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How do I tactfully deal with fellow students who provide low-quality answers to my questions (which dissuades instructors from answering)?
My sister's studying at a UK university remotely because of COVID. Each course has an internal online Q&A forum  for students  to ask questions, and instructors or students to answer. I don't want to divulge any more detail for the purpose of confidentiality.
The issue is that 2 particularly hotheaded students don't actually answer the question asked — let's call them Gunners. This is cumbersome because:

my sister must add a comment, asking an instructor to answer the question.

These Gunners' useless answers can mislead instructors astray and away. She notices that instructors prioritize unanswered questions, and disregard questions that a student putatively  answered.

She doesn't know why these Gunners spree to answer - perhaps they are bored by COVID, or they want to impress the instructors. But she wants to tactfully request that the Gunners:

stop answering her questions. She wants solely instructors to answer them.

answer a question only when they are absolutely certain they read the question properly, and their answer is correct.

How could she best formulate such a request?
Here's an example of a useless answer. My sister asked

Does anyone have access LexisNexis or WestLaw for Canada and the USA for free? I already know that our library offers free access just to LexisNexis and Westlaw UK - I'm not asking about this.

One of these Gunners answered

Hi! Yes. Our library has links to  LexisNexis and WestLaw for free. I hope this helps!

Obviously this doesn't answer her question, because she already knows  about the libary's free links the UK versions of LexisNexis and WestLaw. She wants the North American versions!

What is the problem with tagging the instructors? If answering questions is their job, this should not be any problem. Furthermore, I dont think that it is problematic to politely comment the answers of the two hotheads and to state that their answers were „non-answers“. I would assume that directly contacting the hotheads will only lead to bad blood without any chance of improvement.

Welcome to Academia.SE! This is an interesting question, and an unfortunately common problem that has many different manifestations (including on this site!). I suggested a change to your phrasing because we don't generally accept requests to write e-mails for you, but we can give more general advice on how to tactfully handle the situation.

Your sister should not complain to the Gunners about their behavior. While they are not being helpful, they seem at least well-intentioned, and in any case have no obligation or responsibility to help your sister (or even to avoid unwittingly sabotaging her efforts by using the Q&A platform in a way they perceive to be consistent with its intended use). And perhaps the reason they are posting these useless answers has something to do with a grading or other sort of incentive structure (e.g., some professors will offer bonus points for online participation, and some students will go to great lengths in an attempt to impress professors they later plan to ask for letters of recommendation from, even when their behavior is actually far from impressive). If that’s the case, it seems potentially unfair — or at any rate unlikely to have much of an effect — to ask the Gunners not to behave in a way that they think benefits them, even if it doesn’t fit with your sister’s plans.
Rather, the responsibility for running the class efficiently and avoiding a situation where some students are undermining the ability of other students to make effective use of the course resources lies with the course instructors. This is who she should address her complaint to.
I’d suggest to your sister to email the instructors, politely explain the issue with the Gunners and how it is preventing her from getting the help and answers she needs, and ask for their support. This can be phrased as an explicit request for help (“please take action”, “please suggest a way I could avoid having my questions ignored”) to something more vague like “please advise” or “do you have any thoughts about this?”. The specific choice of how explicit to make the request depends on the local culture and the personalities involved, so I don’t have any recommendations.

+1 for "email the instructors, politely explain the issue [...] and how it is preventing her from getting the help [...] she needs"

On piazza and edstem you can mark questions as private. I assume you must be using canvas or another platform without private questions as that is a trivial solution.
Honestly the "gunners" are just scapegoats in this situation. The instructor(s) should be monitoring the course discussion page and reading all the questions and any answers there. If they don't answer a question, then that implies that they either agree with the student answer or aren't doing their job.
You should consider also that perhaps the questions being asked aren't appropriate. The example you give for asking about links to some subscription - that sounds like something to ask a librarian, not the professor.
As for directly answering your question, you need to clearly explain why the answers they give are wrong going forward. Students are not going to want to reply if they realize it's just making them look bad.

Maybe the instructors should be mailed or otherwise contacted? Instructors have to keep track of many different streams of information, maybe they have a preferred one, and this class discussion isn't it. I tell my students to mail me, as this is the only channel I track reliably.


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Sharing an article under review within research group
I am conducting research with a Ph.D. scholar under a supervisor, who is, in some sense, heading this group. The Ph.D. scholar and the supervisor have two articles undergoing a second round of review. The comments received in the first round of review were fairly straightforward to address, according to the Ph.D. scholar. For this reason, it is believed that the papers will be accepted soon.
The research I am conducting can be seen as an extension of the work done by the Ph.D. scholar and the supervisor. For this, it would be useful for me to have access to the articles they have under review. However, the Ph.D. scholar is reluctant to share these articles, and would prefer to wait until they are accepted before sharing them with me. Note that the papers are not available anywhere online, not on a website and not even on a preprint server.
The question is not about what I can do: I am okay with the situation, and have no real problem waiting till the articles are accepted. I am however interested in the more general case.
I assumed that since I was, in a way, a part of this group, any research that they may have under preparation or review would be accessible to me, in some manner, and I wouldn't have to wait until the published versions came out to access the research. Is my assumption correct, and is my experience less common than the alternative?
Simply put, is it common for members of a research group, including graduate students, to be able to access articles that are under review, so that they can obtain insights to further their understanding of a topic, and hence increase the pace at which they conduct research? Or is the alternative more common? For obvious reasons, I am particularly interested in fields/topics in which preprints are not widespread, and the usual publication process holds.

Where are you based out of?  I realize a lot of these are culture / context-specific so it'll be good to know more info.

India. (See the comment on Bryan Krause's answer.)

I assumed that since I was, in a way, a part of this group, any research that they may have under preparation or review would be accessible to me, in some manner, and I wouldn't have to wait until the published versions came out to access the research. Is my assumption correct, and is my experience less common than the alternative?

It would be typical in any research group I've been part of that everyone in the group have access to collaborative work within the group. Our in-progress papers are all readily accessible by anyone in the lab. It's common to pass things around among people not immediately involved in a specific project to get their "outsider" perspective on it, as well.
The only time I see people being secretive is usually students who are a bit nervous about their early-stage work and like to hide it a little bit until it's "ready for consumption". That's fine, too, but it's limited to things that are still far from ready to submit.
The secrecy you describe seems very weird to me, but maybe just a personality quirk of this one individual. When patents are concerned, rather than publications, things might be a little different because of the complex legal circumstances around patents.
It seems like a serious impediment to your work to not have an up-to-date history of what else is going on in the research group. Although you are okay with waiting, I don't think it's in your best interest. I would ask this PhD scholar to have a conversation with you and the supervisor to resolve this. Perhaps they are nervous about something they shouldn't be, perhaps they fear the supervisor wouldn't want the work shared when in reality they have no issue with it. I would not recommend simply waiting for it to be published. Science is a constant incremental project, and being in a lab where related work is being done you should be ahead of the broader community on the next steps.

This seems odd and unproductive to me, also. I am wondering if the person got burned in the past by sharing. They aren't doing their long term career much good as they are probably not open to collaboration.

@Buffy Yeah, seems mostly like just a bit of paranoia, whether from their own past experience, reading some horror story from someplace else, or just unfamiliarity with how things normally operate.

@BryanKrause I will mention that my experience is not very uncommon in certain parts of the world. (I am in India.) I have conducted research under multiple professors, and each of them had their weird quirks on sharing work that was in progress, and these could just be filed under "conservative behavior arising from reading/hearing horror stories".

Contrary to Bryan Krause's experience, none of the research groups I've ever belonged to had any kind of structured shared access to work-in-progress. I also had not heard of other groups with such arrangements before. If any of my colleagues would express entitlement to see my work-in-progress, I would be rather peeved. But polite requests by people I know (whether at the same institution or not) typically lead to access to even extremely stage drafts.
That said, an article that has already been submitted for publication is supposed to be something ready for others to read. I personally would make such articles publically avaiable via the arXiv anyway. But even if there are reasons to keep the work confidently, not sharing it with a close colleague who has a good reason for needing to read it seems very weird to me.

That is quite interesting. Could you clarify which country and which field you are talking about?

@RamPadmanabhan I'm in mathematical logic/theoretical computer science. Positions were in the UK and Belgium.

Okay in the west it’s fairly common to have access to the group’s research as you go along, build up partnerships and write papers together. However, based on some of the horror stories I’ve heard about in Indian universities, that might not be the norm because of part experience of stealing research and whatnot (this also happens in the west, but only with a confluence of a lot of major wrongs).
But if this paper is already in the review process I don’t see why your colleague would be so scared to share the research. You should talk to your advisor and find out what the norm is within your research groups so that you know how to operate going forward. If scooping each other’s research is a concern you all might try and putting out preprints at arxiv in the future to ensure that a public DOI is generated and is couple to active manuscripts.

