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When I started graduate school ten (!) years ago, a more senior Ph.D student at the time told me about MathOverflow and encouraged me to participate. Ever since then I have found it immensely useful. Some responses (see Richard Stanley's answer: Some restricted weighted sums, and the paper that this answer appeared in) have directly helped me with my thesis work. Many answers I received here have been very helpful for my subsequent research.

However, I feel like MO today is quite a different place than what it was ten years ago (this perception possibly influenced by nostalgia), and in particular, possibly less friendly to newcomers. In the early days one could get away with asking questions that are borderline research level and it's less likely that the question has already been asked.

My question is: would you still encourage graduate students to participate in MO, and if so, at what stage of their PhD would you recommend that they start participating?

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    $\begingroup$ The main problem I have with MO is that I am unable to define what is a research-level question. Sometimes, some questions are closed without I can understand the reason. In mathoverflow.net/q/187609/24563, I asked a question about Boolean algebra and the poster added "per request for more detail" and I think that for him the answer was obvious and probably not research-level (see also the comments). I am not specialist at all of this field (Boolean algebra). $\endgroup$
    – Philippe Gaucher
    Commented Sep 8, 2020 at 7:41
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    $\begingroup$ @PhilippeGaucher I mean, he wrote a book about it. $\endgroup$
    – Harry Gindi
    Commented Sep 8, 2020 at 19:23
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    $\begingroup$ Just a personal anecdote that does not answer the question: I came to the game a little late (less than 5 years ago), and I did struggle at first. Like any other place in mathematics, I think reputation (in a broad sense) plays a role in how someone's work (e.g. questions) is evaluated, and I feel much more at home here now that I have established myself (not just in terms of points, but also that people know what sort of things I post). These feelings are not so measurable, but that doesn't mean they aren't real. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Sep 10, 2020 at 17:35

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I guess I still would. The clear case is when one can answer a question. The regular practice of picking up a question that is accessible, answering it, and communicating the answer can be very beneficial. Even I am learning from it a lot and I'm by no means new to "random problem solving". So, if a student (or anybody in general) can answer questions on MO, I would definitely encourage them to participate. Just warn them that it may become addictive, so they should balance it with their regular duties. I have also never seen any "unwelcoming behavior" with respect to people who post answers, at least, as long as purely mathematical discussions are concerned.

The issue arises when somebody's contribution is reduced to asking endless questions of the type "I cannot figure out this place in the textbook, can you help me?". There are questions of this type that are still eligible for MO, but many of them are on the borderline and would be better redirected to MSE or just to the student adviser. So, I would suggest that in such cases the students view MO as the last resort, not as a help hotline.

Any questions about student research (i.e., something that is not readily found in the textbooks or, if you prefer, something that falls under the "missing-lemma" tag) have always been welcome and I try to answer them and to protect them from closing as much as I can even if they look a bit "homeworkish". I would encourage such questions any time. Some of them are now being asked on MSE, which is a pity, because they usually just get flushed by the stream there unless somebody attaches a bounty.

As to "MO being less friendly to newcomers", my feelings are mixed. On the one hand, we introduced a lot of friendly perks like the automated reminder to (try to) be nice whenever responding to someone who posts for the first time. On the other hand, there still seems to be a strong movement to eradicate everything that is not "research-looking" when judging from pure appearance (though I would say it was much stronger a year or two ago). I mean such things as, for instance, voting to close for "the lack of motivation". My own stance is that any clearly posed mathematical problem that I cannot solve in under 10 minutes should pass and the OP has no obligation to explain in detail where it came from and what it is related to. It especially applies if the OP is a student: quite a few of them have enough trouble with merely understanding and stating what exactly they need.

