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Several mistakes in published work: Advice
I am a beginning scholar. I recently published my first article in my field in a very good journal. The peer review went incredibly smoothly with basically no need for revisions (one reviewer gave a particularly glowing review). I even recently got an email from a scholar praising the work.
However...after setting aside the paper for awhile I picked it back up and I have found a handful of errors that escaped my notice. I'm really dissappointed that I didn't catch these when editing. The article is quite long, well over 10,000 words with a plethora of citations. I noticed I made errors in page numbers in about 6 of them -- the pages are mostly off by one. For example 305 is 304 or 27 is 28 etc. in another case, I type something along the lines of "see pages 150-151" and it should be "150-153", etc.
I also noticed a (very) minor factual error and an error in a reference (though the work can still be located easily). Altogether something like 10 mistakes. Obviously none of them affect the arguments or conclusions of the work.
The journal does not do errata, so there is no correction. I've already been thinking about how to change my editing and revision process so this kind of thing does not happen again. I've learned from this and will do better in the future (particulaly in managing sources, triple-checking details, etc). But I am feeling an incredible amount of imposter syndrome and intense anxiety. I feel like the praise I've gotten is unwarranted and that this paper is a complete mess from all of these mistakes.
I guess what I would like to ask is: is this going to be something I have to cary with me for a long time or are most people not going to notice/care? I am currently in a position where I have no one I can ask for advice, so I thought I would try my luck here.
EDIT: I should be more clear about a detail. The incorrect page numbers are on citations in the text. For example "see (John Doe 2000: 324) for comment on X." When it should be 323 etc.
I found that I had typos of this sort a couple times in the article, and what has worried me was readers deciding to look up the page numbers for the corresponding discussion and finding that that the page numbers were wrong. Particularly if this happens more than once. That's what I meant by several mistakes and the kind of thing that was really bothering me, because I wasn't sure if that was going to influence opinions of the work.

Sorry I should have been more clear. One of the reviewers asked me to add a paragraph and a few footnotes to clarify something. The other asked to add a footnote to clarify something (and to add a detail in two places). But other than that really nothing else. Still, I am surprised at how smoothly it went.

Honestly when reading the title I thought your whole research was faulty. No one is gonna mind some page number mistakes.

How often do people actually find articles in hardware journals these days? Virtually never... I don't even include page numbers on most citations anymore. As long as you've got the Name, Year, Journal and DOI (if there is one) correct, you're fine.

@ScottishTapWater: I don't think the point of the page numbers as mentioned by the OP is to locate the article within the journal, but to specify the particular part of the article that one is referring to.

I don't know about your field (actually, you haven't mentioned it), but I am under the impression that you are really overthinking this.
For instance, many people would probably describe "confusing page numbers 27 and 28" as a typo rather than a genuine mistake.
In general, it is simply impossible to right [edit: I meant "write", of course - but no, this was not on purpose... ;-)] a lengthy text without making any mistakes. Checking everything multiple times can reduce the number of mistakes, but:

Most likely you won't get the number of mistakes down to 0, no matter how careful and often you proofread everything. (This is just how our human brains work: when you proofread something that you have written, you simply read what should be there, not what is really there. One way to deal with this problem is to put your article away for a week or two before you proofread it; this will help a bit, but it will not solve the problem completely.)
Personal anecdote: I'm working in pure maths, and I made it a policy for myself (during my PhD even more than now) to really very, very carefully check everything I publish, first before submission and then again before publication. At some points I was close to annoying my co-authors because I wanted to do one more double-check of a paper. But still, I'm sure that in every single of my papers you will find something which is not completely right. This will most likely not be a serious flaw in an important argument, but in some cases it might indeed still be a small mathematical inaccuracy, which is far higher on the "error scale" than the things you mentioned.

Beware of the law of diminishing returns! The less errors are left in your paper, the more effort you will need to find one more of them. When you're talking about things like page numbers off by one, you will somewhen reach the point where it's just a waste of time to search for further mistakes and typos.

You might even reach a point where, when you ge ahead with proofreading, you will probably introduce more errors than you fix - simply because at some points it becomes very difficult to focus on even more proofreading, and you are thus more likely to make mistakes.

The journal does not do errata, so there is no correction.