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How do I deal with a collaborator who stopped working at the final stage of a research project?
I am a graduate student who started to collaborate with a postdoc last year on a research project which we both came up with. During the past year we had several discussions about this project. I mostly did the calculations and proofs and he focused more on giving general advice and comments on the big picture of this project. The collaboration seemed to go well.
Early this year we decided that the results are strong enough and we should write a paper soon. I finished a paper draft containing all results about three months ago. There are still several sections (e.g. introduction) left blank because he insisted on writing that himself.
After I sent him the draft, he didn't start to write his part but asked me to change the structure of the paper and I also finished that pretty soon. Then he had a vacation for two weeks and I didn't bother him during that time. Every time I asked him about the progress, he was either busy finding a job or having a family emergency and he didn't start to write his part until now. He said once at a time that he could finish the paper in two weeks but then two weeks later nothing was written down.
I didn't want to rush him because job/family seem to be more important things than writing a paper, but I couldn't help but worry if he was just making excuses to procrastinate. And since he seldom wrote anything down from the beginning of this project, I have no idea how long it will take him to finish the paper.
How do I deal with the current situation? I don't have a high expectation that the paper can be finished soon, but I also cannot stand waiting forever.

Is it necessary that it be finished soon? Or just your expectation? On the one hand, you finish it yourself. On the other, patience is suggested and that you work on something else for a while.

It may be that putting off dealing with a serious family issue in favor of writing the paper would be the true procrastination. See the tag description.

Did you try asking your advisor for advice?  Is the postdoc a postdoc of your advisor?  If so, they might have some perspective on the situation/postdoc in question, and can possibly help mediate if necessary.

I'm 90% sure this is a dupe.

So, just do it. Write the missing parts as best you can. Keep the other as co-author and ask permission to publish. You have an issue that may not be resolvable otherwise. The co-author may not have any options about this.
Keep them informed, but push forward. If they come back in, then do what you need to do to merge the work. Preferably ask advice (say, in editing).
This isn't an especially unusual situation, actually. Things happen. Life goes on. Some obligations are absolute, but working on a paper seldom is.
And, if you can't do it on your own, then your only other option seems to be patience, unless you can expand the collaboration.

If he can't/won't do the work, then he has to compromise and your judgements prevail. If you're not a player then you're a spectator.

@Jane: You will offer him to either write the introduction, accept your introduction, or take his name off the author list and be relegated to a "The author wishes to thank".

@einpoklum, NO NO NO. Taking them off the author list if their intellectual contributions appear would be unethical and actual plagiarism. You can't shut out a valid author like that. Dropping out of the collaboration before completion happens. It might result in not publishing the paper at all, since all "authors" need to agree. But there is nothing preventing the OP to complete it. Alternatively, all contributions of the "dropped" author need to be removed, which, in the present case, seems impossible. Don't play with plagiarism. You get burned.

@Buffy: I mean't let the post-doc choose. OP can't take them off the author list, but if the post-doc doesn't like the resulting paper, and doesn't want to help improve it - that's basically the only thing he can do.

@einpoklum. Sorry, but that doesn't work. If their intellectual work is part of the paper and they are not an author, then the other author(s) are in danger of accusations of plagiarism. I don't see how you can finesse it. Every idea contributed by the other author singled out as in a citation? I doubt that they can be separated if it was a true collaboration (as the OP suggests) at the start. Write the paper, list all authors. Do what you have to do to make it work. What if a co-author dies or gets a debilitating illness? Don't suggest dangerous actions, especially to new academics.

@Buffy: The other authors will not be accused of plagiarism if they indicate the withdrawing author contributed to the work. If the withdrawing author perfidiously decides to accuse OP of plagiarism, that will blow up in his face immediately, since he himself had explicitly authorized them to take his name off. Why do you believe someone would do something like that?

Why do you think someone would agree to be removed, leaving their ideas in place? Why do you think it is such a bad thing to just leave them in place? Where did perfidy come from suddenly?

@Buffy OP writes "I mostly did the calculations and proofs and he focused more on giving general advice and comments on the big picture of this project."  Sounds to me like the collaborator did not contribute much more than what you would expect for an acknowledgement.  Leaving them off of the author list, especially if they're not committed to making any actual contribution to the paper, seems reasonable in this case.

This is a variation on @Buffy's answer.  I would ask the collaborator if it would be helpful for you to provide a draft of the missing sections that they could then edit as much as they want. You could also list out the bullet points that you think should be in the missing sections so that they can respond before you write that draft.
This is what I almost always had to do when I was a doctoral student writing with faculty. Frankly, most of the time they didn't change much but it was still the respectful way to make it clear that they own that section.

Nice suggestion on a plan.

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Informing advisors on a tangential sole-authored paper
Over the last year, I conducted research under two professors A and B on two topics A and B. In the last few months, I have come up with an idea that uses concepts from topic A to present a solution to a problem in B, i.e. the idea could be called "Solution to B using A". As far as I can tell, the idea is new, and (if I may say so myself) quite elegant. I intend to write this up and submit to a decent, normal-ish conference. Note that neither of my advisors are aware that I have been working on this, and hence I intend to put my name as the sole author. I have, however, drawn from research I conducted under them.
Question: Am I under any obligation to show the two advisors my work, and inform them that I plan to submit to a certain conference?
I am a little afraid of negative feedback if I show my work, feedback I would rather get anonymously from reviewers rather than from advisors under whom I continue to conduct research. (This is possibly irrational, and I need to deal with it, but for the moment, it is what it is.)
Neither advisor is familiar with the other topic, and hence neither will be able to judge the paper as a whole. I am also a little worried that by the time I obtain feedback from either (or both) of them, the deadline for my preferred conference would've passed. While I do realize that consulting with them will likely improve the quality of the paper, I will mention that:

I am confident that the mathematical details are correct.
Both advisors have been very complimentary of my writing style in previous paper(s) I have published with them. The final, published manuscripts do not differ from the drafts I wrote by more than 5%.
I don't believe that asking both of them to work together on looking over my paper is a viable option, as it's unlikely they could spare the time. Also, they don't really get along with each other.

If I don't consult with them, I am worried that either (or both) of them would be a little offended, since I have drawn from work I conducted under them. I don't wish for them to think that I have an over-confident or arrogant streak in me, and hence did not consult them. Is that logical?
EDIT: I am not a Ph.D. student. I have just completed my undergraduate degree, and will be joining a master's degree in Fall-2021. The research I conducted under both advisors was not to obtain a degree, it was voluntary. Advisor A is at my undergraduate institution, advisor B was at that institution until last year. There were no institutional rules I was bound by while conducting research.

Can we assume you will arrange funding to attend the conference (online or in person) without the assistance of either professor?

I will not be depending on them for funding to attend the conference.

You have more to gain from showing the professors your work, than directly submitting it. Make good use of their mentorship. Don't be afraid of potential negative feedback. They're likely to be nicer than anonymous referee reports.

In addition to Buffy's and Arno's excellent answers: Strategically, you could think about presenting this work at the same time that you also present some other work that you intend to complete together with them, possibly something you discussed together in earlier conversations, and ask them for input for that. That could "soften the blow".

"Neither advisor is familiar with the other topic," but if they have a good academic network, they can (and will) quickly find people who are familiar with both fields. And those are the people you should be talking to about your idea.

If I don't consult with them, I am worried that either (or both) of them would be a little offended, since I have drawn from work I conducted under them. I don't wish for them to think that I have an over-confident or arrogant streak in me, and hence did not consult them. Is that logical? - No, it is not logical.  The only reason I can imagine they would be upset is because of their own feud.  While it's possible that you could get caught in the crossfire, that would be illogical on their part.  Presumably they already know you're working with each or them, yes?

@Kimball Yeah, both of them know I'm working with the other one. The "Is that logical?" was actually to do with "I'm worried they would be offended" rather than "They would be offended".

I'm saying the only reason it makes sense for you to be worried is they've show ill-will towards you working with both of them, or if they disparage each other's work or the like.  If they just don't get along personally, and can separate this from their academic views, I wouldn't think they would get upset with you.

Even if I submit the paper without consulting them? I have drawn from work I conducted under both of them for this.

There is a delicate balance here that only you can resolve. It is good to keep advisors advised, so to speak, about your activities. Most will support you provided that the tangential stuff doesn't detract from your main work.
OTOH, you need to consider personalities. There are certainly those who would object, but you can probably judge that from past interactions.
Ultimately the purpose of doctoral education is to have the student eventually surpass their advisors. Only then does science make breakthroughs. But not everyone can deal with that idea.
It might be that one or the other would react badly to any action you take. It shouldn't happen that way, but think about your relationship to these people.
One thing you might consider, but need to judge for yourself, is to tell them that you've been working on a side project that has come to fruition, but is different from your main-line studies. You'd be happy to share it if asked, though it is outside their main interest.

Edited to add in response to the edited question. I think that the balance shifts strongly toward informing them. I'd guess my suggested format is even more appropriate in your case. And you are lucky to have people supporting you at an early stage of your career.
I'll leave the answer in place as a general one that applies to the more common situation where advisors are also "supervisors" in some sense.

Thanks a lot. The penultimate paragraph is extremely useful.

A few things to consider:

Consult the rules for PhD students at your university (I assume you are one). The university where I did my PhD had rules stating that a student needed permission by their supervisor to submit stuff for publication. My understanding is that such rules are rare however.

