text:
How should a reminder e-mail asking for a PhD position be written so that it is not considered rude?
I live in an Asian country and I wrote to three professors in France asking to discuss PhD research opportunities in Math.
I introduced myself and told them my background in the particular branch of mathematics (courses done, title of my master's thesis) and e-mailed them my CV along with mentioning how my research interests match with them.
I e-mailed them on 22 December and 23 December and haven't received any reply.

I think now is a good time to send a reminder E-mail. Am I right?

Question : How should I frame my reminder E-mail so that the E-mail is not interpreted as rude or impolite or disrespectful? I am not a native speaker although I have a good command over English. But still I thought I should ask, as such impressions really hold value if done badly (I think).
What I thought of writing:

Subject:  A very polite reminder of my previous E-mail regarding
discussing PhD research opportunities in ...
Dear Dr. ABC,
This E-mail is a very polite remainder of my previous E-mail written
to you on December 23  regarding discussing PhD research Opportunities
in ....

After this line I will copy contents of my previous E-mail and attach my CV.
Is this way of writing fine?

(1) if those were cold emails (never communicated with them before), getting any answer is a bonus. They don't owe you a reply. (2) you emailed them right before the holiday break, which many still might be on - classes at universities in my area don't start until next week.

December 22 and 23rd is right before important European holidays (Christmas, Boxing Day, New Year's, etc.). I don't know the European calendar precisely, but some American professors might just be getting back to email today.

@JonCuster Are you from France?

@JonCuster ya, I have never communicated with them before.

@YannicMuller As others have stated, here in Europe some might have been out of office until the 6th of january, and then the first thing they do will not be responding to (potentially unsolicited) PhD applications. If you haven't heard back at the end of the month you might send a reminder.

@Sursula do professors don't reply when they are not in the office?

@YannicMuller - not in France, have worked in continental Europe. Same principles apply where I'm at, except that actually not doing work over holidays is more respected in Europe.

Further, while I did check my email (national lab, not university) over the holidays, I only replied to one high priority item. I would not have replied to any unexpected external communications.

One tip: Although plenty of people seem to do it, it has always been a bugbear of mine when people explicitly self-describe their own email as "polite" or "very polite" within that email. The recipient will be their own judge of whether or not they find your communication to be polite, so it is potentially a bit presumptuous (and therefore ironically impolite) to self-describe your own email in these terms.

@JonCuster ok thanks!

Not directly an answer to your question, but you should note that in France most PhD position funded either through doctoral scools or through grants received by the supervisor are advertised at some point of the recruiting process. Thus if you sent an email to professors simply based on seeing their work, they might be interested but not have any possibility to offer a PhD position unless you have figured a scheme to secure funding through your home country. So such emails can simply be ignored or classified low priority since they receive many of these and cannot give positive answer.

@Sursula Doesn't writing by the end of the month be too late because I have to apply for scholarships as well ?

For the subject of your next email, a simple "Re: " is probably enough.

The 6th of January is also a public holiday in France, so it could be that the professors only went back to work yesterday (i.e. the 10th). Give them longer than one day to respond!

@astronat I didn't knew that there are holidays in France from 23 Dec to 10 Jan

@astronat So, atleast how many days should I wait before writing to them?

As an alternate strategy to Ben which had a good template, I would propose to send a follow up email which does not directly refer to your original email at all. First, based on your proposed emails for the follow-up, there's some odds that your original emails were dismissed for cause (i.e., they had issues). Second, if they did not see your prior email, then including the prior email on the thread is a nudge, even if you never refer to it. It says: "Hey, I tried before and you didn't get a response." Sometimes this is what you want, to add a bit of pressure, such as when bothering an admin where a paper trail of non-responses will help light a fire if you need to escalate the issue.
In the current case (and for professors in general), you're asking for a favor you messaged during the middle of the most prevalent religious holiday in their country, so including the prior message in the thread will definitely not help you. I would say don't include it and don't directly refer to it. There is no upside. They either already dismissed you (and will again) or you will be highlighting to them that you messaged them days before Christmas and actually expected a response.
As a proposed contact.

Dear Professor [NAME],
I hope you had a good holiday break and are doing well.
I hope it is not a bother, but I was writing to see if there may be
any opportunities for a PhD candidature under your supervision in the
upcoming year. I have attached my CV and previous experience.
If you think I might be a suitable fit for your program then I'd
appreciate an opportunity to talk more about this and to read any
articles or whitepapers that reflect your current directions.
Alternatively, if you think my background is not suitable, I'd love it
if you could send me back any critical appraisal of weaknesses or
skills that I could build which would make me a better candidate (or
even just reply to let me know I'm not a good fit).
Thank you so much for your time.
Yours sincerely,
[NAME], Pestering PhD wannabe
[No prior email chain. Nothing here. But attach your CV and whatever
else you're giving]

This is sufficient to alude to the prior email (it references the break, which excuses them for not responding) and it apologizes for bothering them.
Then, before you send anything, have two people look it over: someone who is ideally somewhat familiar with the academic culture of where you are sending it, and a second person who is good at French.
With that said, rather than starting by cold-calling professors: I would encourage you to look for the CV's of the students who are currently in the labs you are applying to. Take those CV's, cross out everything that happened since they started the program, and compare your CV to those CV's. What are you missing? Are you actually at all competitive for where you are inquiring? It's fine to inquire at a few places that are a reach, but you need to know where your standing is: if you're well-below where the average accepted applicant is, you need to own that and know where you stand when you contact. People are more likely to respond if you're actually closer to a fit and if you have reasonable metacognition of your strengths and weaknesses.

"[NAME], Pestering PhD wannabe" ...really?

@TSF If you would look at the other answers, this was done because it was an adaptation of another comment using that phrase (which I referred to). Don't actually use that line.

Though this is really a matter of opinion, in my view every and any reminder for an unsolicited communication is rude. It carries with it the presumption that your original contact warranted a response. Yes, it would probably be most polite for people to graciously decline, but unless they've solicited contact by advertising an open position I don't think they are being rude by not replying.
As pointed out in the comments, you've sent your message immediately before or right after the start of a typical holiday break period in the country you are targeting. To expect a reply while people celebrate the primary holiday season in their country is even more rude. Although working styles vary immensely from person-to-person in academia, my impression from the French academics I have worked with is that they are far more serious about separating their work from their vacation than the average US academic.
I personally would not recommend sending any reminder email after unsolicited emails like this. If you do, follow the general guidelines in How should I phrase an important question that I need to ask a professor? and please do wait for at least a couple more weeks past the holiday season for people to catch up on more important things first.

I completely agree with the second paragraph... But the conclusion I draw from it is: Yes, absolutely do send a reminder email. Chances are, the email sent during the holidays has been completely dismissed. If you don't send a reminder email, you'll never get a response.

@Stef It's still rude. Most people have a system for going through their emails after holidays; prioritizing what needs to be done now, bypassing things that can be delayed. Getting a second unsolicited email while the first one sits in your to-do list isn't helping the problem of having too many emails.

I wouldn't say it's rude. Professors and university researchers are in some respects public figures: we have public emails because it is acceptable and expected that people are going to contact us (about PhD's, news stories, whatever). I can't always respond to every email, but I don't think it's an offense to try or to retry (3 tries would be the limit though, if there's no response by 3 tries spread over maybe 6 months it's not going to happen).

Don't frame it as a "reminder" at all
For any unsolicited emails where the recipient has reasonable grounds to ignore the email, it is not a good idea to send a "reminder" (and especially not a "very polite reminder" --- yuck!).  As Bryan Krause points out in his answer, a reminder could be taken to imply that the original email demanded a response, and this can come across as presumptuous.
Instead of sending a "reminder" you should just send another email that makes it clear that you're still really keen to work there.  Remember that the inevitable effect of the email is to remind him you exist so you don't have to explicitly tell him that it is a reminder.  There is also no need to explicitly mention your previous email, since you can just put your new email as the next part of that email chain and speak in a way that assumes previous contact.  The goal here is to get the recipient to draw the conclusion that you are eager and ambitious, not that you are bossy and presumptuous.
If it were me, I'd wait about six business weeks from the previous email (not counting the holiday break) and then I'd send another email, in the same email chain (i.e., with your previous email showing below) just saying how keen I am to work with him and how much I'd appreciate it if he could consider me.  If I felt like I was being a bit pestering/annoying then I'd probably even ham this up a bit with some self-deprecation.  Something like this:

Dear Professor [NAME],
Hello again --- I hope you had a nice holiday break.
I'm sorry to keep bothering you, but I just thought I'd write again to let you know that I'm still really eager to see if there are any opportunities for a PhD candidature under your supervision.  I hope my CV and previous experience looked okay to you.  If you think I might be a suitable fit for your supervision then I'd appreciate an opportunity to talk more about this.  Alternatively, if you think I'm not suitable, I'd love it if you could send me back any critical appraisal of weaknesses in my skills that are holding me back, or just reply to let me know I'm not a good fit.
Thanks so much for your time.
Yours sincerely,
[NAME], Pestering PhD wannabe
[My previous email in the email chain appears here]

I agree that this is the best approach (if a reminder is to be sent at all).