As I've said, I don't know your field - but in pure maths I have never seen anybody submit an erratum for any "mistake" similar as the ones you described. Please beware that publishing something (like, for instance, an erratum), adds information to a very full scientific communication channel - so make sure that what you publish adds more information than noise. If people have read your paper and see an erratum in a journal, then they will probably expect that there is some serious mistake in the paper. So they will feel obliged to check the erratum to make sure that they are not relying on false information - and if they then find that your "erratum" consists of of few corrected page numbers which were off by 1, they will most likely just be annoyed that you bothered them with the erratum.
(One thing you could consider is to upload a small file on your webpage which you name something like "minor_corrections.txt" or "minor_corrections_and_typos.pdf" and where you list those points you mentioned. But make sure that the name of the file indicates that there is nothing substantial in it, unless, of course, there is something really substantial in it.)
Concluding remark 1. Of course, nothing of what I wrote should be interpreted to the end that correctness weren't important or carelessness were alright.
But it's important to properly gauge your expectations of how much correctness you can expect (from yourself and others) and to note that drawing a reasonable line somewhere does not mean to be careless.
Concluding remark 2. You just published your first paper, so it is simply natural that you put very much weight right now on everything being as perfect as possible. Thus, small issues are likely to annoy you now, which probably won't bother you much when you will find them in your fourth or tenth paper.
I think this is completely normal for most of us - just make sure that you don't go crazy over it.

"it is simply impossible to right a lengthy text without making any mistakes" - an example to prove this!

@BryanKrause: Ouch, but "right" instead of "write" is a bad one... :-)

Also "annecdote", "no matter how careful and you often you proofread" :)

@BryanKrause: Of course it's important to illustrate everything by as many examples as possible... :-)  (Oh, but "annecdote" was a genuine mistake rather than a typo - I actually thought the spelling were correct until I read your comment.)

Yes, and I'm sure I could proofread your entire post and still miss some error someplace, further illustrating the point!

Thank you very much for your reply. I think you are right. Its been eating away at me probably because its my first paper. My field is in philosophy, but I am balancing texts in three languages from different traditions and the paper was written while I was on lockdown from COVID when I didn't have access to a library and other resources (I had to use electronic sources or find other creative ways to solve problems). So I think that added to me making more mistakes than I think I normally would. I have been thinking of making an errata page with minor corrections. Thank you!

Also, one difficulty with proofreading, especially your own, writing is that when you reread it, your brain reads what it expects to see, not what your eyes see.  That's why having other eyes helps, but they are not always easy to come by.

@JochenGlueck Except that "righting our wrongs" is a commendable thing to do, so "righting a text" isn't completely out of the question, either. :)

There is nothing to do and nothing to worry about. Readers can find things if the articles are correct.
Take more care in the future, but we would all be happy to make only minor errors.
Using a local proof reader is a good practice if you can arrange it.
If you have a personal website you can put errata there also.

You are right. Nothing can be done and there is no reason to dwell on these things. I've certainly figured out some methods to use to make sure I find these kinds of errors in the future. I also plan on putting the errata on my personal site. Thank you for your answer. Everyone here is right, take more care in the future and just keep moving on.

One thing you can do to help with getting references is use a tool like JabRef to manage your bibliography.
In particular, you want to make sure that you utilise DOI lookups (JabRef can do this for you with the clikc of a button) to populate the content of each entry. This way, the contents will be programmatically filled with the data coming directly from the publishers (via the DOI system).
You can then auto-generate your bibliography from the  .bibtex database and you never have to worry about this again.
Unfortunately, not everything has a DOI (particularly older papers) but the vast majority of modern papers will.

I'm bringing this back from the dead...
I ended up taking a bunch of time to make corrections to my article and put the corrections on my personal website (found far more errors than I anticipated).
But!
I just got very positive reviews on a new manuscript and I feel like I will be able to do a much better job with the revisions and proofs. I am now much more conscious of how to proof a paper and keep all of my notes, sources, and other things in order (also actually have access to a library now). So I won't be making the same mistakes again.
I think I've really learned from the previous experience. The previously published paper was my first project in the field. I think the shoddy mistakes (those page errors, citation errors, reference problems, etc.) really reflect that fact. I've now got methods to triple-check my notes and everything else.
Just wanted to send my thanks to people who offered support and advice in the thread. It really did help me when I was feeling down and helped me put things into perspective. If another beginning scholar runs into the same issues, hope that reading this thread helps you out as well.

Just check the page references more carefully next time. That will not be difficult, and it will deal with most of the errors mentioned that would cause any problems for readers.

Depending on the app or platform used for writing, there should be a way to have page references generated automatically and thus updated whenever revisions are made (LaTeX has this feature, for example).

@GregMartin: This is certainly the right thing to do for cross-references within an article, but I think that the wrong pages numbers mentioned by the OP (or at least some of them) rather referred to articles or books they cited.