If I put myself into the shoes of your supervisor, I would consider it rude if I only learn about this paper in hindsight. I would not necessarily expect to be ask for permission, but I would like to be informed (and the chance to provide feedback). It's perfectly fine to say "I will submit this paper to conference X on the YYth. I know you're busy, but if you have any feedback for me please let me know".

It is also rude to the referees. Since referees perform a voluntary and mostly thankless duty, there is a responsibility on authors to submit stuff which is as good as possible. Its your supervisors job to read potentially horrible drafts and tell you how to fix them, not necessarily the referees.

Neither my point 2 nor my point 3 have the conclusion that you absolutely must not go ahead with your plan, but that it would be better to overcome your reservations.

Could you please see the edit to my question? Apologies for the confusion.

@RamPadmanabhan Then Item 1 does not apply to you, but 2 and 3 still do.

3 assumes that the supervisors are better writers than OP. I did not find that this is necessarily true for any given supervisor-supervisee pair.

@lighthousekeeper You don't need to be a better writer to potentially give some decent advice.

But decent advice is not necessarily what lifts a paper from a horrible to a decent draft.

@Arno "I would consider it rude if I only learn about this paper in hindsight." Isn't "rude" a little too strong? Clutching at straws, but I would've thought they'd be more surprised than hurt that I didn't consult them. My (naive) thinking was that it would be very "cool" to show them an accepted manuscript rather than one in preparation. As I mentioned, no institutional rules, so this essentially comes down to a point of morality.

@RamPadmanabhan No, "rude" definitely fits.

I've been in a very similar situation about two years ago. Now I'm a PhD student.
While you are not under any obligation to tell them, I would  strongly advise that you do. There are multiple reasons why:

It will most likely improve your paper.

Talking to people about your research tends to improve your research. Especially if the people are more experienced than you. For example, maybe B can tell you that your result has other cool consequences in their reseach area, or A sees a way to generalize some part of your proof, or....
In my case, input from one of my advisors helped generalize my theorem, greatly improving the quality of my paper, and I got my paper into a much better journal than my original version would have been accepted in.

It can help improve your chances of a career in academia.

Math that connects two areas of research is very interesting. You finding these connections yourself is impressive and will make you a very good candidate to offer a PhD position to when the time comes. At the very least, the more promising you show to be, the more likely they are to show attention to you, involve you in future projects, give extensive feedback on other projects, etc. So, showing them your research is good.
In my case, I got two very good PhD offers out of it, and I could choose my advisor.

It will show that you are a good person to work with.

A couple of comments on quotes from your question:

As far as I can tell, the idea is new, and (if I may say so myself) quite elegant.

As an undergrad (and even as a grad student), you are probably not a good judge on this. This is why we have advisors, because they have a much better overview of the field, and can tell whether something has been done before.

I intend to write this up and submit to a decent, normal-ish conference.

Again, you probably can't judge whether that's the appropriate format. Maybe the result is so cool that it should be submitted to somewhere better. Maybe it doesn't stand a chance for acceptance at even a decent conference because the result is so niche. Another reason why talking to someone more experienced could be helpful.

I intend to put my name as the sole author.

You can do that even if they read through it and provide feedback.

I am a little afraid of negative feedback if I show my work, feedback I would rather get anonymously from reviewers rather than from advisors under whom I continue to conduct research. (This is possibly irrational, and I need to deal with it, but for the moment, it is what it is.)

Me too. But I can tell you that your mentors will most likely not judge you, even if you give them complete nonsense, or a very niche result. I can't tell you how many times I showed mine math that was just plain wrong, or that they found uninteresting. I can assure you, the fact that you are an undergrad  trying to produce (from what it sounds like) sensible independent research will be far more important than any mistake they could find in your draft. (And most mistakes are fixable.)

Neither advisor is familiar with the other topic, and hence neither will be able to judge the paper as a whole.

I wouldn't be too sure about that. Most professors know a huge amount of math, and even if they might not be experts on that topic, they could very well be fluent enough to provide meaningful opinions.

I am also a little worried that by the time I obtain feedback from either (or both) of them, the deadline for my preferred conference would've passed.

Why are you rushing to get it submitted? Why do you want it submitted at that exact conference? I can't answer that for you, you might very well have good reasons. But you should probably ask yourself whether an increase in quality would remedy a later submission.

A few points I wish to respond to: (1) The field isn't mathematics, it's a subfield of electrical engineering. (2) Some conferences are a fairly big deal in electrical engineering, similar to CS and unlike math. (3) I wish to get it submitted a little early because I will apply for Ph.D. programs in Dec. 2022, and getting this published will make a significantly positive impact on my profile.

I'd like to add to the second-to-last paragraph that a paper combining topics A and B should ideally be written so that both experts on A and experts on B can understand it and see its relevance. Both professors could help with this.

@RamPadmanabhan Re(1): I think most of what I said is universal to all STEM fields, even though there might be slight differences in cultures. Re(2): Even in math, in some subfields conferences are a big deal. What I'm saying is: You probably can't accurately rate the quality of your paper, and the quality of conferences/journals it should be submitted to. Re(3): If you have some research publication already, (as it seemed from your question), another one might not make a big difference - you've shown you can do reseach already. But you need letters of rec as well, so don't annoy your mentors.

Since you're an undergrad, I suggest that you're not the most experienced with the dynamics of presenting at a conference, the subtleties about which conference is the best for your type of work, and a whole bunch of other things.
Also, since you're an undergrad, you don't have any real allegiance to any PI.  Your time is largely your own (assuming you haven't used any resources you're not really entitled to use).
If somebody experience is willing to review your work before submission, this would be a favor, and worth pursuing.

I would add a key point, that it is hard for an undergraduate to detect scam conferences.  This is a big danger that professors can help one avoid.


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How to join PhD in Mathematics in France?
I am interested in joining a PhD program in Mathematics in France. Now I am a Master's degree student in India.
My research interests include Differential Geometry (in particular, Hyperbolic Geometry). I want to take admission into Université Paris-Saclay (more especially, Université Paris-Sud,  École Polytechniqué), University of Strasbourg etc.
Can you please advise me how one could join the institutes as a PhD student?
Thanking in advance.

Don't each of them have a web site that explains admissions to doctoral programs?

I suggest you have a look at this page : https://www.campusfrance.org/en/how-to-enrol-Doctorate-France and then specific universities pages.
I recommend you act swiftly as the process tends to start much earlier in the year, several offers have now been filled, but there are probably still some good opportunities if your search is wide enough.
Edit : here are some more details. On the following page you will see the results of the Masters students that have been accepted to a PhD position in mathematics at the University of Grenoble, which is strong in differential geometry. https://www.adum.fr/as/ed/page.pl?site=edmstii&page=alloc_modalite
As you can see :
A) the deadline to apply was in june
B) the results show that all but one students whogot accepted had done a Masters in France. This is not surprising since in mathematics most PhD positions are reserved for this case, i.e.  this natiowide process where applicants with a french Masters are numerous and usually take up most of the positions since coming from a french Masters  allows professors to get the grades of these students and the opinions of the colleagues who thaught them during the Masters. It's difficult to judge the level of an international student unless their candidacy has strong backing from a researcher whom the french ones know.
So to achieve your goal you'd have more chance to do a Masters in France next year, try to be among the top students, and apply. To succeed this year your Masters thesis should be really good, and you should have a recommendation letter from a good researcher, and finally you should try to see if a university, or a researcher with a special kind of grant, has some unfilled positions.


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Strategies for the selection of graduate programs for borderline/weak applicants (STEM)
This question is aimed at the countries where the graduate program admission decisions are normally made at the department level; the primary sources of funding are TAs/RAs and there is at least one year of coursework prior to the transition to full-time research (e.g., the US).
It is likely that I will be looking to apply to graduate programs in applied mathematics next year (however, the question can be generalized to certain other STEM disciplines, I believe). I have filtered a number of schools based on my research interests (somewhere around applied dynamical systems, control, bifurcation and stability analysis, applications in mechanics), but it is still a reasonably long list. I would like to understand how can I select a (fixed) number of graduate programs from this list so that my chances of admission somewhere are maximized.
I am looking at schools in a wide range starting from around 30 down to 150+ according to the rankings in the subject area, all of them being within the R1/R2 range. However, I have a concern that there may be little correlation between how selective a given program is and how well it does in the rankings. For example, some schools state explicitly that they admit exactly 5 students to their graduate program each year (while there seem to be others that admit well over 20 graduate students each year). In other cases, I would speculate that the location of the school might make up for its lower “reputation” and warrant fierce competition for positions. I could also speculate that private universities may be more selective than the public ones at around the same “reputation/quality level”.
Is it possible to devise a reasonably generic methodology that can be used to filter schools that, in all likelihood, will have less stringent or even reasonably lax admission requirements? Otherwise, I am looking for any helpful hints/advice.

A few notes:

It is likely that I may be classified as a “borderline/weak” applicant, primarily, because I will be trying to make a transition from my undergraduate engineering degree to graduate studies in applied mathematics, not because I did not do well during my studies as an undergraduate. Also, by now, I am far from being young (I have spent quite a number of years working in the industry since graduation).
This is not a shopping question, I am asking for a methodology for the selection of programs.
I am familiar with the questions How does the admissions process work for Ph.D. programs in the US, particularly for weak or borderline students?, How should I choose which graduate programs to apply to for the PhD? and a few other similar questions on this website. However, I believe that they do not provide an answer to my question (in a certain sense, my question is a specialization of How should I choose which graduate programs to apply to for the PhD?).