Maybe not "love", but "appreciate"? :)

I like this one, with some minor polish. Honestly though, I might not even have the prior email included and not refer to it at all, because then I could retry with a nudge on a better email like this which intros with that I hope they had a good holiday. You really can't easily send 3 emails to someone in a chain with no response unless you want things to get awkward.

"[NAME], Pestering PhD wannabe" this is not helpful to include.

I disagree; it blunts the annoyance of the unsolicited email and shows that the student has a sense of humour about any unintended adverse effects of their own actions.

It seems to me that:

the recipients might not have yet read your email as they are still on holidays, OR
the recipients have read your email and are not interested for whatever reason.

I would consider it rude to send a reminder.  These people do not owe you anything so if item 1 applies it is indeed rude, and if item 2 applies they will be annoyed and not answer back anyways.

Hero Ya, they don't owe me anything but certainly I need to apply and get PhD position.

Sure... but you're asking if it is rude to send a reminder email, and my answer is yes.  With due respect: these people don't care about that you need, and viewing this from your perspective is not useful: they might not reply to an unsolicited email as a matter of course given the volumes of such emails, and if they are not annoyed by the original email they might be by a reminder.  I certainly am when this happens to me.

Or 3. They have read your email, might be interested but did not have the capacity to respond immediately - in which case a reminder may be alright.

@Lodinn I actually think that if one or more are interested they are unlikely to need a reminder, but YMMV.

@Lodinn so a remainder by the end of this week is fine? What do you say?

@YannicMuller I would not do this, and I am not alone in thinking this is inappropriate.  I would certainly annoy me to the point of diluting any good will may have had.

@ZeroTheHero Ok!

Don’t follow up. It is rude if you’ve never met before. At my university, PhD students are frequently students and paid employees (research assistants or teaching assistants). If a professor had a position open, they’d be required to post it in an official way and would probably link to it from their lab’s website. If you’ve checked all the right places, that professor probably doesn’t have any student positions open right now. Might be time to look around at other institutions and other professors.

Sending an email the 22nd of December, just before holidays, is not the best idea. You can leverage on that to rephrase your second email, if you want to write a second one.
Also keep in mind, when I was looking for a PhD, about 80% of professors I have tried to contact have not answered my email (a single one). And at the end I ended up working with a professor I sent no emails at all before the interview.
The official channel, which is the PhD application form, is a better idea.
After applying via form, it may help to have a phone call with one of
their assistants, that is NOT directly involved in the decision
process (that would be then your co-supervisor). Prepare some humble
questions about what they expect from you and if you would be a good
fit for them. This would help you both to understand if the future
years of collaboration would go smoothly in the department. Also if
the phone call goes well the assistant would support you at the
moment of the decision.


text:
Do you worry about using certain words too frequently in your research paper?
I'm working on a paper and found myself using certain words very frequently. For instance, the word "condition" or "study" or "group" or "participants" or "the results suggested"....?
I mean I was just looking at a paragraph and realized I had used the word condition 14 times!
Do you have a list of synonyms you use so as not to be repetitive? I've never been good at writing, so I'm finding this very frustrating. Thanks for your help.

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

Some of my students have found this useful: Academic Phrasebank

Does this answer your question? Repeating technical terms using varied ones

For instance, the word "condition" or "study" or "group" or "participants" or "the results suggested"....?

Synonyms in technical writing are not always good. If you have been calling your groups "conditions" in the entire paper, switching to calling them "groups" or "categories" would be jarring. Are you referring to something  else now? I would stop and scan back to figure out what you are talking about. Same for "study" or "participants."
You might have a bit more wiggle room with "the results suggested." Recall though, that most papers are not read beginning to end. It should be clear what you are referring to for someone to pick up your paper, find the paragraph they want, and get the information they need.

I strongly agree. Please don't use different words to refer to the same thing in order to have more variety in the language. It is confusing as it makes the reader wonder if you are talking about something different.

This is good advice, but does not prevent the problem. The repetitions may be present because of an underlying problem with the writing. It may still be possible to avoid the repetitions by changing the structure of the sentences. OP should consult a native speaker or a more experienced writer.

@toby544 in fact, let's liven up equations at the same time. Why would we always use the same symbol to mean the same thing? Booooring. There's no reason the gravitational constant has to be 'G'; why not make it 'Q' this time, or even 'Bob'? The novelty will help hold readers' attention.

There's a difference between terminology, whether established by and for the paper itself or more broadly, and the general language in which the paper is expressed.  The former should be used consistently, but a certain degree of variety in the latter makes the paper easier and less tedious to read.  Moreover, to the extent (which varies) that one is at liberty to choose one's own terms, there is an art to choosing terms that -- used consistently -- contribute to the paper's readability.

@john I answered the question as written. Many more questions about writing could be asked and answered.

@Louic I don't see any evidence OP isn't a native speaker?

Would you go so far as to say research papers are supposed to be repetitive?

@Jay No, I wouldn't say "Research papers are supposed to be repetitive."

At least in math and STEM fields, it would seem difficult to avoid repetition of keywords without damaging the meaning!
In a STEM context, especially mathematics, I'd strongly recommend just saying, literally, what you assert. Don't try to be "cute" or any other rhetorical device (unless you're a bit more senior, and know your audience, and know what you're doing).
c. 1978, I asked Serge Lang about what seemed to me a very repetitive use of "thus" and "then" in others' and my own mathematical writing. It was a minor revelation to me when he told me to worry about bigger things, and not to avoid that. :)

and not to avoid that. :) -- The story sounds better ending with "and then said thus one need not avoid that. :)".

@DaveLRenfro But... but... "then" and "thus" and "that" all sound so similar :(

The advantage of clear writing far outweighs the disadvantage of repeated words.

@MaxD Then use a different word. Swap “thus” for “hence”, and “then” for either “therefore” or “so that” (depending on the intended meaning) – but don't try to alternate.

@DaveLRenfro: Thus saith Serge Lang then.

To echo, but also to supplement what others have said: clarity comes first; but for me producing nice-to-read prose is also an important criterion. If you do it right, then these two demands will go hand in hand; but if they don't, then clarity and precision should not be compromised on.
And yeah, writing well is very hard!

If you are referring to the same thing(s), use the same term.
This isn't creative fiction writing, or dramatic historical writing. Your readers want to understand quickly, clearly, and unambiguously.
You can help by ensuring they don't have to stop, backtrack to figure stuff out.
If you called them a sample, then every time you refer to the same thing/s or people, call them a sample again. If you called some set of people a group, use the term "group" every time you discuss something similar, such as that or a parallel group. If you refer to a study in a paper in 2018, then use the word study for an equivalent study in a paper in 1993. And so on.
(At most, check if you can condense a bit. If its obvious what you mean, can you use "that/it/those condition(s)", instead of "the above/the same condition", for example?)
You get points for clarity, unambiguity and ease of fast reading. You get points docked for making readers have to work out if the "group" of people  in section 2 is the same as the "set", "study population", or "target cohort" of people in section 3.
Similarly, you get points docked if you use the word condition in section 4 para 3, but a different synonym for the same kinda thing in section 4 para 4.
Using that term 14 times means you saved your reader 13 occasions of having to check what it meant and learn a new group, subset, criteria or whatever, to understand your work.

If you have a tendency to use a certain word or phrase very frequently, you can come across as a more eloquent writer or speaker, and your writing can flow better, if you actively try to limit your use of that word or term.
But this applies to English/language in general and I wouldn't worry about this when writing a research paper any more than I usually do (but I'm usually semi-formal and fairly pedantic about my use of language). The goal of a research paper is to clearly convey the methods, data and results of some research. Eloquence and well-flowing text is easier and more pleasant to read, which is helpful, but this is secondary to clarity (it doesn't help to have a paper with text that's easy to read, but where the actual research isn't understood or it's misunderstood).
Using synonyms is one way to avoid the overuse of certain words, but, like others have mentioned, you can't necessarily do this in a technical research paper. It can cause confusion to use different terms for the same thing and often it would be more clear to just stick to the same word. If you were replacing e.g. "likely" with "probably", that's much less likely to cause an issue, given that different instances of those words are typically not so closely linked (although they do mean slightly different things).
Rephrasing is another way to avoid word overuse. You could possibly avoid reusing a word by combining sentences. Although long sentences tend to be more confusing, so try not to overdo it.
Using pronouns like "it", "this", "them", etc. is also a common technique to avoid word reuse. Although this may also lead to more confusing sentences, as it's not always clear which noun a pronoun is referring to, so approach with caution.
As an example, this would sound very unnatural (not only due to the repetition of "equation"):

The equation is based Newton's laws. The equation was determined in consultation with leading experts. But no-one fully understands why the equation works.