It might be a reasonable solution to publish a pdf with the errors corrected on the internet with a comment (if that doesn't violate the publisher's copyright).  In physics, you can use arxiv which allows you to add a comment regarding the errors being corrected.
If you happen to have a personal page on the web (which is seemingly a common thing nowadays, given the existence of GitHub pages and things alike) it's likely a good idea to publish a pdf there as well.  Google Scholar, for instance, crawls personal pages of researchers searching for the pdfs of articles.


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Is it appropriate to use acronyms when citing a source in MLA?
I'm writing an MLA research paper for a school assignment. When citing a source in-text, is it appropriate to use an acronym? For example, I'm citing the Centers for Disease Control and Prevention. Would it be acceptable to cite the source as (CDC)? If I'm citing the source at multiple points in the paper, could I cite it with the full name the first time and with the acronym for any subsequent citations?

From the 7th edition of the MLA handbook.

Abbreviations are used regularly in the list of works cited and in
  tables but rarely in the text of a research paper (except within
  parentheses). In choosing abbreviations, keep your audience in mind.
  While economy of space is important, clarity is more so. Spell out a
  term if the abbreviation may puzzle your readers.... The trend in
  abbreviation is to use neither periods after letters nor spaces between
  letters, especially for abbreviations made up of all capital letters.

The following points are from a MLA style guide on the doc scribe site.

If an acronym is commonly used as a word, it does not require explanation (IQ, LSD, FBI, ESP).
A term must be fully written the first time it is used, thereafter just the acronym is used.
If an acronym is not familiar use an expanded abbreviation; for MLA write Mod. Lang. Assn

Also from the same source (Page 8)

The common practice in research writing to spell out the name or
  phrase to be abbreviated followed by the acronym in parentheses.
  Thereafter, just the acronym is used. For example, the Modern Language
  Association (MLA) publishes a journal. The reverse is also works—the
  MLA (Modern Language Association) publishes a journal.

So in your case Centers for Disease Control (CDC) or CDC (Centers for Disease Control) for your first citation and CDC afterwards should be fine.

First time in full with the abbreviation in parentheses, then use the abbreviation to your hearts content. (In longer works it may be advisable to reset for each chapter, but I doubt that's the case here).
Spell out in full for the works cited unless abbreviated original in the title by the source.

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Is there a way to register now for the GREs, or attend grad. school some other way?
I have my Bachelor's and want to go to grad. school—and I want to go as soon as possible. I recently quit my industry job (I hated it) and am now searching for a job at a university. I can't register for the GREs now, it's after the deadline, but there are some other options I've thought of:

Take classes in a Master's program, take the GREs in April and try to get into a Ph.D program
Apply to universities that don't require the GREs (some in the US, but most in other countries)
Find a way to register for the GREs

My degrees are in, and my target field of study is either Physics or Mathematics. Do you folks have any suggestions, or judgments regarding the viability of my above presented options?
(P.S. - This question is not off topic. Any single person who did not register for the GREs, but still wants to begin grad. school could use this question. Nothing depends on "personal preferences" because my personal preferences [outside of wanting to begin grad. school ASAP] are in no way relevant. The answer accepted is appropriate, and easily generalized.)

For any particular university, the question of whether you can get admitted to graduate study on short notice depends on the particular rules at that university.  Such a question is off-topic.  There might be a university somewhere that might be willing to accept you into a graduate program without GRE scores, but shopping for one is off topic.

For a lot of European countries (but not all, probably!) you're 11 months out of phase. However, they don't require the GRE. That said, you'd need a Masters first for most of them.

@BrianBorchers I've clarified my post a little. While attending grad. school within the next couple of months would be great, I was not expecting this. I was expecting to apply in Nov./Dec. and attend next year. This would be fine and dandy if it were not for the fact that I can't register for GREs until April, which would effectively postpone my attendance until two years from now. It's that two-year wait I'd like to avoid.

Have you taken the general test? That is more commonly required than the subject tests. You should be able to find schools that will consider you based on the general test and your degree transcripts, without a subject test.

@PatriciaShanahan I have not, but I certainly will; I'm able to register for these! Thank you.

Am I so old that they don't do standby testings anymore? You just show up to the testing center, and they generally have a few no-shows and they let you take the test in their place.

@BMD how many people normally take these things? I took my GRE a year ago and there were literally two of us there for GREs :(

@BMD They still do standby, if it's a paper-based test: http://www.ets.org/gre/revised_general/register/paper/arrangements?WT.ac=gre_26150_register_pbt_accommodations_140529

I recommend you go for a Masters degree first. Some schools admit Masters students without GRE. While doing your masters, you can prepare for GRE and take courses that would eventually count for your prospective PhD degree which you are intending for. This will benefit you getting into a good PhD school which otherwise may be very difficult to get in.