Your age and time since graduation should have no impact on your admissions chances, provided you still have at least one good referee in academia and haven't spent the intervening years twiddling your thumbs.

In my view, the best methodology, which also applies to top students, is to cast a wide net. Make sure that the schools you apply to have faculty in the subfield that most interests you, but don't parse it too finely.
You have a better chance in a large program. And you have more options at a large place if you get in. You may get good advice from your local faculty. Some of them might know people in some places so that a recommendation will be given a bit more weight.
You might try to expand your search even beyond your "range", however, with a top school or two (in case you have a bit of imposter syndrome), and a bit farther down the list as well.
Don't put artificial constraints on your search. In fact, no constraints that aren't actually necessary. Even things a bit outside your desired field might be worth a look.
But the key, I believe, is a broad search. Most schools that are relative low in the overall rankings still have some excellent faculty. Ultimately your advisor may be more important than the ranking of the school you attend.

Thanks. Overall, it’s good advice to give, but it doesn’t address my concerns. I don’t have an impostor syndrome: a few years ago I applied to graduate programs in the area of my major (engineering) and received offers only from 30+ schools (I decided not to attend). So, it’s senseless to aim higher while trying to switch to applied mathematics. In any case, I’ve already selected several schools based on their compatibility with my research interests. However, I would like to narrow down my search to a selected few, given that my primary goal is to maximize my chances of admission somewhere.

Sorry, but narrowing the search decreases your chances. But if it is to be only a few, make sure that they cover the range. The net needs to be wide. Exhaustive is a different thing.

Don't lose track of the fact that it is very competitive.

As per the premise of the question, the total number of applications is fixed. I don't know what this number will be yet, but at the moment I am looking at the list of 30+ schools. It would be both impractical and time-consuming to submit so many applications. In any case, thanks for trying to answer this. One useful advice that this answer contains is that I "have a better chance in a large program". This is something I was not entirely certain about.

Again, you don't have to apply to "every" school. Just pick from a broad range, not a narrow one. I'm not suggesting that you make 200 applications.

@human "So, it’s senseless to aim higher while trying to switch to applied mathematics." Perhaps, but maybe a professor at a top school also came from engineering and sees a lot of promise in you

I have a concern that there may be little correlation between how selective a given program is and how well it does in the rankings.

Well, if you compare the #30 ranked school to the #31 ranked school, then yes, you might find out that the #30 ranked school admits X% of its applicants while the #31 ranked school admits X/2% of its applicants.  But if you consider groups of schools very far apart in the ranks, say consider the group of #30-#40 versus #130-#140, I would be extremely surprised if the lower group was on average more selective than the upper group.

Is it possible to devise a reasonably generic methodology that can be used to filter schools that, in all likelihood, will have less stringent or even reasonably lax admission requirements?

If you want to apply to only, say, 10 schools total, then apply to the bottom 10 ranked schools, i.e. the absolute lowest ranked (or the lowest ranked schools that have faculty in your interest area and meet your other filters).  Although one or two of them may be more selective than their ranking would imply, I'm sure that at least one of them will have reasonably lax admissions requirements.
That's not what I would recommend you do though.  I'd recommend you apply to a few "safe" schools, the bottom of the list that you are pretty sure you can get into to, a few "stretch" school, the top of the list that you are pretty sure you cannot get into but are trying anyway, and then a good number in between.  E.g. if there are 100 schools, apply to a few ranked 10-20, a few ranked 30 - 50s, and a few in 80-100.

I am confident that your idea of applying to 10 lowermost schools would work well for undergraduate admissions. However, I doubt that this is a good strategy for graduate admissions because funding is involved. My guess is that the lowermost 10 schools may not even hire graduate students every year depending on the availability of resources/need for TAs. I could speculate that large departments that need many TAs in the 80-120 range may be safer bets than 190-200. However, I am asking this question precisely to find out about such things.

"My guess is that the lowermost 10 schools may not even hire graduate students every year depending on the availability of resources/need for TAs."  Remember that TAs are cheaper than professors.  If there is a class to be taught it is always cheaper to hire a TA than a professor.  So I'm not sure if your assumption is correct.  The schools with less resources might actually have more TAs (and less professors) than schools with more resources.  This is just a guess though, no data to back it up.  My original advice still stands: apply to a range of schools, a few each at top, middle, bottom.


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Is arXiv really a lower form of publication than the majority of computer science journals?
arXiv, by itself, and not as a preprint server, is already moderated by top computer science researchers at Cornell.
I understand if it's not the top journal, but isn't at least getting into arXiv actually more reputable than submitting directly to a lower-tier journal? Especially if 90% will just publish it anyway.
There is also evidence that arXiv papers are cited more, although this isn't conclusive.

See here for what moderation means: http://arxiv-export-lb.library.cornell.edu/help/moderation It is very different from peer-review.

The number of P=NP proofs on ArXiV should be a clue that it's not a replacement for journals.

First, arXiv is hosted at Cornell but this does not mean it’s moderated by Cornell people (that may be true in CS but it’s not true in physics, where the pool of moderators is quite wide).
Next, different disciplines have different traditions, so beware of self-selection.   In my field of physics, there is now a growing trend where authors upload on arXiv only after they have received comments from a journal referee.  This  avoids major updates that may result from the refereeing process, minimizes the number of arXiv versions, and also prevents the material from going stale before the next submission if it has to be resubmitted (nothing worse than a paper that has been “sitting” in arXiv for 18 month).  Thus, some fraction of authors only submit to arXiv after third-party validation through usual refereeing channels.
ArXiv is fantastic to get pre-publication feedback - someone will provide additional references, will point to possible improvement in a proof, will ask for additional clarifications.   However, be reminded that moderators keep out the obviously bad and crackpotty stuff, but that’s about it. Since arXiv is easily searchable and open access, it is no surprise that submissions there will have greater visibility than if only published in a journal behind a paywall.
Very few active researchers will referee more than one paper per month because refereeing is quite time-consuming.  A fortiriori no moderator has time to do any refereeing-level checks on their share of daily submissions to check: one would spend 24hrs of everyday checking papers.

Not the original question, but I'm puzzled by this trend of only uploading to arXiv after reviewing. In my field (theoretical CS) this looks like the opposite of what we'd like to encourage. It just wastes all the benefits of pre-publication feedback that you mention in your third paragraph, and the arguments for not doing it (avoiding an arXiv version) really don't seem convincing to me. If you think your work is ready to be refereed by expert researchers, taking them several hours/days/weeks, then why wouldn't you spend 10 minutes making it available to everyone on arXiv?

@a3nm It is a trend.  I suppose in my personal case it’s 50/50 depending on preferences of the co-authors.  Methinks there are too many submissions hastily uploaded to arXiv; my personal preference is to submit a later version, wait a few days for independent comments, and then submit full corrections.  This means the arXiv version is closer to the final published product and has fewer bugs.  Some high profile journals explicitly ask not to submit to arXiv until the paper is published (which I will agree will negate the value of feedbacks), others are fine with uploading rough work.

@a3nm Some also feel the rise of satellite discussion boards (v.g. scirate) means the referee might be contaminated by comments from casual readers.  Again… I am ambivalent myself.  I prefer to wait but in some cases co-authors need the arXiv submission for an annual report or a grant application.  I can’t recall ever objecting.

Given that arXiv is not moderated, I don't see the value in self-moderating. You can write in the arXiv Comments field that the work is still a draft and not peer-reviewed yet. In my area, the one reason I've heard to not submit to arXiv is the risk of being "scooped", i.e., publish paper X on arXiv, other people notice it and improve the result in a paper X' (citing the preprint X), the paper X' gets published in a conference/journal faster than X, and now getting X in a conference/journal becomes difficult. But again I think that this is a bogus reason, in terms of making science advance.

@a3nm speaking for myself, most of that you bring up are reasons to withhold overly agressive submissions to arXiv.  You never quite know what you’re getting since drafts are subject to (sometimes substantial) changes, v.g. I’ve had papers where the title was changed to accommodate referees.  As to scooping, in fact in physics it’s the reverse: people will upload obviously poorly prepared manuscripts just to establish priority.  If all arXiv submissions were properly updated with a version close to the published version, maybe I would trust arXiv more than I do now…

… Others presumably have their own reasons, and some people I know basically use arXiv not as a pre-print server but more as a post-print service to fulfill the open-access mandate of their grant.  (I have one specific colleague in mind.)

The standard for the arXiv is "looks like an actual paper on the first glance". So the fact that a preprint appeared on the arXiv isn't really adding any respectability to it that a researcher in the field couldn't spot themselves. The point of the filtering the arXiv does is instead about avoiding spam.

"The point of the filtering the arXiv does is instead about avoiding spam." Wouldn't it also filter out a lot of cranks, since they need an actual scientist to vouch for them, or has there been enough incautious invitations that cranks have begun inviting other cranks and their presence has metastasised?