It can be rephrased to something like the below, which sounds much better.

The equation is based Newton's laws and was determined in consultation with leading experts. But no-one fully understands why it works.

But you'll need to figure out how to do this (if it's even possible or a good idea) largely on a case-by-case basis.

Given the unavoidable presence of statistics and probability in almost all of experimental science, I'd avoid using the word "probably" without reason. I also would be very wary of pronouns. It might be obvious to you what a pronoun is referring to. But it's not going to be obvious to the reader. Having to determine which word in the previous sentence is referred to by "it" or "this" or "that" in the current sentence is just one extra layer of obfuscation.

@Stef  Banning pronouns is a ridiculous idea. "Probably" is a common and useful word, and can easily be used without readers getting confused about whether it refers to a probability.

@toby544 There is nothing ridiculous about using explicit words. That doesn't mean banning pronouns. Just be wary, and don't be afraid of repetitions. In case of doubt, always prefer an explicit word.

I second the recommendation to minimize the search for synonyms. Azor Ahai -him- explains the reason very well.
My first bit of advice would be to go back and look at papers you thought were especially clear, and see what they did. Of course you can't copy their wording, but you can learn from how they phrased things. This is a big part of becoming a better writer.
My second thought is that, if you are using the same word 14 times in one paragraph, then it may be organized poorly. If the groups are in the hot condition and the cold condition, then maybe you are ping-ponging back and forth between talking about those two conditions more than you need to. Perhaps you can list some properties of one condition, and then the other.
Clarity, however, is the rule. Ideally, reorganizing the paragraph will actually make it clearer. Sometimes trying to rephrase things to satisfy some arbitrary-seeming constraint will lead you to come up with wordings you otherwise would have missed. But if you are losing clarity, go back to what you had before.

Don't use synonyms to avoid repetition. It may create an impression that two different words refer to two different things. This might be confusing, especially for the readers who don't know the language very well.
Synonyms often have only similar meaning, not exactly the same. Searching for synonyms can help to find a word to describes something the most precisely. But if you exchange synonyms at random, you disregard part of the meaning and add noise to your message.
As Mark suggested, excessive repetition may signalize there is some other area where you can improve your writing.
There is a relevant article on Wikipedia:
https://en.wikipedia.org/wiki/Wikipedia:The_problem_with_elegant_variation

Elegant variation is the attempt to relieve repetition by replacing words with synonyms. (...)  Elegant variation distracts the reader, removes clarity, and can introduce inadvertent humour or muddled metaphors. It can confuse readers who are unaware [that two different terms refer to the same thing]

Also, a related question (possibly a duplicate, but I don't have enough reputation to flag): Repeating technical terms using varied ones

I do worry about that indeed, but it depends on the part of the paper these repetitions are encountered in.
In Materials and methods, the language would be very dry and precise, and sentences terse. Does one need to worry about having 20 instances of "bolt" on a single page of a furniture assembly instruction? I believe not.
In the more free-form parts, however, you might want to avoid these repetitions. Again, this does not have to do with substituting technical terms, rather rephrasing the whole thing if it does not sound fluent or eloquent enough.

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text:
How much is PhD admission procedure in France different from that in UK?
I live in an Asian country and I have planned to apply to PhD positions in France and UK in Pure mathematics.
I wrote to one professor saying that I am interested in his work and would like to work under his supervision for PhD research.
Here is an excerpt from his reply:

"In France we have essentially targeted funding so we need students
and advisor to know themselves well in advance so that a project can
be defined before applying for funding. It can work only with at least
a couple of months preparation before the deadlines for funding
applications."

So, there is quite a difference between PhD research in UK and France. In UK the admissions are centralized and other 1st year is taught advanced courses and funding is for 4 years while in France the funding is for 3 years and I don't think there are taught courses, admission is also not centralized.
So, can you please tell me some other differences  which I must keep in mind (and which I am unaware of) before writing to professors in France?

I suspect that your understanding of the UK system is quite lacking, too. It might thus be better to ask HOW stuff in France works, rather than asking about the differences to the UK. It would also make the question more broadly applicable.

@MassimoOrtolano Let me have a stab at that canonical question for UK and France. I agree that the answer to this question would fit there perfectly.

@penelope If you have time, that would be great! :-) We surely have a number of users from UK and France, but probably thus far none have had the time (or dared) to write something.

@MassimoOrtolano I never noticed the question :/ I did my PhD in France (so might be a bit outdated) and now hold an academic position in the UK, so I think I should be qualified to answer both.

There may be some informal advanced lectures during UK PhDs, but they are typically not mandatory and there will be no exams. The focus is on research from day one, like most (all?) European PhD degrees.

@astronat That is very untrue for an increasing number of PhD positions in the UK. Most of them now include a first year of coursework, which is graded, and includes a final project. They also sadly do not consider themselves particularly European.

@penelope Thank you so much! Even if you are a bit out of touch with modern realities in France, you know how it is in the academic world - it's much easier to get people to critique your work than publish their own ;)

Does this answer your question? How does the admissions process work for Ph.D. programs in Country X? I have now cast a vote to close this, as I've provided very detailed answers for both the UK and France to the other question.

@penelope Thank you very much!

@penelope interesting, in which field? I have never heard of this in physics. I finished my PhD in the UK last year.

@astronat I'm in computer science/robotics. It is mostly in the last two years that we had a major re-balance of the type of PhD students in the group: from individually funded 3 years research only programmes, to 1+3 years (1 year integrated thought masters) programmes funded through larger grants awarded to institutions for several PhD positions 3-5 years in a row. Feel free to read my UK answer to the linked duplicate question for more details.

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text:
Cultural differences and etiquettes to be kept in mind for a PhD interview
I am a student who has completed his master's and I took an year break to prepare for PhD and living in a poor Asian country. I applied to PhD programs in Germany and France this year in Pure mathematics and I have received 3 interview calls in January last week and February.  1 interview is with prospective supervisor alone and two are with admission committee of 4-5 members.

What cultural and etiquette should be kept in mind during Zoom interview?

Also, suppose In case I am not able to answer question of a professor (suppose being not able to recall or not so good  in that field) then how should I handle that situation politely and without getting embarrassed ?
(I have only given one interview in my life (in person, not remote, for master's entrance) and I must say that I panicked because I was not able to solve the questions completely and hence said to prof. I don't know and in fact I felt a bit relaxed when  that interview was over (despite being in good command over the topics that I was asked questions about and the program being one of my favorites). I have a problem of panicking in front of authority and have self esteem issues.)
I am putting emphasis on culture and etiquette because my country is one of the most racist country where people are not known for their politeness and are very sarcastic to each other.
So, what etiquette should I keep in mind while giving interview to be polite  and well mannered person as per Western European standards?
Any advice, reference you give to me will be immensely useful to me and I shall be really thankful for that.

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer.

@Community Done.

If you do not know something, just say so.

If it helps, the interviewers are likely to be very used to talking to people from other cultures and countries, so will probably be forgiving of any accidental faux-pas.

Your problem does not sound specific to your country of origin.
In my language, there is an idiom with a rough equivalent of "there's no way to make up for lost time". It helps people hold their heads high when facing anxiety, particularly when it comes to exams or interviews. You have arrived at this point in life, there is no need to pretend being something you are not. If you do not know something, just say it. If you get a feeling you have offended someone by some cultural quirk, bring it up, get an explanation, proceed from there. People make inappropriate remarks even growing up in the very same country; learning from your mistakes and being able to move on is more valuable than wasting precious resources on self-flagellation. So just try your best to not overthink it, observe and learn.


text:
What should I expect in PhD interview in Pure Mathematics
I am a person who completed his master's last year and now I am preparing for PhD admissions in Europe. I am from a poor country in Asia.
I have 3 PhD interviews scheduled in last week of January to February for universities in France and Germany.1 interview is with prospective supervisor alone and 2 are with admission committee of 4-5 members.
Question: What exactly should I expect in PhD interviews of Pure Mathematics? Are questions related to Master's thesis asked? How long should I expect the length of the interview?
Any advice will be highly appreciated.

The process is likely to be different in different places (even within the same country, or university), depending on the person interviewing you. I currently work in Germany. At the University of Bonn, the topology group interviews candidates together. The interview consists of a 30 minute presentation by the candidate, usually on their masters thesis, followed by 30 minutes of questions. The scope of the questions is informed by the presentation -- the first half is usually based on the talk, and the second half on general questions in topology, of the type we would like the candidate to already know before starting a PhD position. Sometimes we ask the candidate whether there are specific topics they feel most comfortable with, but usually this is clear from their application. Yes, we do ask the candidates to solve math problems.
Even within my university, the process varies from group to group. Others might just have a quite informal (virtual) conversation with the candidate, the talk component might be unique to the topologists. The length of the interview is highly variable.
My advice to you:

Ask the person who invited you to interview what the planned format is and what you should prepare.