The MO may be not exactly what it was 10 years ago (all is changing, all is flowing, and some part of the discrepancy in perception may come from the fact that we ourselves are not exactly what we were 10 years ago), but it still looks like a nice and interesting place to be at to me. As soon as the student is ready for a mathematical discussion, he or she is welcome to enter. That may happen at any "stage of PhD" and at any age. I wouldn't be surprised to learn if some participants of MSE who ask there questions that might be suitable for MO as well are, in fact, high school teenagers or undergraduates. If in doubt, just try MSE first, but if you see that your questions merely earn you reputation there but are left unanswered, switch to MO.

Those are just my two cents, but I'm an old-timer. It would be really interesting to hear what the newcomers have to say, especially the young ones.

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    $\begingroup$ My chief impression is that MO has become less friendly to newcomers only in the sense that (compared to when I joined, as an undergraduate ca. 2009) there are much fewer elementary and/or simple questions to answer. This is partly because math.stackexchange is gobbling them up, partly because lots of "canonical" questions that are on people's minds have already been answered (and MathOverflow makes it fairly easy to identify a duplicate), and only to a minor extent due to the "not researchy enough" close brigade. $\endgroup$
    – darij grinberg
    Commented Sep 5, 2020 at 18:39
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    $\begingroup$ I also think that questions shouldn't be closed just based on lack of motivation or wording reminiscent of assigned homework. I think that this especially hurts non english speakers. I remember when I first started learning to formulate mathematical questions in English they mostly sounded like "prove that __" because I was trying to mimic the wording of textbook problems. $\endgroup$
    – Gjergji Zaimi
    Commented Sep 5, 2020 at 19:09
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    $\begingroup$ "any clearly posed mathematical problem that I cannot solve in under 10 minutes should pass" I strongly agree with this. It really bothers me when I see MO users voting to close questions as not research level, and they clearly haven't thought through what it would take to answer them. (Yes, I can think of exceptions -- if someone posts a 100 digit integer and asks me to factor it -- but I think it is a very good standard for problems that a reasonable human would ask.) $\endgroup$
    – David E Speyer
    Commented Sep 6, 2020 at 12:03
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    $\begingroup$ Strong +1's to Gjergji's and David's comments. $\endgroup$
    – Sam Hopkins
    Commented Sep 7, 2020 at 1:14
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    $\begingroup$ "any clearly posed mathematical problem that I cannot solve in under 10 minutes should pass" I strongly agree too, and I would agree with the same statement with 10 replaced by 2 -- "I" being, as in the original answer, fedja. $\endgroup$
    – Joël
    Commented Sep 8, 2020 at 1:28
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    $\begingroup$ @GjergjiZaimi I think motivation might sometimes make an otherwise inappropriate question appropriate. If someone says they are working in field X and as a result now have a problem in field Y, the resulting problem might not be research-level for work in Y but still research relevant for their work in X and, therefore, appropriate here. $\endgroup$
    – Michael Greinecker
    Commented Sep 12, 2020 at 11:54
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Since no one has mentioned it, I would like to suggest that one thing that might make MO somewhat "less friendly" than 10 years ago is the vast number of questions that have already been asked and answered. In particular, graduate students or more advanced researchers who ask a question without doing some MO searches on key words and phrases may well ask an already-asked question, which then gets closed. So I encourage my graduate students to first use MO as a reference source, then after they've seen the sorts of questions that are asked and answered, they're likely to be able to ask their own good questions. And this applies to everyone, of course, I've often started to draft an MO question, done a search, and lo and behold, exactly what I want to know is already in the MO database. Which is great, of course.

To answer the specific question posed in this thread, I'd encourage graduate students at any level to participate after they've become comfortable with MO by reading a bunch of questions and answers. However, I also suggest that they should be somewhat careful, since for example, posting the primary problem for one's thesis and having it solved on MO is not a good way to get a PhD!