@nick012000 I'm counting cranks as a type of spam here. The point is, it helps the knowlegable reader to not waste a short moment reading the crank stuff; rather than providing a non-expert reader with justified confidence that what they read is true.

Moderation on arXiv is extremely quick - papers typically go out within 24 hrs of submission, except for weekend submissions. The moderation process is described in this arXiv blog post:

The team performs checks with software support to confirm that the submission is classified to the appropriate subject category, that it is in compliance with our technical requirements, including formatting and metadata information, and that the content is appropriate, which, generally, describes content that would be refereeable by a conventional journal and in the format of a research article.

In particular they are mainly checking that the author has submitted it correctly, including proper classification, as well as filtering out anything obviously inappropriate, e.g. if it isn't even an attempt at writing a research paper.
The moderation process is not about whether the paper is correct or not, it's whether it's a well-formed paper. Saying "this paper got on arXiv so it must be decent" would be like saying "Microsoft Word didn't find any errors in this paragraph, so it must be true".

I  strongly encourage you not to trust anything on arXiv (unless it has a gazillion citations).
I had the chance, many times, to (officially) review papers that I "noticed" beforehand on arXiv.
As someone else mentioned "at first glance they looked OK". Then, after reading  into them, I started to notice a lot of issues. And I'm not talking about small inconsistencies; I am talking about significant problems that completely invalidated the papers' main claims.
Just to give you two examples.

A paper once stated in the abstract "Our method is superior than the state of the art method X by y%". Then, after reaching the experimental section, I discovered that the authors used a completely different setup than what was used to evaluate the original method X.

Another time the abstract stated "our attack breaks the state of the art technique X". Then, later, the authors wrote "we did not implement X as it should be done".

Any reviewer would spot such issues right away and reject these papers even in low-tier journals because they are flawed.
Note that I'm not saying that any arXiv publication is trash. There may very well be (actually, there is) good source material on arXiv. However, while you can take any peer-reviewed publication (almost) at face value, when it comes to arXiv you need to thoroughly question its validity first - and use not just a grain, but an entire bag of salt.
My bottom line: "publishing" on arXiv is not devoid of meaning, but the scientific value of arXiv papers is either negligible, null, or negative. I would take a low-tier journal paper over an arXiv publication any day, if given the choice.

To be clear, this answer addresses submissions on arxiv that have not been published elsewhere.

@MateenUlhaq yes, but an article that has been published elsewhere then it is not an article "published on arXiv". The question specifically asked if "publishing on arXiv is more reputable than a low tier journal" - which is not.

"Any reviewer would spot such issues right away and reject these papers even in low-tier journals because they are flawed." Stated without evidence. Indeed, there is significant evidence to the contrary.

Citations are not an accurate indicator of quality.

@AnonymousPhysicist: Of course citations don’t infallibly imply a paper can be trusted, but they’re a very good first-pass metric, especially if you look a little closer and filter on “citations from reputable authors, and/or in reputable journals”.

@Xerxes. And its corollary "you can take any peer-reviewed publication (almost) at face value". Hahahahaha

Trust it just as far as you would trust the authors making the claims to you in person.

arXiv itself isn't intended to be treated as a publisher. Submitting a paper to a low-tier journal helps its credibility at the very least because at least it receives some peer review. Posting a paper on arXiv just means you satisfy the requirements of their site and you have some credential (i.e. a university email address) to post papers. A good rule of thumb is to verify a paper has undergone peer review and publication in some trustworthy journal, and then read its preprint on arXiv.
Regarding your point on citations: Many documents have been cited from arXiv because authors post their preprints there before formal publication in journal or conference proceedings and the formal publication is not available. Or, it's just because people don't want to pay to read a document on a site like IEEEXplore and just cite the preprint on arXiv.


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Should I mention about the time wasted (6 months) due to depression in my CV when I will apply to PHd programs
Very brief of the situations that happened from Jan 2021 : Living in a 3rd world country. I completed my masters in math in Jun 2020. Then I wanted to study some additional topics before applying in Europe. So, took 1 year break and went home. Studied fine till Jan 2021, then constant verbal abuse started by my father and occasionally by my mother (they earlier also used to verbally abuse me but I didn't do my bachelors and masters in my native city, so less impact was on me).
Admitted to mental hospital for 10 days in Feb and misdiagnosed with schizophrenia (I have depression). Verbal and emotional abuse continues and again admitted to different hospital in March for 11 days, diagnosed with depression. Verbal abuse again continues after some days of discharge. So, I move out taking a part time job to sustain myself in different city on 1 July and learning some new math topics alongside. Thinking about suicide since March 2021 but postponed to try 1 more year if I am not able to get a phd position as learning and doing math is the only thing I like.
(Reason of verbal abuse: parents wanted me to do full time job and not math research so that I can settle early and also wanted me to take a career like medical doctor or engineer which is more earning and takes less time to settle time). I could only study 1 month of math in these 6 months.

How much should I mention the situation due to my 6 months of my life are being wasted in my CV when applying to PHD position in Mathematics? Will the admission by me that I have suffered from depression affect my selection for Phd i.e. how it would be taken by selection committee?

I don't think I should lie as I would be really weird to make up what I was not doing as what I was doing. I really need advice.
Also, I think if I don't mention it in CV then certainly I will be asked in interview about the gap in studies (and then I will have to explain in very few lines which will obviously change the course of interview to an entirely different thing if it is not known to the prof. beforehand).
Certainly, I am not mentioning it for any sympathy but I think I should write about it very briefly as I write CV in a bit timeline form; if I just left a gap or don't mention it then certainly it would be thought that I have been less productive because of the lower number of things I studied. (But there is a very valid reason of being less productive).

Edit : I feel that I should at least mention it because it is due to abuse by my parents that I have been so depressed and I wasted precious time and I would have certainly been better if my parents were able to understand what I want to do with my life.

@No-OneThanks  to you  to add a bounty on it.

What country? As attitudes toward mental health and the reception you might get may vary by culutre.

@AzorAhai-him- I will  apply only  in Europe: France, Germany, Finland , Denmark.

Based on the description, it sounds like everything that happened, occurred during this period:

Then I wanted to study some additional topics before applying in Europe. So, took 1 year break and went home.

That's all you need to say. It's true. You aren't under any obligation to go into details about what happened during that year in your personal life.
I think you are overestimating how suspicious people will be about this one year. On the other hand, disclosing personal information like what you say in your description, unprompted, will almost certainly make admissions committees nervous (at best it is an extra piece of information they have that doesn't contribute positively to your case). Focus on your conveying your background and studies. These things happen, be easy on yourself, and remember we can move on from these personal events that seem very important to us in the moment -- don't tie yourself to it and define yourself by it.
Edit Dawn also made an excellent point in the comments that people are aware the previous year has been difficult for everyone because of the pandemic. In my mind, this goes even further in the direction that people are more likely not to weigh one "gap year" very strongly, and you should not go out of your way to answer a question that wasn't asked by providing intimate details about your personal situation.

"Based on the description, it sounds like everything that happened, occurred during this period" : Ya

Agreed. I would ignore this year for purposes of evaluating candidates due to the pandemic. If the rest of your record is strong, best not draw attention to it.

@Dawn That's an excellent point, I've added a comment about this to the answer.

@Andrew Certainly total wastage of 5 months is not a short time. I am planning to write it in CV  in the following way , not mentioning depression, please have a look: " I had planned to apply to phd positions in January 2021 and subsequent months but my father in January 2021 became really adamant that I take a job so I had to take a job. But learning more math and doing research in it is my aim , so I left the teaching Job and moved to different city in June 2021, self studying more math topics full time with my mother financially supporting me"

@Andrew Your advice will certainly help me a lot

@res In my opinion (having served on PhD student admission committees) this is still too much personal information. Even if you feel you could have done more in those 5 months, the programs you are applying to don't know that you were originally planning to apply in January. Plus many deadlines are actually in the fall, so applying now is not suspicious. My advice is that I would not say more than "I took a year from 2020-2021 to work and self-study additional math topics. I focused on topics X, Y, and Z by studying resources A, B, C." But, you should do what you feel is right.

Of course, it is different if during an interview you are asked for more details about this year. You should never lie in any application materials, especially the interview. But in terms of the materials you submit for your own application, I would advize not offering details about your personal situation (this is actually advice I would give anyone). Just focus on what is objectively true (you took a year to work and self-study math), and on what mathematics you have done. Again, you need to do what you think is right, but I'm just hoping you don't self-sabotage.

If the gap is short enough that it is unlikely to be noticed, then ignore it for the CV. For a longer break give as few details as possible. Certainly nothing like what you write here. "Medical issues" or "Personal/family needs" is all you need to say.
Reserve the CV for those things that show your productivity and skills, not for those that explain problems.
If you are asked about it in an interview, I also suggest that you give as little information as you can.