Regardless of the answer, be prepared to answer questions about your masters thesis, your general area of research, and the direction you would like to pursue for your PhD.

Just wanted to ask the following: I have done quite a lot of pure mathematics courses in my Masters which were heavy in proofs. I wanted to ask whether Prof.'s ask about proofs of results done in courses or just results. While revising, I revise only statements of proofs and definitions in a course. I don't remember a lot of proofs even of courses done in last semester ( 3rd semester). So, will it be a setback to me or not in the interview?

A reasonable interviewer wouldn’t/shouldn’t ask about technical details of proofs, but questions about “ideas” of proofs of big theorems are on the table I think. (E.g. proof of the Seifert van Kampen theorem. I wouldn’t ask anyone to write down the details, but would like to know that they know the general idea).

But you should also consider proofs of the type you probably had to do in homeworks or exams, those are definitely on the table. Also, well known counter/examples.

They will ask you about your research, your masters thesis (possibly also about your masters advisor), and whether you know what to expect from the PhD program. Usually, unlike interviews outside academia, they will not ask you to solve math problems. But they may ask you what classes you took.

" They will ask you about your research"  but I haven't started any research yet?

" whether you know what to expect from the PhD program"  I am expecting publishing 2-3 research papers and then join a good institute as post doc fellow.  Is there anything else I should expect?

If you have not done any undergrad research, you chances of getting into a PhD program at a good research University are slim. Publishing 2-3 research papers (or even one paper) in good journals would require a lot of work, and you seem to not expect any hard work.

@markvs When you say 'research', do you mean original research or simply having learnt about the topic you want to study at PhD level? In the UK most people will be interviewed whilst part way through fourth year of an undergraduate degree, and they very often won't even have started their master's thesis by that point

@Person: By "research" I mean what here is called REU, research experience for undergraduates. It is usually done after the second or/and third year. It is an ordinary research which may or may not be related to future MS or PhD research.  The point is that the interviewers should determine whether the applicant is prepared for PhD work and the applicant should show that (s)he is.


text:
Motivation letter / Letter of Application for 1 year Research Master's in France
I have done my master's from an institute in Asia in 2021 and I planned to  write for PhD in Pure Math in 2022 in France. One prof suggested me to also apply to 1 year Research Masters(M2) in France called Fondamentales Mathematique.
I have to supply a cover letter for scholarship applications and I am confused on how to write them.
My approach:

Dear Sir/madam, I am applying to M2 Research in Mathematics program of
University of X because this program is very well suited to my aim of
doing PhD research in [Branch Y of Math] after completing master’s
studies. M2 program will allow me to gain more specialized knowledge
in [Branch Y of Math] and other branches of Pure Mathematics.
Although, I have completed my Masters in Mathematics in 2021  from
institute Z but ...  . France has a lot of researchers working in
number theory and so it will also help me in getting in contact with
them and discussing PhD research opportunities.   2 paragraphs
describing my interests and motivation in [Branch Y of Math].
2 paragraphs describing work done by me in my Master's thesis in
institute Z in 2nd year of my master's.
last 1 para.
Sincerely
ABC

The length of the letter of motivation is 2 pages and 70 % of a page is taken by explaining the work I did in my master's thesis because I did very detailed study of 5 papers and I can't squeeze an explanation of that in less than 70% of a page.
Is it a good way of writing a motivation letter?

General answer
Points customarily expected, when applying to a master's program in France, are:

Are you a good fit for the master, from an academic point of vue? Do you have all the pre-requisites? Will you be able to follow? This is especially important for foreign students - if you already have a master, you should cover its contents quickly in your letter. Also, did you check what language the classes will be in? Universities with classes in English only are rare, so your knowledge of the French language should be mentionned.
Is the master a good fit for you regarding what you want to do next? This is where you talk about your research interests, why this specific master is intersting for you, and so forth.

Points to consider

1 Prof suggested me to also apply to 1 year Research Masters(M2) in France called Fondamentales Mathematique.

There is no master called "Fondamentales Mathematique" in France, but there are masters in "Mathématiques Fondamentales" (it is a field, not a lab name!). What makes this specific masters program interesting for you, compared to all the others existing on the same topic?
Can you mention this teacher? Does he know the people on the committee? Did he already send students here?

Letter structure
To go back on your specific letter, here is what I would suggest:

Sir/madam,

My name is A, and I just obtained a masters in Mathematics from institute B, specialized in X. As I am interested in field Y, and intend to do a PhD in it, prof Z suggested I should broaden my knowledge of the field by coming to your program.

I am very interested in coming to your masters, as it teaches bla, bla and bla, which are essential to learn about Y. (You need to be specific about the masters!) I am also particularly enthusiastic about meeting researchers working on Y, and discussing with them about research and PhD opportunities (if you can, be specific about the researchers!).

Field Y interesses me because ... (though it's likely not the most relevant paragraph)

During my previous master, I have studied the necessary prerequisites (Add info about your grades and achievements here! This part is important and missing.) During my masters, I also did my master's thesis on ... Lastly, following classes in French will not be a problem, since...

If you can add a paragraph on what you can bring to the lab, like relations with researchers you know or worked with..., it's a plus.

A version of "Thank you for reading, I am at your disposal for any further information you might need".

Sincerely

I am not sure if using sir/madam is usual in France. That's Indian subcontinental style. Usually, in western countries, supervisors are addressed as Dr. . So, first of all, I would ask you to make sure of this thing.
In the first paragraph, instead of saying France has lots of researchers, I would say more about the institution and/or the supervisor. This will make an impression that you are interested in the institute itself, not any place in France.
The rest of the structure looks good. The overall letter will depend on how you write your experiences, motivation, and other staff.

It’s common in France to start with « Madame, monsieur… ». But it is good to address the person with his/her title.

I would not write “France has a lot of researchers”; you should specify which researchers you want to work with at the university you’re applying to. So while much of your cover letter will discuss your thesis work, you also need to include how your interests and goals intersect with the work and labs of your potential professors.

do you mind writing a detailed answer?


text:
How should I follow up with this professor?
I e-mailed a prof in France (Paris) on 10 Jan that my research interest align with his work and I would be willing to work for a PhD under his supervision in Pure Mathematics.
I am a student living in Asian country. His reply was the following:

Dear X,
Sorry but to get a grant for a PhD is very competitive in Paris. You need to be supported by a mathematician who knows you and is convinced that you have a top level.
I don’t know you and I have already 2 PhD students

How should I follow up to this email of his?
Should I ask my referees to send him LOR? Should I send him a copy of my master's thesis? I have already attached my CV along with this e-mail.
Edit 1: I also wrote at the end of the mail that if Prof. doesn't thinks that  I might be a suitable  for  supervision then I shall be really grateful if he could write me back briefly about the skills I am lacking.

Telling the professor you'd be "willing" to work for them instantly got you off on the wrong foot. If a student told me they'd be "willing" to work for me, I'd be insulted by their presumptuousness and tempted to respond, "Great, but I'm not willing. Please go away."

@NicoleHamilton That was not how I actually wrote in mail

(regarding the edit) ...now you are just making matters worse. First you cold email them without LOR or anything, and you are trying to get them on board to help you with your application. They might not even have read your CV if they know it's a no-go without a LOR, and spending the time actually reviewing your application in-depth might be a serious waste.

@NicoleHamilton If a student from a non-English speaking country wrote to me that they would be "willing" to be my student I would understand that the intended meaning is "interested" (note that in many languages the distinction between one's "willingness" and "desire" is not as explicit). There may be other aspects of the student's email (which we haven't seen) that conveyed a sense of presumptuousness, but professors should not ascribe so much meaning to the choice of a single word.

@Max "Many" languages where there's little distinction?  That sounds suspect.  In any event, the OP was writing here in English, where there is a distinction, with what appears to be sufficient proficiency to know the difference.

@NicoleHamilton You're not supposed to be able to read Shakespeare or catch willing-vs-interested nuances in an email if you're doing a PhD in Pure Mathematics. I'm pretty sure no one (specially not a non-native English speaker) would take offence with an email saying "willing" instead of "interested". Not because you're a native speaker you can expect everyone to speak English on the same level as you (or catch that "suspect" in your comment is not a grammar error).

@user4052054  It'll be interesting to see if the votes shift over time but right now, it looks like more readers agree with me than you about whether "willing" might quite reasonably be taken as insulting.  Either way, if you reach out to a professor who doesn't know you from Adam, asking they take you on as a student, they are free to say no for any reason or no reason.  For many people, I think this would be a reason.

