notation and terminology from each other: imagine what a disaster it would be if everyone had their own "non-plagiarized" notation for an integral or cross product...

@PeteL.Clark, we shouldn't?  That explains the puzzled looks when I ask my calculus students to calculate $\times \vec v\int\vec w\textup dt$!  (EDIT:  Oops, maybe it's less funny without MathJax.)

Between this answer and @PeteL.Clark's comment, this covers pretty much everything I would have posted myself. The only point I would add is that it's very common to say something like "In this section I follow the derivation of [source]" and then include the derivation from that source - but this is really an acknowledgement that you are copying the overall structure of the derivation, rather than individual equations. Also, a nitpick: I would note that the distinction between ideas and expression is important to copyright, not so much to plagiarism.

@LSpice Even with MathJax turned on, it has an error (\textup is not recognized).

@aeismail I like your reply and agree with it. Many months ago, on this site, I also discussed "explaining things in your own words" and I was told it is plagiarism. Most persons' "explanations in their own words" are inevitably derived from somebody else's explanations. I don't believe it is plagiarism and I think people are REALLY too hyperactive about plagiarism these days.

I do believe, like @aeismail and other commentators, that equations are knowledge, and not subject to accusations of plagiarism. Yet, you should be cautious about your mathematical notations and their correspondence with the copied equations.
If you have adapted the proofs, or the calculations, with your own symbols, and typed the equations yourself, you are fine.
Be cautious about some mathematical notations (imaginary unit, convolutions, ordinal variables, greek letters), that  may be more or less standard depending on the  scientific field. For instance, the imaginary unit is often "i" in maths, and "j" in electrical engineering. The reuse of other field's notations should not look too artificial.
However, I have seen persons who have cropped blocks of equations from some source in slides or reports, and who did not bother change their notations or cite the source. This behavior is closer to laziness or dishonesty than to a genuine personal work. But working on equations, their variables, to make them more understandable, clearer, is a plus.
Finally, one side of plagiarism is in automated plagiarism detection tools. You may be worried about texts, but so far, they do not seem very efficient with equations.  Unfortunately, since it is difficult to search formulae online.

"Your own symbols" sounds ambiguous. One should of course ensure notations in one's work are internally consistent. However, it is totally fine (and recommended) to use the same symbols as another work, because many notations are (semi)-fixed for some concept, down to the letter to use (say, a standard deviation is always a $\sigma$).

@ Blaisorblade I meant that inn the context "Yet, you should be cautious about your mathematical notations and their correspondance with the copied equations". Additionnaly, symbols may vary from one domain to the other: E.g "i" vs "j" for  the imaginary unit

@Laurent Duval I think yours is a good answer. You shouldn't copy blocks of equations without citing the source. It gives the appearance that you wrote the equations or even developed them.

You should remember that academic plagiarism is not plagiarism in the sense of copyright. Academic plagiarism is mainly the "stealing" of other people's ideas. As such rephrasing is normally not enough. Mathematical formulas are a bit of a gray area in that. Much of it is (very) well known and also subject to possible independent invention. Once you have a lot of them though, the probability of the same approach without awareness becomes unlikely.
In all cases of gray areas, especially if they concern aspects of professional ethics (such as plagiarism), be safe and  reference. Also be aware that not all readers are familiar with all parts of maths and may not recognise that you are just describing a reasonably well-known concept, not introducing it yourself (I have reviewed a journal paper with this issue, directing the authors to add the reference was sufficient, it was clear that they actually didn't come up with the idea).

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Weird/Dumb question: What does a thesis do when the non-thesis option means more courses?
Actually, this is my question: which option should i prefer Thesis vs Non-thesis for MS in CS? but I hope this post will not be closed as either off-topic or duplicate because I hope to be specific. There's some more context than the little context below, but it's kinda long to explain and maybe unnecessary. So, suffer with this short weird/dumb question (which could be made only less weird /dumb instead of non-weird/dumb with context).
Observations:

I notice some curricula of master's programs in pure math, there's like a thesis option and a non-thesis option. It seems like the only difference is in the non-thesis option, the thesis is replaced by 3 additional courses and a comprehensive examination.

In the other question, it says thesis option is more suited to those doing phd afterwards (or the converse: those planning to phd is more suited for the thesis option...idk. or both).

Weird/Dumb Question: Ok so then my confusion is like... I see thesis option has less courses compared to non-thesis. Ostensibly, more courses means more knowledge. But obviously this is wrong because of the previous answers about thesis option as more suitable for future phd applications (or conversely or both). What does a thesis do when the non-thesis option means more courses? I mean, will I actually gain more knowledge working on the thesis as compared to taking those 3 more courses?