"If you are asked about it in an interview, I also suggest that you give as little information as you can" Why ?                                                                                I think if i don't explain it in a concise but complete manner I think the interwiewer will think that I am j=hiding something Or I am not competent enough

Many people need a time out. Medical and family issues are perfectly valid without detail. No one need to evaluate your relationship to your parents. No one will hire you because they feel sorry for you. Including this in a CV doesn't show competence. If the rest does, you should be fine. Concise, yes. Complete, though, I doubt the value. In some cultures, people get a lot of parental pressure to do things that only the parents want. Others in that culture probably understand how powerful that can be.

@res "I was dealing with some mental health issues during that time but have been doing a lot better since then" is sufficient. You can even leave the word "mental" out and just say "health issues", or make it even more vague and say personal/family needs like Buffy wrote. All you need to convey is a) I wasn't able to work during that time, and b) Things are better now and I can work. If that information puts someone off, you don't want them to be your supervisor anyways.

@BryanKrause, I'd leave out "mental". It is a red flag. But "under control" is a vital point.

@Buffy I think there are reasonable differences of opinion on that, depending on what someone is comfortable with. Certainly I don't think anyone should need to disclose any details, including "mental"; on the other hand, I wouldn't want someone to work for an advisor for whom any mention of mental illness was enough to red flag them and deny admission. Details of the specific nature of the mental illness, though, I'd agree is not really anyone else's business.

@Buffy Gap is 5 months which is not  short. So, I think I should tell them . Also, If I wrote it  in  the email which I am writing ( to which I will attach the CV)  I think it will demotivate the prof  to consider me with seriousness. Can you please give the implications of the adding the fact that i have been in depression in detail

"Depression" can mean a lot of things, including a diagnosed medical condition, treatable by a physician. If you are being treated for whatever condition you have, I say, again, that you should reveal the minimum. If you fear that the prof will be demotivated, all the more reason. Note that you aren't unique, but that people's reaction is hard to predict. Don't lie, and keep up your therapy/meds/whatever. But as @BryanKrause suggests, consider whether you want to work with this prof.

@Buffy Ya , I am confused , to me situation is complicated.

@Buffy Thanks for very much  your precious guidence. Centainly 5 months is not short and I will not skip it in my CV. I am planning to write this in my CV for months January to May(2021): " I had planned to apply to phd positions in January 2021 and subsequent months but my father in January 2021 became really adamant that I take a job so I had to take a job. But learning more math and doing  research in it is my aim , so I left the teaching Job and moved to different city in June 2021, self studying more math topics full time with my mother financially supporting me".  ....

@Buffy ( I hope this is a better way of writing). Kindly have a look at it. Your advice will really help me!

@Buffy Can you please reply if you have some time to spare?

When I am making a decision on a PhD application (for my department, not as prospective supervisor, in the UK) the question I need to answer is:
Will this person, if admitted, in all likelihood do well in our programme and earn their PhD (through a good dissertation)?
The question is definitely not Does this person deserve a shot at earning their PhD? Hence, what an applicant has done is the past is relevant only for its predictive value for what they would accomplish in the future. If I believe a PhD candidate needs skill X but is lacking it, it does not matter whether it is the fault of the candidate that they haven't learned X or not.
The decision is meant to be somewhat objective, made in a rational mode of reasoning. Application material overtly meant to elicidate an emotional response is thus going me me somewhat uncomfortable, as I will need to work to separate this from the decision process.
While I recommend disclosing mental health issues that require accommodation to a prospective supervisor early on, only in very limited circumstances would it seem to be in the applicant's best interest to bring this up in admissions. This situation is when an applicant has temporarily underperformed significantly in the past due a mental health issue that is under control now (and will stay that way); and where the applicant now has the relevant knowledge and skills they should have.
In the concrete situation described in the OP: The mental health episode happened outside of formal studies. There are no bad grades to explain. The impact on the profile of the applicant is that they have learned somewhat less of additional math than they would have otherwise, but that is what it is. There is absolutely nothing to gain for OP by bringing up mental health in their application. If in an interview the question of what exactly OP learned during this year comes up, it makes sense to express that it was less than anticipated due to adverse circumstances. But even here I would not bring up mental health, but rather state that OP is much more productive when at a university.
As pointed out by Dawn and Andrew, the last year was a bad one for lots of us. Most academics will not question a simple 2020/21 was NOT a productive time in my professional life claim, because they either share that sentiment or have heard it from lots of colleagues they respect.

As total wastage of 5 months is not a small period, I will certainly mention it in CV, but will not mention anything about depression , but in a following way, please have a look:" I had planned to apply to phd positions in January 2021 and subsequent months but my father in January 2021 became really adamant that I take a job so I had to take a job. But learning more math and doing research in it is my aim , so I left the teaching Job and moved to different city in June 2021, self studying more math topics full time with my mother financially supporting me"

There is no mention of depression or any  mental health.Any advice/ comments of yours will really help me.

@res Needlessly personal, doesn't make you look good. A simple "January - June: Teaching job at X", "June - Now: Self-studying maths (primarily books/topics A,B,C" would do.

If you believe it affected your grades or would otherwise affect your application package in a significant way, you can go ahead and briefly mention it.  If you go overboard on mentioning it and focus on your depression too much though, it won't necessarily be a good thing and could even backfire.
If it's something like you took a gap year after graduating undergrad to mentally recuperate, maybe even take a job during that time, I wouldn't even bother mentioning it (unless you were productive during that time, then mention what you did to be productive and not your mental health), as others have said.  Unless the gap caused by this would be that significant, and even then I'd maybe spend a single sentence or two talking about that and then move on.  (I wouldn't even say a gap from March 2020 because of the initial outbreak of COVID to be that significant given how many people the early stages of the pandemic did affect).  In your case, 6 months isn't that big of a gap (in industry, at least, it isn't uncommon to have a 6 month gap between jobs, or between graduating and starting your first job); I personally can't justify, mentioning, let alone going overboard over this small of a gap.
On the note of something mental that would require accommodations, that stuff isn't meant for the grad school application package.  If and once you are admitted to a few places, THEN you can bring it up with the disability department at your prospective universities and try and work something out.

"On the note of something mental that would require accommodations" :I am working fine now, studying and looking foreward to my future, and I think when I will leave far away from my father( not even talking to him , like i am not doing now , and didn't talked with him in masters) , I think I will remain healthy , and so in my case I don't think it will be a disability.

Just wanted to bring it up in case, but if so more power to you

Thanks for your advice. As total wastage of 5 months is not a small period, I will certainly mention it in CV, but will not mention anything about depression , but in a following way, please have a look:" I had planned to apply to phd positions in January 2021 and subsequent months but my father in January 2021 became really adamant that I take a job so I had to take a job. But learning more math and doing research in it is my aim , so I left the teaching Job and moved to different city in June 2021, self studying more math topics full time with my mother financially supporting me"

Kindly let me know your opinion on this.

Don't even refer to the 5 months "wastage of time".  You had a small gap, or break.  That's it.  Nothing more.

To be honest, most people don't care about personal stuff. I will suggest writing doing freelancing, getting training for any subject etc. People will show sympathy once, but overall, they will criticise it. Yes, exceptions are there.
My opinion sounds negative, but this is reality.

Kujur As total wastage of 5 months is not a small period, I will certainly mention it in CV, but will not mention anything about depression , but in a following way, please have a look:" I had planned to apply to phd positions in January 2021 and subsequent months but my father in January 2021 became really adamant that I take a job so I had to take a job. But learning more math and doing research in it is my aim , so I left the teaching Job and moved to different city in June 2021, self studying more math topics full time with my mother financially supporting me.

Kindly let me know your opinion on this. Any comment / advice you give me will certainly help me a lot.

@res I understand your situation, you can mention that you were learning some course, you don't need to be master of that subject.


text:
Posting an author's accepted manuscript on arXiv
Many journals allows one to post an author's accepted manuscript online, provided some embargo period has lapsed and an acknowledgment is included. For example, for Springer Nature journals, an acknowledgement of the following form should be included:
“This is a post-peer-review, pre-copyedit version of an article published in [insert journal title]. The final authenticated version is available online at: http://dx.doi.org/[insert DOI]”.
I have two question about arXiv.
Question 1:
Is it enough to include this acknowledgement in arXiv's Comments section, without including it in the source file?
Question 2:
Is it OK not to write the sentence above altogether, but to fill out arXiv's Journal reference and DOI fields?

Probably going to depend on the journal. And also the country the journal is published in. And maybe the country the author is in. Copyright is a messy complicated and varied subject.

About question 2, the journal says you should include the statement, so why would you think it’s okay not to include it?

No need to do too much hairsplitting. The journals want a link to the published version and a clear statement of the status of the arXiv version. Was anyone ever sued for incorrectly formatting this statement or putting it in the wrong place?

Question 1: Copyright statements should go into the full text. Copyright statements should not appear in arXiv's metadata (scroll down a little).
Question 2: Please do fill in the metadata with the Journal-ref and the journal-assigned DOI.


text:
Automated proof-checking at mathematical journals

But Timothy Gowers, a mathematician and Fields medalist at the University of Cambridge, wants to go even further: He envisions a future in which theorem provers replace human referees at major journals. “I can see it becoming standard practice that if you want your paper to be accepted, you have to get it past an automatic checker,” he said.

Stephen Ornes, How close are computers to automating mathematical reasoning?, Quanta Magazine, August 27, 2020.