@J... depends on the school and country, this is almost exactly how I got into grad school.

good question. actually re 'You need to be supported by a mathematician who knows you and is convinced that you have a top level' - what if you convince a mathematician you are top level but the mathematician does not know you prior? how will they know you if you don't e-mail them anyway? so what you really have to have worked/studied in the school already or be some kind of celebrity or something?

wait what exactly is the school's procedure for applying as a PhD student? like e-mail the professors is the 1st step of the application?

You don't. That email is a clear rejection, and any attempt at a follow-up will only waste everyone's time. If the professor were open to get to know you and to be convinced that you are a top-level candidate, they would have indicated so. Instead, they are pointing out that they already have two PhD students (and, tacitly, are thus not overly keen on investing effort in a small chance of acquiring a third).

Too strong. You are making assumptions. Following this advice absolutely closes off an opportunity.

A clear rejection would require far less thought and no hope of acceptance: "Sorry, I'm not accepting students at this time.". Instead, the professor shows a path, hard though it may be to follow.

@Buffy  Sure looked clear to me.  I agree 100% with this answer.  I would not split hairs over the fact the response was more than one sentence.

@Arno Kindly have a look at edit.

Looked clear to me as well but Buffy seemingly suggests that OP should reattempt the contact properly, which - fundamentally - implies not following up on this email exchange immediately or, ideally, in the near future at all. It is not entirely unreasonable IMHO. Arno's answer, however, is better as an advice - to act upon Buffy's, OP would have to first change their perception of the admissions process and power dynamics involved.

@Avenger The info in your edit doesn't change anything. Almost everyone will ignore such a request anyway.

@Lodinn Can you please elaborate on "OP would have to first change their perception of the admissions process and power dynamics involved" ? What is wrong with my perception of the admissions process and power dynamics involved?

We can theoretically argue if this means reject at 99.99% or 100%. But in practice, it means that it will be more fruitful to contact a different professor (with a different approach).

@Buffy I am not sure if they are showing a path, but rather stating what kind of requirements would be necessary, and that they are quite hard to achieve. OP writes "I shall be really grateful if he could write me back briefly about the skills I am lacking". French academia in pure mathematics is very competitive, so taking this stance is not a good idea. Very good candidates may not get a position, even if they have LORs from local professors, etc.

@Buffy: You suggest this isn’t a clear rejection, by saying that a clear rejection would be simpler and more direct.  But to my ear (and to some students), your example would also come across as so blunt that it is unkind.  As I read it, the OP’s professor means what you say, but wanted to say it in a way that was more tactful, and with more explanation, to be helpful to OP.

@Buffy I don't think the professor is showing "a path"; he's just stating the requirements OP doesn't fulfill

@Avenger Like everyone else has pointed out, they basically don't care about you, your grades or your thesis - maybe unless you have a truly impressive publication record. You have to have strong results and/or someone vouch for you if you haven't established contact already - rubbing shoulders directly at a conference would be another option. You insist on them being willing to spend time on you. You are not likely to have doors closed in front of you because of that (unless you truly persist) but surely not getting them opened, either.

@Buffy: the prof is probably French. A blunt answer such as "I do not accept new candidates"  is too strong to my (French) ear. The answer is a clear "no", with some extra hints if OP is interested in applying elsewhere in France.

It's a rejection.  A "thank you for your time" letter would be ok, but that's it.

I agree with Arno that this is a clear rejection. I work at a university in Germany, but I suppose the situation in France is similar.
The problem is that professors get such e-mails all the time, usually from students who live in an Asian or Arab country. They send CVs, grades, even theses.
Now imagine a professor who gets an e-mail like yours every few weeks. They simply do not have the time to check if one of these students might a prospective candidate. Open positions are rare, and if there is an open position, they select one of the master students they have already known for 2-3 years. This is what your professor means by saying that he does not know you. You are lucky that he answered you. I know that there are professors who simply delete e-mails like yours.
Sending a thesis or a letter of recommendation will not help here. You are one among hundreds. If you really want to do a PhD in France, you have to establish a direct contact. If the university where you want to work offers a summer school or some postgraduate courses, try this.

wait so re 'You need to be supported by a mathematician who knows you and is convinced that you have a top level' - what if you convince a mathematician you are top level but the mathematician does not know you prior? how will they know you if you don't e-mail them anyway? you said 'you have to establish a direct contact' but how is what OP did not direct contact? by direct contact you mean actually attend the school eg 'summer school or some postgraduate courses' ?

I suggest that you don't send a thesis. It is too much, too soon. Instead, find a sponsor locally who knows you well enough to be a fair judge of your likelihood of success in a competitive program.
Have that sponsor contact the professor, preferably personally, with a recommendation. Even a LoR can wait until the initial contact is made. Given the assumed level of the Paris institution, someone with a known reputation would be best, but someone active in research certainly.
After initial contact by the local sponsor, other things can be sent as requested and required. But let someone else speak for you initially.

Can you please tell what exactly you mean by sponsor ?

Sorry. in this context, a local professor that will speak highly of you. Not a formal thing. An advisor is usually the right choice.

There is a local prof. who I did a seminar and master's thesis with.

That sounds about right.

But how exactly to write local prof. to " speak highly of me". Isn't that same as asking local prof. to write a reco.?

Basically yes, but personal informal contact in this case is probably better than a LoR. If that isn't possible, then a formal letter is needed.

So, how exactly should I write to my local prof to make him aware about the situation? ie how should I write to him to make him personal informal contact?

Kindly also have a look at edit.

1 - 'Have that sponsor contact the professor' --> wait do you mean the SAME professor? or for future potential professors to apply to? 2 - @Avenger what exactly do universities in paris say? Does they say like 'step 1. contact our professor step 2. get a professor from your school to contact professor if need be. step 3. submit requirements if our professor is willing. step 4 etc' ? 3 - Back to buffy do you think anything that anyone's answer should probably depend on the specific requirements/procedures of the university/ies that OP is applying to?


text:
How to write a research proposal for a university in UK?
I am applying to a PhD position in Pure math in UK.
Application form asks for research proposal but I don't have any specific problem now which I would like to work on although I have a good background on the specialization I want to pursue.
I emailed my prospective supervisor the following:

I was completing the PhD application form and now there is only
Funding and research proposal left. Should I fill the research
proposal at this stage? I mean, I want to work under your supervision
in X but certainly I am not certain what exact problem I will work on.
The background I have in X and other branches of Mathematics
(including Master’s Project) has been mentioned in my CV and Personal
Statement.
I have been googling about it but I don’t think I can decide what
exactly I should do about it.   So, can you please guide regarding
what exactly should I write in it or should I leave it blank?   I
shall be really thankful.

His reply was following :

I am not sure about exact the requirement here, but I don't think you
are supposed to know exactly what to work on at this stage. I think
just an indication of your research interests should suffice.

The issue I am facing is that I don't know what problem I should work on. I could write the title as "Topics in X" and I don't know what I can write in the abstract and other parts except the topics in X I have studied and my interest in particular branches of X. Pure mathematics is really specialized.
Can you please guide how should I approach the problem I am facing?

I'm in Engineering not Maths, but from my experience (both when I did my PhD and now as a PhD supervisor - both in the UK), the potential supervisor would provide guidance on what to write in the research proposal such that it aligns with their research interests. But this may vary by department/university. Having a good research proposal is important if you want to be competitive for scholarships. It's unfortunate that the supervisor you contacted was not more helpful. Perhaps try to email others.

Universities in UK do not usually ask for a full research proposal, but rather a short summary of what topic/questions you are interested in (although it might be different in the university you are applying to).

With the caveat that (within UK system) I have only observed things at my own university, but know them well (in mathematics) - the requirement to have a proposal is often part of the central university admissions process rather than something desired by, or important to, the selection commitee within the department. Certainly, it is recognised that applicants in your position usually won't have a focused research proposal, and so I would follow the (vague) indication of your potential supervisor

This is mainly a synthesis of what's written in the comments, with some addition of my own.
It's likely that the application form is not specific to the maths department, and is used for PhD applications across the whole university. This means that the research proposal section is there for subjects in which it's more common for the student to have identified a very specific question they want to research before they apply.
This doesn't mean you should leave it blank. As your potential supervisor suggested, you should write about your research interests and give some evidence that backs up that interest, as well as a reason why you want to study that topic at that university.
For example, you may write something like

I am interested in pursuing research in the topic of Advanced Y
Problems. In my undergraduate degree, I greatly enjoyed the course on
"Introduction to X", in which I achieved a grade of 99%*. This led me to
doing a summer project studying Applications of X to Y. During that
project I found that X can often be applied to Y, which built on the work done by NBH et al. (1995). I am keen to
continue studying X and Y, and especially Advanced Y Problems, due to
my strong background in Y from the courses "Topics in Y" and "Topics
in Y Part 2" which I took in the final year of my degree. I have
chosen to apply to the mathematics department at the University of A
because of the active research group in X, Y and Z and especially
because I am interested in studying if there is a link between the applications of X to Y and Y to Z, as recently proposed by MTW et al. (2019).