Scope: It's not mandatory, but my ideal answer is for math. Next best thing is for theoretical research. Third best thing is in general.

A little context:

My background in applied math: I did bachelor's and master's but in applied math. I want to now go into pure math. I didn't do a thesis in undergrad. In grad, we did have, in my opinion, a 'thesis', but our professors don't actually call it a thesis. (You can refer to my previous questions if you want.) But, Hell, it was tough. It was also so rewarding seeing it hard bound and in those fancy colors. But I didn't quite really learn many new things there. It was more of like experience applying things we already knew. Well it was 'applied' math, so perhaps I shouldn't be so surprised that our sort-of 'thesis' was about 'applying' things. Oh also, it was just a master's (not a PhD or even an MPhil).

Recently in pure math: I was rejected for pure math phd program in Dec2018 and then again in Mar2021 (in Country A, where I currently live. I'm from Country B, and I might have to go back here for a second master's). Both reasons were simply insufficient knowledge. There was no mention of any lack of 'research ability' or 'skills' or anything (see below re 'research ability' or 'skills').

I actually remember reading online (at least in the applied math case) that doing a thesis for master's or even in general taking up master's or PhD in applied math demonstrates to potential/future employers (the context here is banks or financial institutions) 'research ability' or something.

My preferences: Anyway, I don't mind doing thesis or non-thesis. It's about the same time to complete, and I wouldn't really care if one was longer than the other. I just wanna do whichever gives me more knowledge to eventually apply for a pure math phd (in a european-style university. I'm good enough for some us-style countries, I believe. But I want to do my pure math phd in a country that is unfortunately european-style. The country is Country A, btw). I plan to ask the universities of those master's programs, but I also wanna ask here on stackexchange. Perhaps here on stackexchange, I can even ask the universities better, or at least less dumb/weird, questions.

Maybe related questions:

Why doesn't a master's thesis get read by a PhD admissions committee?

Will an off-topic Masters hurt my chances of a PhD place?

2a. In particular, I notice there's an answer here that says

"People don't expect miracles from your master's thesis, rather they want to know whether you have the necessary knowledge/skills to embark on a PhD.

In relation to this, I am interested in acquiring not necessarily 'skills' or 'research ability' but 'knowledge'. (In this way, I hardly expect 'knowledge' and 'skills' to be separated by a '/', at least in my context.) I sincerely doubt that the professor, or any other professor in the university, who/that rejected me cares about 'skills' or 'research ability', or is probably already convinced enough from my applied math background. I believe the only thing I'm lacking here is knowledge. (Well, I was told I was lacking knowledge, but I wasn't told anything about lacking 'experience' or 'skills' or 'research ability'.)

I can't answer for pure maths, as that is not my field, so what I say might not apply there I guess, but a PhD is a PhD.
The first thing to note is that formal examinations and theses test very different things. The test at the end of a PhD is a thesis, not a formal exam (in the sit down and take a paper sense). Thus, your performance in a thesis is more representative of what your performance in a PhD might be.
Doing a PhD is about research, it is not about knowledge. In a PhD you might be learning things, but a PhD is not about learning, it is about the generation of new knowledge. A thesis will test/develop your ability to:

Find new directions to take something to places no one has ever taken them before
Think about the wider disciplinary context in which your work takes place and use this context to guide which directions will be most fruitful/most useful to the discipline. Be able to explain the relevance of your new knowledge to others in the discipline and recognize how it fits in with their work.
Advance several different lines of research simultaneously
Deal with the fact that most of those lines will be dead ends, and you will have to discard most or all of them.

Now you need a knowledge base to be able to do these things. Or rather, you need to demonstrate that you have the necessary skills to go away and learn what you don't know when you need it. But success in a PhD is not primarily measure in the amount of knowledge you have.
For most students, they have spent a long time in education, learning things they have been told to learn, and, if they are lucky, applying that knowledge to new situations. For this majority, skills/experience at generating new knowledge/scholarship is the weak point. Being good at exams =/= being good at research..
That said, it might be that you have already demonstrated these skills in the master's degree you already have, and for you it is the missing knowledge that is holding you back.

Thanks a lot Ian Sudbery! I forgot to mention scope in my post or put math for a tag. I edited post. I didn't and still don't necessarily expect math only answers. But I'm hoping for an answer from a theoretical research perspective (also only ideal and not mandatory). So stuff like theoretical computer science, theoretical physics, etc. But anyway this general answer I think is very good.

Anyway, I was actually quite surprised to see an answer like this. I really thought I was wrong. (Well, of course, we'll still wait for commenters or other answers.) So non-thesis options really give more knowledge in general as compared to thesis options? (Again, I'll of course ask the specific universities.)