Is anything like this being done? I'm inclined to say it is, if they already have the tools, even if they don't publicly admit it.

There's literally zero chance any journal is doing anything like this.  There's been exactly one instance of serious computer verification of a substantial portion of cutting-edge research within the timeframe of the refereeing process, and it's the Liquid Tensor Experiment where a group headed by Johan Commelin verified a key portion of new results of Clausen-Scholze.  It was a big surprise to a lot of people that this was possible, and it's certainly not being done routinely or secretly.

Math papers are full of sentences like "By a routine argument similar to that in the proof of [13, Theorem 2.3], it follows that..."  A human who is familiar with [13, Theorem 2.3] can come up with the argument.  Computers are nowhere near that skilled.  You might ask why mathematicians are allowed to write these vague hints at arguments in their papers instead of actual arguments - the answer is that you really don't want to turn what are now 30 page papers into 300 page papers, and it's actually easier for an expert to reconstruct the argument from the hint than to read it.

@JoeD - You're misinterpreting what happened in this case.  What happened is that some mathematicians who are not the authors took the proofs from the paper, expanded them into a form that could be understood by the computer, and then had the computer check them.  This is far from an automated process.

What Timothy Gowers suggests can't be done in secret or as a simple add-on to an otherwise unchanged review process at the current stage of technology.
The suggestion, as I understand it, is that at some point in time, journals may require authors to submit along with a traditional article a machine-checkable proof of the correctness of the claimed results. The author would, hence, work with a proof assistant to produce a formal proof of a formalized version of their main results. Successful verification of the formal proof would then allow reviewers to focus on assessing the novelty and importance of the results, without having to worry about correctness too much. They would still have to be convinced, though, that the formalizations of any results given are really formalizations of the theorems shown in the paper under review.
I think for this to happen the work factor of formalizing a typical mathematical proof would either have to fall to a level not much larger than writing up the paper (assuming that at least one of the authors is skilled in working with a proof assistant), or the production of a formalized proof would have to become a routine part of doing research. In the latter case, submitting a formal proof of one's results would be similar to submitting source code as supplementary material to a computer science paper.
For either of these to happen, proof assistants would have to become much more powerful and intuitive to use than they are today. That does not necessarily imply, though, that sufficiently useful proof assistants will only be available in the far future. Fully automated machine translation, for instance, has gone from being regarded a fairly intractable problem to being a problem with good, widely available solutions in just a few years.
The problem of automatically verifying claims in a paper written in natural mathematical language is substantially harder still. I would be tempted to say that achieving this probably requires human-level AI, but on the other hand one has to admit that previous claims of AI-completeness have failed in various domains.
The romantic notion that mathematical reasoning is based on some miracle that can only happen in human brains is, however, nonsense of course.

That verifying math requires human-level AI is perhaps incorrect; the vast majority of humans cannot verify say, a proof of the fundamental theorem of calculus.

Probably not.
Scientific papers, even papers in mathematics, are written in natural language - sometimes fairly idiosyncratic natural language, at that. In order for an automated theorem solver to solve a theorem, the theorem will need to be rendered into a form of structured data that the computer is able to understand. While I'm not too familiar with them, I imagine that it probably involves the mathematician manually entering all the different variables and parameters into the solver, right?
Natural language comprehension by computers is currently fairly primitive, and is largely limited to the computer determining things like that certain words are more likely to occur in combination with other words, rather than demonstrating any true understanding of their meanings. As a result, getting a computer to parse a scientific paper written in natural language would be a very difficult task!

But aren't there some parts of it that would work? Not the entire paper obviously.

@JoeD Even the parts of the paper involved in describing mathematical claims would still likely involve a lot of natural language; even if LaTeX might be more structured that plain text, how is your automated proof checker supposed to know what sort of meaning you've given to your various mathematical symbols? Maybe "delta" means a derivative, maybe it means the difference in values between subsequent values in a series, maybe it means something else altogether. That's where context comes into play, and computers are really bad at reasoning about context.


text:
What do graders expect in a master thesis in pure mathematics?
I am writing a master thesis in pure mathematics the upcoming year and want to know what graders look for in a master thesis. I am in the US.
How do they grade the thesis? What are the things that can potentially lower my grade and what should I look out for?
Basically, in a pure mathematics master thesis they won't be expecting original research, and since I will be copying everything from the books in the bibliography, how can this go wrong?
I assume the presentation must also play a role, as for the grader to know whether I mindlessly copied the content without understanding the concepts or if I understood the topic.
Can you explain what the graders are looking for in a pure mathematics master thesis in order for me to ace the thesis?

Which country is this? In mine, a master's thesis in mathematics is very much expected to contain new results.

USA, I forgot to mention this is about pure mathematics not applied, do universities in your country expect new results in pure math, I would imagine this to be very hard, that is to prove an unproved theorem that professional mathematicians may struggle to prove in such a little time and limited experience.

UK here. I don't expect a new result, but I expect it written in your own way, new examples, your synthesis, your proofs and overall your understanding.

"my proofs" so should I find new ways to prove already proven theorems or just present the proofs given in my own way?

Can you have a look at theses written by previous students in your department?

At the American institutions where I have experience, a masters student has an advisor (and a committee).  The advisor (and committee) are responsible for setting the guidelines for a masters thesis.  You should talk with your advisor (and committee) about what they expect from you.

This is completely unanswerable on this forum. In the US, master's theses for mathematics are unusual, so it's even more true that it's up to your institution and your committee.


text:
What is the right time of year to write to professors in France asking for PhD positions?
I am a citizen of an Asian country who completed his master's in math in June 2020. I wanted to study more topics before applying for a PhD (in one of the pure mathematics subjects) in Europe, but got ill in Jan 2021, and couldn't apply to a single grad school.
But I did write to some professors, and in Jan 2021, one professor in France replied the following:

We essentially have targeted funding so we need student 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 couple of
months preparation before the deadlines of funding applications.

Question: In which month of the year should I start writing to professors in France asking if they are willing to supervise me for a thesis, so that I can get funded (as my country doesn't offer any financial assistance for students studying abroad), and so that deadlines are not missed.
Edit: I also want to tell that although I tried my best to ask professors and take their guidance (in my university from which I did my master's), I wasn't helped much despite having good grades and really good knowledge of the branch I want to pursue a PhD in. The research and general  academic environment is not so good and that is one of the reasons I want to go abroad for further studies.
Thank You!!

Keep an eye on sites like: https://scholarshipdb.net/scholarships?q=&l=France and https://academicpositions.com/jobs/ and https://www.mathjobs.org/jobs/job

Question: In which month of the year should I start writing to professors in France asking if they are willing to supervise me for a thesis, so that I can get funded (as my country doesn't offer any financial assistance for students studying abroad), and so that deadlines are not missed.

Any month.  Firstly, different funding sources have different deadlines.  Secondly, different professors are busy at different times.  Some may be able to make time available on short notice, others will need to plan many months ahead to schedule time for defining a project.  Some funding sources require that money starts to be spent immediately, others may allow a time window for when the PhD candidate starts.  PhD candidates may not necessarily need to start at the beginning of the academic year, either.


text:
In what month should I write to professors in Germany regarding PhD positions?
I am a native of a South East Asian Country (3rd World) and completed my masters in mathematics in May 2021. I am looking to apply to some German Universities for PhD Program  for session of 2022.
So, I am thinking of writing to those professors whose interests align with mine asking for funded PhD positions.
I want to know that   which month of the session is the right time to send an email to professors in Germany. I want to work with as a PhD student for funded positions.

This is not an answer to your question, but it might be useful information anyway: Your chances of landing a PhD position in Germany might increase if you consider applying for a DAAD scholarship to fund your PhD studies ("DAAD" is an acronym for "Deutscher Akademischer Austauschdienst", in English: "German Academic Exchange Service"; link to their webpage). This might be interesting for potential advisors, too, since a DAAD scholarship for you means that the funding does not need to be taken from their regular positions nor from one of their grants.

The majority of funding for PhD students is coming from third-party funds (or from the states), for which there is no hiring season. The exceptions are graduate school scholarships, which are very rare.
You write that you want to write directly to professors whose interests align with yours. You can do that, but you have to stand out from the crowd here. Professors get tons of badly formatted e-mails asking for PhD scholarships because their interest supposedly aligns with the one of the professor. The majority of these e-mails do not state what this interest would be, however. They are deleted in 3-5 seconds (rough estimate). So if you write an e-mail without prior personal contact, do this well to ensure that you are not seen as belonging to this group:

Use a proper sender e-mail address. Don't send the mail from luckygeese123@vip.163.com, but rather from firstname.lastname.14@gmail.com, or even better, using your current university of alumni e-mail address.
Use your full name as sender's name.
Sign your mail with the same name as your sender name.
Start the mail properly. E.g., "Dear Professor XYZ,"
Should you use HTML e-mails, make sure that they do not look like copied and pasted together. Uniform font and uniform font size are a must. No color. Allowed are itemizations, enumerations, bold, italic. No other formatting elements.
Write early in the mail (first three sentences) what the area of interest is that you have in common with the professor. Be as precise as possible, but choose topics in which the professor has at least two publications. This shows that you actually had a close look, which puts you ahead of 99% of the crowd. This will make the professor have a closer look at your "case" if they happen to have some funding. Also mention in your first three sentences what your existing expertise in the area is. Did you do research in it already? Did you write a thesis on it or a closely related topic? You want to demonstrate that you are a low-risk candidate.
If you happen to be in Germany already, you want to demonstrate that you know already how doing a PhD in Germany works. The fact that you talked about "session of 2022" shows that you don't, however. So look it up!