Obviously you would give more details on the particular topic you're interested in, courses you've taken related to that, any projects you have done, papers you have read and so on. I would aim to write about a page in total.
Good luck!

*This is not a suggestion that you need such a high grade to succeed in PhD applications! Just an example :)


text:
How to properly manage unexpected questions in screening interviews?
There are various sources in the internet in which different general questions one may encounter in screening interviews are reviewed. However, sometimes an applicant may face with some questions whose answers can be totally unexpected, and often opinion-based. In such cases, as the applicant (unfortunately!) can't read an asker's mind to know what perspective appeals them more, is there any strategy (for example, based on the clues captured during the interview) based on which an applicant's answer properly converges to what is expected to be heard?
Example*:
A while ago, I was invited to an interview (for a tenure-track assistant professorship position) conducted by a search committee of an aerospace department (in an R1 U.S. university). A panel member asked me the following question:

We have a large body of undergraduate students who are potentially
interested in your field of research. What sorts of plans would you
have to successfully attract them to this stream of research?

I, initially unprepared to be asked such a question, thought for a couple of seconds and then (kind of spontaneously) responded like:

Nowadays, the surge of coding and programming among undergraduate
engineering students is growing. They often, regardless of their
majors, get attracted to computing by learning some programming
languages. Since the background of the majority of them to do
cutting-edge research in my field may not initially suffice, I would try to
define some numerical projects for interested students through doing
which they can gradually get familiar with the principal components of
the theory in my field. Once they are equipped with some solid
background, as well as the magnification of their interests in the
field, I may hopefully plan deeper steps for the involvement of those
students to the mainstream research in my field.

The asker then rolled his eyes staring at me for almost 10 seconds when he finally said:

Well... I don't know about that!

I don't claim that my answer was the best possible line of planning for what that question sought. What I am concerned with is some strategies to handle such not-that-much-standard questions so that the result would be less embarrassing that what happened in that experience.
* The quotes may not represent the exact word-by-word passage of conversation.

I don't see anything in your answer that should make anyone roll their eyes and embarrass you, and an interview is a two-way assessment: you might not want to work with certain people.

I am not sure the real issue is that the question is "opinion-based" - it is the fact that the question is unexpected/non-standard.

Did you ask for an explanation on their response? Or did they move on to the next question/topic without giving you the chance to respond?

@Dawn - at the places I did my undergraduate and graduate studies the departments put a premium on involving undergraduates in research with professors. Such a question would definitely come up in interviews. So, no, such a question should not have been unexpected if the OP had done a little homework on the university ethos.

@JonCuster My university is also interested in this as a SLAC.

Isn't that the whole point of an interview? To peel back some of your formal, studied "face"?

@MassimoOrtolano I rolled eyes here, it is just hot air and a stream of buzzwords (probably because they didn't know how to answer this).

speaking directly, the asker got what he wanted, 1. asked a placeholder question, expected you to give something outrageously sounding positive or something 2. you gave him a rational answer 3. now he acts like you were that nerd in the class. Zero points to him.

When an interview question is non-standard or otherwise unexpected, I fall back on two types of answers:

Reframe the question/your answer more generally. So, if a question is around how you would teach a particular class, you would answer by describing your teaching philosophy or your approach to syllabus design more generally. Often, by the time you have done this, you have developed at least a couple of thoughts specific to the class so you can end with some more specific information. So in your situation, you might start by giving a minute or two about why you are excited about undergraduate research and your general philosophy about working with undergraduates. Then maybe you would have a thought at the end about the specifics of the issue of recruitment.

Turn the question back around (Note this works at most one time per interview.) In this type of answer, you say something like "I think the norms and culture of the department play a big part in how I would approach this. I would probably start by checking what other faculty are doing and what has been done in the past. In general, my goal would be to align with the standards in the department. How are other faculty [handling the situation currently]?"

If I were in your situation I would have asked for clarification instead of trying to answer right away. In your case, there is ambiguity in what the interviewer meant by

successfully attracting [undergraduates] to your field of research

Are they talking about undergraduate research projects (your answer seems to assume this)? Or do they mean attracting undergraduates to graduate programs? In the latter case, is that at the masters or PhD level? At their home department or a different program?

Sorry if this starts with jumping into the subject used as your example rather than giving the generic thoughts first - doing this as I feel something useful may be able to be derived from this.
I would not try and attract all of "a large body of undergraduate students who are potentially interested in your field of research" to research in that field. I feel doing that would be prone to attracting, and having to deal with and tediously sort out later, a lot of people who are only slightly interested and/or just looking for "something" to do and who are not really excellent, or have the potential to or interest in being excellent, in the field or specialty. Also running increased risks of missing the really interested and excellent ones by drowning them in a mass of others, and/or turning them off by having to be drowned in that mass.
Thus, if I had to answer that question I would along the lines of saying that I would specifically not try to attract all of these people to doing research in the field, but rather attract those that can be derived to be particularly promising. In order to achieve this, I would probably ... (plan designs here)
I might or might not add, "of course in alignment with the policies of this school and department" but I feel I'd rather be inclined to not add that because I feel strongly that showing some determination and backbone counts more in a responsible position than offering to be a total push-over just in case.
Also, it seems to me to be taking a side-path approach to follow the path you sketched out in your answer (get people to do X by getting them to do Y in an environment revolving around X then hope doing Y will make them good at doing X.)
I wonder if these points may offer some of the expectable explanation of the eye-rolling you got.
Avoid overdoing the urge to give a pleasant answer, if that could be an issue. Focus on what you would want to shape a situation into, or if there's too little time to make up your mind about that in a moment within an interview situation, on what you think a situation should helpfully be advanced into to create advancement to the field.
That said, if you find in an interview situation there's something you are having too little time to make up your mind about in a moment, you may find you have just harvested from that interview situation some insight into the fact that there's a thing you simply haven't thought about yet. When going for a position where you will be able to shape things, I feel you want to have given a lot of explorative and constructive (quasi engineering) thought to the goals and designs you would like to pursue in shaping things, seen from both angles, i.e. both for the good of the field, school and department and, as well, what you as being who you are would really like to do in that quarter.

Another option to add to the excellent suggestions by Dawn:

Decline to offer a solution — Just say you would think carefully about it: (This method should also be used sparingly.) Sometimes when you get a question about how you would solve a certain problem, the best answer is that you would not solve it by making a snap decision in an interview.  Tell the panel that you are not inclined to try to develop a solution on the spot; in leiu of this, tell them about your decision-making process and how you would go about thinking about the problem and developing a solution.
This is a particularly useful response if you have a demonstrated track-record of experience in the field, and a demonstrted track-record of making managerial decisions.  (It is less likely to work if you are inexperienced and applying for an entry-level position.)  By declining to solve the problem on the spot, and explaining your decision process instead, you can show the panel that you are a decision-maker who takes their time with a problem and is not going to be rushed into making a snap decision when that is not necessary.  If you like, you can say to them that you are happy to implement your decision-making process after the interview, and come back to them in a day with some proposed ideas for a solution.
This method can work well in some cases, if used sparingly, and if you can back it up by showing that you have demonstrated experience making decisions on these types of topics.  Sometimes an interviewer may react sceptically if you have not answered the question, and press you for specifics on how you would solve the problem.  In this case you have to decide if you can wiggle out some reasonable tentative suggestions (without looking wishy-washy on your original stance) or if you want to hold your ground.  If they keep pressing you for an immediate solution you can even be brave and go further --- tell them that any candidate who gives them an immediate solution in an interview is doing them a disservice, since they are making a snap decision on an important matter instead of implementing a longer deliberative decision-making process.  Reiterate your managerial experience with this kind of problem and use this as a basis to make an adverse judgment on the proposal that you provide a snap assessment of the problem.  But beware: Depending on the attitude of the interviewer, this can go badly and sink your chances for the position (which has happened to me), or it can impress the hell out of them and make you the lead candidate (which has also happened to me).

Hmmm. "Refuse" is pretty strong. Probably to strong.

@Buffy: I've tempered this to "decline".  (Also note that the answer allows some wiggle-room to offer tentative suggestions if pressed.)

Still too strong. I wouldn't be happy to hear such from a candidate. "Sorry, I don't have an answer (handy)." is better.

That is one reaction; I would be happy to hear it.  Anyway, I'm happy with the strength of the answer now; your critique is noted.


text:
How many days should I wait for e-mail of Prospective Supervisor in this condition
I am a citizen of an Asian country who e-mailed a prof in France 4 weeks  ago that I am interested in discussing PhD research opportunity in Pure mathematics.
He replied 2 weeks ago asking me to send him my master's thesis and my bachelor's and master's transcript. I sent him that the same day.
How many days I should wait for his reply? I know prof's are really busy and he must have a lot of work to do. But certainly I think it is only my responsibility if he has not replied anything after a give time to remind him.