@JackBauer I don't think it makes sense to think in terms of "more" or "less", it's...different. In courses you focus on more foundational, settled content. In research or when writing a thesis you're more focused on the boundaries of knowledge. In some ways this means "less" is learned when you are at the boundary, since every piece of knowledge is harder to obtain there. But in other ways it's the stuff that matters for the future. If everyone only becomes an expert in what is already known there is no progress.

@BryanKrause is the difference like depth vs breadth? by which i mean: depth: the research is studying more into 1 or a few specific thing/s. breadth: several things but not as much depth. thanks

@BryanKrause ' If everyone only becomes an expert in what is already known there is no progress.' --> you know i'm talking about master's and not phd's right? i think the idea at least in this situation is like ok let's be beginners at what we don't yet know instead of just experts at what is known, but i think we do need some expertise of known stuff 1st before becoming beginners at the unknown. idk

Clearly you need both the background knowledge AND the practice at being a "beginner at what we don't yet know". Unlike Lewian below, I am unlikely to actaully read an MS thesis, and don't really care about the topic - I just need to know that a student is capable of this sort of dive into the unknown, and capable of sustaining an extended piece of unguided or semi-guided work. You'll have to look at your record for yourself and decide if your record of foundational knowledge, or research experience is stronger and which could do with re-enforcing.

@JackBauer Your question touched on admissions for PhD and a thesis masters being better for that purpose. If you want to show to a PhD admissions committee that you can push the boundaries of research forward, it's more instructive to show your abilities to think on the edge of what is known. You say you're not interested in skills or ability but these are certainly things a PhD admissions committee cares about, probably more than "knowledge". A key skill there is working more independently as both Lewian and Paul's answers emphasize.

Ian Sudbery's answer is excellent, but here's something more. A general aspect is that students tend to think that things are learnt by being told. This runs counter to most educational research. Discovering things for yourself as required while working with a bigger scope (thesis topic) and applying them in turn with more freedom to make own decisions is usually far more effective for learning than having facts told and having to replicate them in an exam. Admittedly many (but surely not all, probably not even the majority) courses do better than that and use more interesting tasks for assessment, maybe even involving some freedom for own decisions, however by and large "teaching and exam" style learning is normally less effective in the long term than project work.
Things that students should learn when doing a thesis:

Acquiring knowledge from the academic literature, deciding what is important/relevant and what is not, and applying it to a problem that is not of a kind that was taught in the same or a fairly similar way before (note that you don't only learn the knowledge itself in this way, but also how to do this with new material for the next problem),
Scientific writing,
Time planning and structuring the work,
Strategies for solving hard problems that are not tailored to the methods taught in a course; particularly dealing with mistakes and approaches that don't work,
Competence to make your own decisions and take responsibility for them.

Now I have to admit that this is a rather idealist view, and that due to thesis supervision, unsuitable topics, or also student's attitude the student may not have "learnt" all of these in the end; however it is of course also true for courses that the students often don't learn all the stuff that is taught. If a student is interested in doing a PhD with me I'd always ask for a thesis and I will actually read it (if maybe not word for word); for me that's in fact the most important qualification. I'd always prefer students with thesis to those without if they are otherwise fairly equal (and 3 courses more than the one without thesis may have will not change that). Obviously this is from a system where the supervisor decides whether to take on a student rather than a commission of people who ultimately don't have to work with the student. (Actually a commission may decide general suitability but the student needs to find a supervisor who agrees to take them on anyway.)
PS: Your first question was already asked (and answered by somebody else and me) here.

I agree: with courses, as with many things in life, there is a law of diminishing returns. Those three additional courses would not add much, especially if they are not directly relevant to the PhD research topic.

Speaking for the milieu familiar to me, mathematics in the U.S. (probably some people would label the work I do as "pure", as opposed to "applied", despite its applications... but, nevermind, let's go with the cliched labels... I'm well acquainted with them, as inaccurate and misleading as they may be.)
And, having been on grad admissions in a (what some people would call) "joint pure and applied math department", I am well acquainted with our criteria, anyway.
A thesis-oriented M.S. does invariably involve more self-actuated, self-selected work, and practicing doing that is a great thing. It is often less informative about the accumulated historical record of math stuff (which, in some contrast to other subjects, does not become "wrong" (usually), even if maybe less relevant in some cases).
A course-oriented M.S. is more "automated", and can be stultifying, but is (at best) far more informative in a scholarly sense (if the courses are chosen wisely, as opposed to being chosen to make the easiest possible path).
Only rarely would a PhD admissions committee look at an M.S. thesis. Most of what would be informative should be clear from the title and the abstract (and the advisor's comments about it).
Yes, I have observed that some aggressively-self-styled "applied" grad students (whether M.S. or aiming for Ph.D.'s) misconstrued "applied" as some sort of license to not know much... This is a ghastly self-deceit. Yes, there is also much useful computer science and other "natural sciences" to know (and engineering), to meaningfully apply mathematics... but ignorance of mathematics is not so helpful in that! :)
(Being in a hurry is understandable, but often leads to error... :)
A possibly amazing thing about mathematics and its applications in the practical world is that abstraction is very, very useful. A great economy and simplifier in many situations. (Not always...) And, as with many things, a 2-fold speed-up (not to mention 5-fold or better) is indistinguishable from miracles or genius. If that can be acquired by study of standard (if "fancy") mathematics, ... well, whoa, I'd take it! :)
So, no, an M.S. thesis will rarely impress PhD admission committees. It can be a plus, if it shows initiative and creativity and such. But/and the coursework, especially if your earlier coursework background was thin, is also something people do not want to see missing... It's nice to be clever and hard working, but failing to take advantage of all the hard work of all the clever people in the past is ... very foolish.