Thanks for such an great answer.

While all of this is answer is true @YannicMuller, a word of warning is still in order - your chances of actually getting a positive reply to a cold email application, are very slim. So, if you want to apply this way do it like DCTLib says, but be prepared for a negative response (or no response).

@xLeitix Thanks for your advice, "if you want to apply this way do it like DCTLib says, but be prepared for a negative response (or no response)". So, how should I write so that I can get a positive reply. Can you please shed some light on this? Any advice you give me will really help me.

@xLeitix I am using this year full time for self studying and applying to professors so I cannot afford negative responses. My aim is to do research, so can you please give some advice on how should I write about it so that professor comes to know that i am genuinly interested in working with him, which I am to the proefessors I am writing.

@YannicMuller This answer is already the best advice. You might get a positive response if you follow this advice. There is no way to guarantee a positive response.

It probably depends on the field, but in my personal environment, chances would be quite a bit higher if you apply for PhD positions that are actually advertised. I don't know many (any?) professors who have spare money waiting for the perfect applicant to cold contact them... Actually, in many cases, it is illegal to hire someone without first making an official job advertisement. I'm not sure I know anyone who got hired based on a cold contact email.

@YannicMuller The hard truth is there is no way that is likely to work. An email can only do so much.

@jhin It is not so uncommon people get hired on a cold contact email, if they are a good fit and there is some kind of link to the PI (in the sense that the PI can meaningfully judge their abilities). The job advertisement can always be made afterwards.

@user151413 Interesting, I'm always happy to learn. So how does that work; where does the money come from in that case? I essentially only know of two states for PhD positions: a) there is no money, and b) there is money and the position is officially advertised.

@jhin There are many other states. Examples are: (a) nobody had the time to make a formal advertisement yet, (b) administrative details still have to be resolved before a formal advertisement can be made (c) there is a grant coming with 99% probability, but before the funding letter has arrived, no formal advertisement is possible, (d) the last ad led to 0 suitable candidates while a lot of work was wasted on the search process, so a formal advertisement will only be made after a good candidate is available. Formal advertisements are BTW not always necessary for third-party funded positions.

@YannicMuller this is the best advice. In addition attach a CV and a short motivational letter why you would want to work in this field and specifically with this person (maybe you read Paper X, you saw they work in project Y etc). Stand out by Formatting the letter and CV nicely, putting Focus on relevan experience you have.

can this be generalised to european?

@BCLC Hard to tell - not sure if anyone here has the overview of all European styles of doing a PhD. I'm expecting the answer not to hold in countries in which getting admitted to Phd studies is not a substantially less difficult barrier to pass than getting funding.

In Germany, PhD students generally do not have any course requirements (which typically would lead to a standardized starting month). Funding is usually via regular staff positions (which heavily involve teaching), or on grants. The former will be filled again when their previous holder leaves, and the latter whenever available.
As such, there is no clear "PhD application season" in Germany.

... and both are usually filled through a regular process after advertising the position, not when the professor gets a cold-contact email application.

@jhin What do you consider a "regular" process of advertisement? (I disagree.)

@user151413 I mean an official advertisement on job sites, the university page, etc. - aren't universities actually required to do this?

@jhin: I'm a bit surprised by the various claims here that all positions in Germany were required to be officially advertised. Do you have a reference to a law (say, in any federal state) where this is stated? In pure maths, I've seen many people getting non-permanent "Landesstellen" (as PhD students or Postdocs) that were never officially advertised. I've also seen grant applications which named a concrete candidate for a position that should be funded by the grant.

@JochenGlueck See e.g. here. There seems to be a general requirement to make official advertisements, but there are some exceptions from that rule. None of the mentioned exceptions sound applicable to state-funded PhD/PostDoc positions (Landesstellen) to me, though I may be mistaken about that. For third-party funding, requirements seem to vary between funding organizations.

@jhin: Thanks for your response! Hmm, I don't think that the info sheet of Uni Göttingen that you linked implies that every (or most) positions have to be advertised publically in the sense of "accessible to the general public". For instance, Section 1.2.1 mentions the requirement that positions need to be advertised at least "hochschulöffentlich" (i.e., within the university, which does not make the advertisement visible to anybody who is not already affiliated with this specific univesity). [...]

[...] Besides, the situation described in exception g) in Section 1.2.2 seems to applicable to state-funded PhD/PostDoc positions if the preferred candidate is already an employee or student of the university. In my experience, precisely this situation happens very often in pure maths: a student and the supervisor of their Master's thesis plan that the student will start a PhD with the same professor, and the student then gets one of this professor's "Landesstellen". Many positions are filled this way, and are thus not advertised.

can this be generalised to european?

@BCLC No. For example, in the UK a lot of PhD funding is done via CDTs; and most of those will start in October. I'd expect that CDT application deadlines will peak around Spring.

@Arno thanks! (UK i believe is the best example to explain why anything involving a european country besides the UK can't be generalised to all of europe!)

CDT = centers/Centres for Doctoral Training ?

@jhin Quite as Jochen describes: Advertisement can be semi-public (e.g., hang out the advertisement on some department pinboard), and furthermore, it is often the case that it is already clear who will get the position (be it through an inofficial call, an direct application received, or a master student who has been offered a PhD, e.g.)

@user151413 FWIW, the document I linked to explicitly mentions department pinboards as not being sufficient. (The advertisement must be at least university-wide.) On the other topic, I completely agree that it is often clear a priori who is meant to get an advertised position, but that is not an argument for the success chances of cold call emails, I guess? :-)

@jhin You reasoned that the fact that the fact that positions are filled through an advertised process are an argument against the success of cold calls. I just disagreed with that, and still do. I never claimed that the fact is often clear who would get an advertised position would speak in favor of cold emails.

tl;dr: The right time is now.
It doesn't matter what the current point in the recruitment cycle is. It is never too early to write, introduce yourself, express interest, discuss a potential visit, possibly be put in touch with other Ph.D. candidates in the group etc.
It may be the case that the official time to apply is still far away, but you will be making a professional contact; getting potential useful advice; and maybe find opportunity to collaborate and increase your chances of being of interest to the group. Plus, if you are asked to visit - you will get to know an additional academic environment up close.
At worst, the Professor might tell you to remind them in a few months.

Actually, I think that if you write emails like "I am X, have a master from Y and my interests are in Z" where Z are also interests of the recipient, the chances are low that you get a meaningful answer.
The university you graduated from is probably unknown to the recipient, and therefore you need to show that you are really able to do research in Z. If you published a paper or have a master thesis online, I would link to it.

Most PhD positions in Germany are regular (fixed-term) employment positions, and they are offered whenever a vacancy arises, i.e. there is no recurring deadline. Since those positions are often (sometimes must be) advertised, you should monitor relevant job bulletins. I'm not a mathematician, but I imagine the German Mathematical Society or similar groups have a newsletter or liststerv over which job postings are distributed, for example.
Some PhD positions in Germany are funded by stipends, in particular those at graduate schools (less common than in the US), but there are also other foundations that may offer stipends for PhD students. In both cases, positions are offered periodically, namely twice per year (once per semester), mostly in spring and fall. For example, the BIGS graduate school invites applications until 30 November and 30 April each year.

"those positions must be advertise": May I ask for a reference for this claim? The second paragraph of this article in the German Wikipedia claims the contrary. I also made the experience (in pure maths) that many non-permanent positions are never advertised.

@Jochen Glueck personal experience, but I recognize that wikipedia says the contrary.

can this be generalised to european?

@BCLC it also applies to Austria, more I can't say with confidence. But you could post another question.

In general, I think you are much less likely to get a response to a cold email like this, than applying to an advertised PhD program. (When I received emails like this as a postdoc -- which immediately was a red flag since I didn't have any hiring powers -- if I didn't delete them my response would be a link to the institute's website). The advertisements should be posted on the website of the University or institute you are applying to. There is very likely also a "job board" section of a journal or institution in your field that lists these types of ads.

As many people already said, you are more likely to get a response if you actually apply to an offered position. Nevertheless I can confirm that that PhD students are not only hired on offered positions, but also in response to cold emails or just "being around" and fitting into the group while showing good work. So in any case you want to show the professors what you have already achieved. Note that the results of your work (e.g. master thesis) are not as important as how you presented it and how it was conducted. To explain this further, in my thesis I was working on an open question and could not present meaningful results, while I was able to come up with good theoretical explanations for it.
In addition it is highly important that you do not only write to the best renowned institutions, as they usually have many more applicants (per position) than the smaller ones. Also: stay strong and do not let yourself go because of some negative or missing replies. Usually you will get a lot of those, especially when applying via cold emails without apparent positions.