So, what do you think how many days I should wait  for his reply and what exactly should I write in remainder e-mail( which don't come off as rude)?

Thank you very much.

Since its been a couple of weeks you could ask whether he has any additional guidance for you based on what you sent him. You could also ask what further steps it is appropriate for you to take at this time.
Don't expect a reply, however, for a couple more weeks. And there is no particular reason to push on dates unless deadlines are approaching.
But two weeks or so is probably about right for a follow up. Keep it short and easy to reply to.


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How to get most from this meeting with a "possible" potential supervisor
This question is a related to this question which has been closed.
I will explain the whole situation here again and the question is different:
I wrote to a prof. on December 15 in Germany whose specialization aligns with mine in Pure Math and he replied back:

Dear X,
thanks for your interest . We can in principle discuss the possibility
of a PhD with me, but I should warn you that it's unlikely to work out
in the end - first of all, most likely I won't accept any new PhD
students for the next year.
If you nevertheless want me to consider accepting you, please send me
your Master's thesis and ask at least one of your recommenders to
directly email their recommendation letter to me.
Best,
XX

I sent him my thesis and 2 LOR's and told him that I viewed 1 talk given by him and read 3 of his research papers to become more familiar with recent techniques he is using.
To which he replied:

Dear X,
thanks, I've received your recommendations. First, let me repeat what
I wrote to you previously - I most likely don't want to accept any new
PhD student this year, so if I don't accept you, it will probably be
to a large extent for that reason.
However, we could talk over zoom sometimes to discuss things. I'm
quite busy at the moment, so let's do it at the end of January - can
you please email me around Jan 21 telling your time availibilty on Jan
24.
Best,
XX

Question: The prof. might tell me some of his colleagues that I can write to. But I have already written to them (by going through his papers I came to know who he has collaborated with). I really want to work under his supervision and some other professors I wanted to work with will not take any new students this year. I wrote also to 3 professors in France and they haven't replied to me despite sending a remainder mail. How can I persuade (if that is the correct word to use here, English is not my native language) him to take me without being needy?
In his 2nd reply he told me that "However, we could talk over zoom sometimes to discuss things". What exactly can we discuss in that meeting? What should I do to make the most of that meeting?
I can talk about the other place I wrote to / applied to , my master's thesis, my knowledge in the particular branch I want to work in and also other branches or the topics I am studying now.
I think it is very kind of him to give me a chance. He could have simply said no. But he is interested in giving time from his busy schedule.

My impression is that this professor has other things going on in their life that make them uncertain about committing to a new student. I would not overthink this - just meet with them and see what emerges. It is unlikely to be in your interests to 'convince' them if they are not comfortable they have the capacity to supervise you.

By my reading, this statement (and the repeated rephrased versions of the same thing):

I most likely don't want to accept any new PhD student this year, so if I don't accept you, it will probably be to a large extent for that reason.

is meant to make it absolutely clear to you that this professor is not accepting new students this year, that the reason for not accepting new students is not related to the quality of your application in any way, and therefore that there is no way you can persuade or convince or coerce or bribe them to take you.
I know they do not say this explicitly; they use words like "unlikely" and "possibility" and "probably". However, I believe they are using indirect language in an attempt to be gentle; this is common in many cultures, even cultures that overall have a mostly direct communication style. It is hard to say "no"! It can feel mean to tell someone else you can't help them, even if the thing that would actually be most useful to them would be to make it clear they need to look somewhere else. I think this person is doing the best they can to tell you this without crossing into saying "no, absolutely not".
Given that prior information, I think it would be very rude to attempt to persuade them. An overly strong attempt may make them very, very uncomfortable and want to never hear from you again: you'll be forcing them to say more firmly "no" when they've been trying to communicate "no" to you without having to actually say it.
To get the most out of this meeting, I'd recommend focusing on what other ways they can help you: there may be other people they know working in the field (that may not be one of the authors you've identified already) who are looking for students. It's also okay to simply ask them for their help and advice: they're experienced and they've been a prospective student before. It's fine to say exactly what you've written here: that you've been in contact with other professors as well and unfortunately none of them seem to be taking new students and you're not sure what to do next.
As for what to talk about, you should talk about your shared interest in mathematics! There is a very very small group of people interested in any narrow topic within pure math or any other specific field of research. If you tried to speak to every single person you ran into on the street in a big city for a whole year, you'd be unlikely to encounter a single one that's specifically interested in that area of research. I don't think there's much point in trying to brag or show off your knowledge or potential, just talk about what interests you or what you find fascinating, what you don't know yet that you want to know, etc.
You can conclude with a request that they contact you if anything changes about their plan to not take on students; that's a request that they can easily and graciously say "yes" to, and they'll be happy to reach out if it does happen, even though you should plan for that message to never come. This is your best chance, even if it's most definitely an unlikely one.

Good answer and good advice, but two comments: (a) while it may be true for the professor, Germany as a whole is not a gentle culture. I remember high school students receiving an "F+F" because one F wouldn't have done the effort justice, and the worst effort in a biology exam being read by the teacher to the class for general amusement; (b) I think the professor is at capacity (either financially or by what he thinks he can supervise), but if the applicant ends up having 4 Math Olympics gold medals, or such, he might try to make it work.

@gnometorule Yes, I'm aware of the reputation of German culture to be direct, which is why I mentioned that this aversion to saying "no" is not limited to cultures that are overall more indirect. I disagree that there is any chance this professor plans to "make it work" if they are sufficiently convinced; I think they've shown clearly that their reasons are not related to the applicant. We don't know those reasons: they may have too many students, they may plan to retire or take a long sabbatical, they may have cancer or other serious illness in themselves or family. It doesn't matter why.

@BryanKrause I completely agree with your answer, although I would add that a possible outcome of the meeting could be that they offer to take them on as a PhD student in the next year.

@Sursula Can you please tell why exactly you think so?

@gnometorule I don't even have 1 math olympiad gold medal and I think Prof. is aware of that as none of that is mentioned in my CV which I sent to him.

@YannicMuller If there was indeed a 0% chance they take you as a student, they would tell you that. The fact that they are offering to meet means there could be some change in their circumstances, but it's unlikely and they want to meet anyway.


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How should I answer questions about what I'm "expecting from my PhD" and "what I'll do after my PhD"?
I am in the fourth semester of my master's in a poor Asian country and want to do a PhD in pure mathematics in France or Germany.
I am asking how should I answer these two probable questions in my PhD interview:

What are you expecting from your PhD?
I think I will say that I will put hard work in my research  during my PhD and will take part in whatever teaching activities are assigned to me by the department. I expect that I will publish 2-3 research papers.
Is this answer fine?

What are you expecting after your PhD?
I think  I will say that I will do a couple of postdocs in Europe or in North America and continue my research. After that, I will join a university as an assistant professor.
Should I specify country here? I don't want to live or teach in my home country because of terrible political conditions and social conditions, should I say that I would like to work in Europe or North America (I know to  be able to do that is very hard)?
Or I should avoid mentioning that and just say that I would like to join a university as assistant professor. What if then I am asked in my home country or any other country?

Did you consider thinking about these questions carefully and then giving an honest answer? That would probably be the best thing to do. Obviously we cannot do that for you because we do not know your motives. Voting to close.

Looks like they have thought about them carefully, given an honest answer, included it in the question, and are looking for advice on whether there are any accidental red flags that could turn  a potential supervisor away

@Louic I have thought it carefully and added my thoughts in the question itself

@E.Rei ya, you are right.

(i) These answers don't feel very natural but rather like something you think the other side might hear, at least from a european perspectice. (ii) The answer to point 1 does not at all answer what you expect from your PhD - it only says what your future advisor can expect from you. Or is your motivation why you want to do a PhD to do hard work? Not to learn exciting new things, get to know people, and whatever else ...

By the way, I don't think either of these questions is very probable.

A much more likely question would be "Why do you want to do your PhD with us"?

"I will put hard work [...] and [...] take part in whatever [...] activities are assigned to me [...]" - noooooo! Don't say that! At least in European culture, it is perceived as between being hopelessly lost and offensively not caring. This seems to be a common point of conflict, too: a hard worker who is not independent is a nightmare for the advisor. So, the qualities the department would normally seek are: 1) Independent thinking/ability to work solo and 2) Hard working. In that order.