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How much knowledge is expected of a PhD applicant as compared to a postdoc or a research assistant?
My understanding is that postdocs, research assistants (which I understand are master's level analogues of postdocs) and PhD students are all paid positions except for self-funded PhD which I hear is rare, but regardless, PhD students like postdocs and research assistants are expected to assist with the specific research that the professor is doing.
But, considering that a PhD studentship is the pursuit of a degree rather than the application of acquired knowledge from a degree, how much is expected of background knowledge for a PhD applicant?

There was this quora post that I can't find anymore that asked why professors took on PhD students to assist with research instead of hiring a research team or something. I was supposed to link it as an introduction.

Update: I found the question! Why would a professor want a Ph.D student instead of a research staff? What are the advantages and disadvantages?

Anyway, my question is different from the following because I'm asking specifically for PhD students relative to postdocs or research assistants.

How much is expected to know before starting a PhD
What kind of knowledge is required to do a PhD?
How can an undergraduate student conducting mathematical research with a professor expect to contribute?

I'm going to use math for examples.

If I know elementary topics of complex analysis, abstract algebra, algebraic topology, algebraic geometry and differential geometry, can I apply as a PhD student of a professor who specializes in complex analysis? Complex geometry? Functional analysis? Differential geometry?
For location: I don't mind if people post an answer for US or Europe, but in case I really need to say: I live in Country A, which is somewhere in the middle.

I will only attempt a partial answer here as your question is a bit convoluted. Doctoral students on the first day, are expected to know less than a post-doc or research assistant on the first day. But they are expected to know relevant things. How your specific case would be judged would be up to a specific judge (i.e. a potential advisor).
But note that universities (as opposed to commercial/industrial labs) have a dual purpose. It isn't just to support the research interests of the current faculty, but to also educate the next generation(s) of scholars and researchers. Without that second  purpose there would be no need for graduate students at all. But then scholarship would end in a generation or two.
Universities are long term contributors to the future and take the role seriously, both in research and teaching.

The other links in your question seem to be pretty good for answering your other concerns.

Thank you so much Buffy! I wasn't sure I would get a posted answer. About "It isn't just to support the research interests of the current faculty, but to also educate the next generation(s) of scholars and researchers" That's kind of what the answer in the quora question said. Please check if I understand: Bachelor's and master's students are part of the educating, postdocs and research assistants are part of supporting, PhD students are somewhere in the middle and how much in the middle depends on the location, professor and research area?

@JackBauer - no, postdocs are still learning. "Research assistant" is a multi-purpose term, as many grad students are supported as RAs, which is different than a research person (has PhD, likely completed a post-doc) supported (often on soft money) by a professor.

@JonCuster Thank you! Really? Is that even in Europe? From what I've seen locally, research assistants(usually bachelor's/master's level)/postdocs/research fellows(not sure how this is different from postdoc apart from this 1 post I saw that said "master's or PhD")/research associates (usually master's level) are advertised the way industy jobs are and so candidates are expected to have very specific high-level knowledge. Anyway, your answer is that PhD applicants should have about the same level of knowledge as a postdoc or research assistant applicant?

Buffy and @JonCuster , I reduced the examples in my question.

'Without that second purpose there would be no need for graduate students at all. But then scholarship would end in a generation or two.' --> I was thinking about this recently. It's the same thing with pre-university education: It's important, but we don't expect academics to teach children. But we do expect them to teach adults. Like there's a profession for teaching pre-university. And then there's a profession for both research and teaching university. Why isn't it split like: a profession for research w/o teaching and then a profession for teaching w/o research? Where is the line drawn?

buffy, i was also thinking mainly in the context of european-style universities as opposed to us-style universities. were you thinking us-style mainly? or both?