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    $\begingroup$ As to your last sentence -- I'd rather doubt that a problem which can conveniently be solved in an answer box on MO (i.e. on something like one page) would be a good "primary problem" for someone's PhD thesis ... . $\endgroup$
    – Stefan Kohl Mod
    Commented Sep 8, 2020 at 16:21
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    $\begingroup$ @StefanKohl That's a fair point. So maybe I should have said that posting one's primary thesis problem on MO is not a good idea, since someone might solve and publish the solution somewhere. As you say, probably not in an MO box, but possibly in an ArXiv preprint. It's the same advice as for giving talks about partially finished research, one has to be careful, since there are horror stories of someone in the audience who happens to know just the right tools publishing a paper and scooping the speaker. Obviously this should not be done, and isn't done by anyone who's honorable, but ... $\endgroup$
    – Joe Silverman
    Commented Sep 8, 2020 at 18:13
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    $\begingroup$ @StefanKohl There is a situation I've seen which is more nuanced than this. A professor knows that some problem should be solvable with a standard toolkit, but writing up the details will run 30-40 pages, and learning that toolkit would be a valuable experience for a grad student. So the prof assigns this problem to a student, who posts it on MO. Someone writes a one screen answer saying that the problem is solvable with the standard toolkit, and giving links to references. The student is now not much better off, and feels much worse because "my whole thesis was a one screen answer". $\endgroup$
    – David E Speyer
    Commented Sep 9, 2020 at 13:28
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I encourage my graduate students to ask questions on MO when they've asked me and I didn't know the answer quickly and it seemed like a good question for MO. My hope is that since I'm stumped they won't feel embarrassed about asking. Mostly they still don't ask, but this still feels like a good approach to me.

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    – Lennart Meier
    Commented Sep 9, 2020 at 7:32
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I'd say I'd encourage people having on-topic questions to participate, as the questions are mostly judged for their own sake, and not with respect to the status or identity of the OP (which is often not disclosed). For most of the people I interact with here, even part of those disclosing names, I have no idea if they're professors, PhD, people outside academy after PhD, amateurs, etc.

So I'd encourage in participating especially if there are questions that are research-level, and likely to be answered. It might even be by undergraduates.

Then the question boils down to "what are questions are research-level", but it's already been largely discussed, and having an idea of, practically, what questions are closed, what questions are well-received, etc, comes with regularly browsing the site.

To more directly address the question "would you still encourage graduate students to participate in MO", my answer is then: if I encourage anybody to post on MO, it's probably because that person has just asked me a good question.

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Yes. I think they should. Questions of grad students being recorded for everyone to see adds to communal knowledge for free. This is valuable in its own right. I encourage everyone I meet (at a certain level of mathematical maturity) to post here!!

I even (jokingly) call people cowards for refusing!!

Also, in my experience, MSE is much less friendly.

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  • $\begingroup$ Can you specify the level of mathematical maturity? Without that, this doesn't really answer the question of "at what stage" graduate students should participate. $\endgroup$
    – user44143
    Commented Sep 6, 2020 at 19:09
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    $\begingroup$ @MattF. Someone who can have a mathematical conversation at the board rather than a one-way lecture. Anyone who is reading papers beyond textbooks, also. $\endgroup$
    – Harry Gindi
    Commented Sep 6, 2020 at 19:33
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The question "should graduate students be encouraged to do X" can be asked about any professional activity, and the answer surely depends on the student's situation.

The entry barrier at MO is a certain mathematical maturity which in my experience (in the US) is common in graduate students in their last two years of study. Of course, some students get there much earlier, while others never get there (despite getting a PhD).

The main benefit of MO is that it lets you interact with experts from all over the world, and do so anonymously, if desired. Is there any other place that offers the opportunity?

In summary, students who could benefit from talking to experts should be encouraged to participate in MO. Not to the point of addiction, of course.

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    $\begingroup$ IMO participating in MO to the point of addiction as a graduate student is not that harmful, and has some advantages, as long as you can find enough questions that you are capable of answering. $\endgroup$
    – Will Sawin
    Commented Sep 14, 2020 at 22:24
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    $\begingroup$ @WillSawin: careewise, most students will be better off focusing on their research, and as long as MO participation furthers this goal, it is fine. $\endgroup$
    – Igor Belegradek
    Commented Sep 14, 2020 at 23:26

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