Since your ultimate goal is to join academia as a profession, the second question should emphasize that. Say that you'd eventually like to become a professor in Europe or North America. You don't need to talk about post-docs, but say that you will work to stay on track to that goal.
As for the first question, it isn't about what you intend to do so much as what you want to learn and accomplish as a doctoral student. You want to, I hope, become grounded both in your field and in effective research in that field. If you have a small sub-field in mind or a particular person to study with, mention that.
The situation in your home country is not relevant for this, but you might be asked at some point, and you can answer then.
The actual path to your short and long term goals is fairly standardized and hard, but you don't need to spell it out, since it is known already. Talk about the goals.

Isn't a person  from poor Asian Country mentioning about aiming for Professorship in Europe considered as too ambitious ( Also, whether the natives of the country like to employ people from other nations to academic positions is another things that come to mind)? So, should I say that I would like to join as assistant prof in my country ( and then later while I am a PhD student with them I will mention that I would like to be in Academia in Europe or North America?)

Your country of origin shouldn't be relevant. But if you are uncomfortable, just say you hope to eventually become a professor in your field, not mentioning any country and not specifying "assistant". You want to get in to a good program. Focus on that.

@No-One No-one who is clear in their mind will expect that at the beginning of the PhD you need to know whether a career in academia is for you, and what kind of career. If this is what you have in mind, there's nothing wrong with saying that, but you don't have to say this to make a good case. (Unless, of course, you say "I just want to do a PhD to have a PhD".) So I think the issue with sounding overly/underly ambitious in that regard is low. Personally, I find people who know for sure what they want to do in 20 years somewhat intimidating, even more so when they are only 25 years old ...


text:
Two papers on the same base of arguments
I had written a paper (I) on some properties of (real or complex) Hilbert space space valued continuous function to generalize a well-known theorem. I also thought the same about Banach space valued function, but could not do it on that time. After completing the work of paper I, I submitted it to a journal. Later after three weeks, I solved the problem for Banach space valued function in the real case. I wrote this findings in the form of a paper (II), by that time two months were passed from the submission of paper (I). Paper II also conclude the well-known theorem in the real case under a suitable assumption, which I didn't require in Paper I. Also, in paper II, a lemma was needed to handle the complex case, which is not present in Paper I. The main result of two papers uses the same logic with a little modification in the complex case. Both of them were published almost at the same times in two different journals. Is it unethical? If so, What should I do about it now?

I assume paper 2 properly cites paper 1? Then the reviewers knew what they were doing when they accepted it.

I see no issues here. More likely developing the first paper gave some insight into the Banach case. This is fairly common, actually, that a proof of one result gives insight into something similar. Proof methods are a powerful, insight driving, thing.
Relax.
Note that someone else, seeing your first paper, might have also said "a Ha" and written the second.

Thank you @Buffy. I just got nervous.

At worst one could say "if things had been a little different, it would have been better to have written them together in one paper" (if that's even true). But there's a huge difference between this and "unethical".


text:
Asking a collaborator to work on a second problem he is already working on
I am a PhD student in Mathematics and I'm working in collaboration with a more experienced researcher for some time. We are working in a nice topic in which some good results are being attained. On the other hand, there is another problem in the field in which I'm very interested in, and which seems to be very important. I became aware recently that he's also working in that problem and recently published a first paper on it. In particular he looks to be interested in several open questions related to such problem.
My question here is: is it fine for me to ask if it would be OK to collaborate with him in that other problem as well (I would like to keep both, since on the one hand the one we are working right now is going well, and since the other one looks to be very promising)? If so, how would be the best way of doing so?
I mean, my issue is that this is something that he's already doing, and in particular the particular paper he just published was with some other people. I fear it would be rude to ask to get involved in something that has already began without my participation, but perhaps this is just because I'm quite an introvert individual and always worry what people are going to think of whatever I say.

Yes, it would be fine to ask, assuming that you have something to contribute. Tell him of your interest and what your thought are on that problem. Then, depending on his reaction, you can make a request to participate.
Just make sure you have something to add to a research project.
Don't use introversion as an excuse not to make your wishes known. While it may not feel especially natural to you, practice will change that. I once made the mistake of not speaking up for my own interests and it cost me dearly.


text:
My name can be written in two ways: arxiv struggles to list all my papers
My problem is the following:
I have a last name that is composed of two names. In some of my papers, the two names are hyphenated (in others, they aren't).
So basically my full name is either written as:
Firstname Lastname1-Lastname2 or as Firstname Lastname1 Lastname2.
Because of that, it makes it harder for arxiv to find a complete list of my papers. In practice, if I click on my name in the paper preview on arxiv, it will list all papers associated with the either the first or second way of writing my name, but not both.
In order to solve the issue, I linked my arxiv account to my orcid, which is listing all my papers irrespectively of this issue. But this doesn't fix the issue on arxiv.
In summary: is there a way (without modifying my name in my previous papers) to fix this issue? I have an arxiv author id but there is the same kind of issue that it is not listing all my papers there.

What's the issue with the author ID? Can't you select all papers you have authored by logging in?

On a tangent: Are you aware of OrcID? Seems to be a particularly useful thing in your case.

@GoodDeeds the papers are related to my author Id. But when you click on someone name in arxiv it will do a research with its name rather than showing all the papers related to the author Id. Because of that, only either of the two way to write my name will be displayed when I click on my name for a given paper entry.

@OlegLobachev yes I am and my arxiv account is linked to my orcid.

I think your best chance is to spell your last name consistently (possibly updating old papers - it would be sufficient to do so in the arxiv author field, you need not update the paper itself). E.g. most spanish people I know either use a hyphen or just the first last name - the issue with the two last names w/out hyphen is that many people (and software) are unaware that this exists.

@StarBucK That's the default behaviour of arXiv, I don't think there is any solution to that (unless you resubmit your papers updating your name).

Is Arxiv that important in your field?

@AzorAhai-him- yes, it is.

https://academia.stackexchange.com/questions/9382/is-a-researcher-with-the-same-name-in-a-different-field-likely-to-cause-confusio

@AnonymousPhysicist This seems rather orthogonal to the question.

@user151413 Both questions are based on the incorrect premise that ArXiv is an identity management system, so they should be linked to each other.

@AnonymousPhysicist Just like orthogonal vectors are closely connected!

In my experience, the key to issues like this is dogged consistency. Once you decide how you want your name to appear on papers, that's the format you need to use in every paper and in all of your correspondence (at least in your signature block). Literally everyone should know you by the version of your name you selected. This is difficult for many people (including those with hyphenated last names) but is the only solution I've found.

This sounds like a technical problem with the ArXiv treatment of author accounts.  It is unlikely to be easy to solve the problem externally, but presumably it could be solved internally with a reasonably simple change to the underlying code for the author accounts and search facilities on the site.
ArXiv is run by a group of researchers at Cornell University.  It has an IT Advisory Group that meets several times per year to discuss an advise on IT problems relating to the site.  There is also a Leadership Team that makes decisions relating to the site.  I recommend you contact the IT Advisory Group for ArXiv and let them know the problem you are having.  Since this is a volunteer group, they might also be willing to have you attend one of their meetings to discuss your problem with the group.  Many people have multiple surnames or multiple spellings of their surnames (it happens commonly with female researchers who change surnames after marriage) so this seems like a problem that would potentially affect many people, for which there ought to be a reasonably simple internal programming solution.


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Sharing my graduate school application
I am a graduate student at a well ranked US institution A, and my undergraduate was from a very low ranked and small institution B in another country. Sometimes students from institution B (encouraged by my previous advisor there) ask me for the documents I used to get admitted to A. This includes my CV, motivation letter, and even the emails I sent my current advisor at A to contact their for the first time.
I don't feel like sharing my CV is a big issue, but I feel uncomfortable with things like my motivation letter and first-contact email with my current advisor. This is because:

I invested a lot of hard work and time in these (probably months and several rejections until gaining some experience).
I want to help them so I offered to give them feedback on their documents, but they insist on having mine to see how I did it (and because my previous advisor at B tells them to do so).
I am afraid that I could get involved in some problem, most of them plan to apply to my same program (somehow small) and even contact my current advisor.

Should I just share these documents? Maybe I am overthinking something small, but it does not feel completely fine.

Actually, I think you are doing the right thing. Your point 2) especially - offering advice and feedback. And if someone else copies your SoP and such and it is noticed, then it will be bad for them anyway.
But answer requests as we do here, with general advice. Even better share that advice with an old undergraduate advisor who can pass it on without bothering you much.
They are asking for something that isn't their due, actually.

I would also add that the OP should talk with their advisor and/or department chair to see how they feel about it. I'm sure nobody wants to see copies, but showing what good application materials look like might be fine (and might not). It is 'easy' to see what good journal articles look like - you read journals. Much harder to recognize how much work goes into an application if one has never seen one.

Give them a general advice and explain a general procedure but do not share your personal things like CV, e-mails, correspondence etc.
You have invested a lot of hard work and time in these and your experience could be valuable for many but it should not harm you in any way so just guide them in general terms but not very specific.