I basically see three options that overlap quite a bit with the answers you've already suggested:

Apply for programs that don't require GRE scores. Most US-based programs require the GRE but not all do. These might not be the programs you want. These might be MS/MA programs instead of PhDs. It's going to limit your options but these programs do exist.
Apply for programs and simply do not provide GRE scores. You can certainly submit an application without all the required material. If you do so, you should discuss this in your coverletter or similar so folks know why the scores are missing. For sure, many programs will not review your application but some might — especially if the rest of the packet is extremely strong. Because each application can be expensive, it is worth contacting programs in advance to explain your situation and to find out if they will review an incomplete application for someone in your situation.
Travel to take the GRE. GRE testing works different in different locations. Within the US where the test is computer-based, you can register to take the test basically at any point given availability in the testing center. I just checked and it was possible for my to register to take the test in my city as soon as two days from now. Obviously, travel (perhaps even intercontinental travel) is not always going to be possible due to expense, visas, etc. That said, for some people, it might be an option.

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Community College or extension programs as a way of preparing for a Masters in Mathematics
I wanted to get some directions on how to prepare for a MS Degree in Mathematics.
Background:

I'm interested in getting a Ph.D in Statistical Learning or related area in 5-6 years.
I took some courses in Mathematical Statistics and I struggled because I do not have recent coursework in Analysis, Measure theory, etc.
I studied electrical engineering with a very heavy mathematical component from a very decent University 20 years ago however, it is amazing how much I've forgotten.
I've always been fascinated by mathematics and I'm very tempted to build a solid foundation before partaking in doctoral study.
I am working at the moment - my job is flexible and I'm saving to take off a year or two for the final years of my doctoral work.
I have a couple of graduate degrees in the area of Computer Science and Artificial Intelligence

Plan:

I'd like to build up to where I was 20 years ago: calculus, linear algebra, diff equations, calculus of complex variables, frequency domain analysis.
I'd also like to take courses that are typically reserved for math majors like proofs, analysis, group theory, algebra, etc.
I think the best way to accomplish the plan would be a decent community college or extension program like (UC Berkeley extension) that offers online classes -- any recommendation?

I fear that not so many community colleges would offer the upper-division courses a math-major sort of person would want, especially to aim toward graduate school in mathematics. Further, you'd be needing letters of recommendation for grad school, and community colleges would not generate letters that would help you, since the letter writers (by far most often) would not be familiar with grad school from the side of mentoring and supervising grad students (even if they themselves did have a Ph.D.).
It is true that community colleges are usually much cheaper than "universities", but the coursework, context for coursework, and outlook of faculty teaching the upper-division courses you need, and their letters on your behalf, are things that you can't avoid but need.

I think the community college route could work as a Step 1. (Our local community college offers courses in calculus, linear algebra, and differential equations). There's no reason the O.P. couldn't start his journey with some refereshers there.

What about extension programs like "UC berkeley extension"? The prices seem to be pretty reasonable? Know of other good extension programs.

Indeed, as @J.R. says, community colleges could be money-savers on refreshers about lower-division and maybe middle-divion courses. Also, some universities do offer "extension" courses that have essentially the same content, but just not "in day school". (Indeed U of Minnesota formerly offered "evening/extension" courses that were cheaper, very small classes, etc., at a historical point that "night school" was declasse' in comparison to "day school", but, a few years ago, "night school" was "inloaded", and now there's no substantial "extension" alternative.)

The Berkeley extension program doesn't seem to offer any upper division courses either. If you are entering a math masters program you'll really want the background you typically get in upper-divison courses: abstract algebra, real and complex analysis, topology, etc. Community College and extension courses may be helpful if your calculus and linear algebra are rusty, but not much beyond that.

Unless you are using the term "community college" different than I am use to for in Canada and the US, then no.
If you mean a local state or government funded degree (4-year) granting universities then perhaps. Another possibility, if they exist anymore are one or two-year junior college which acts like a feeder, or extension campus to a 4-year university.
Not the 9-months to 3 year diploma granting colleges which tend be vocational oriented. (Similar to a bit lower academic standard than Polytechnical post-secondary schools in Europe)

You should be able to find introductory classes (years 1 and 2) via distance education through out the world easily.
In the interests of cost and legitimacy, I recommend avoiding privately owned/run distance education programs including online universities. This does not mean government owned or run distance education like Open University or Athabasca University, those are great and affordable.
At an university you should be able to register as a "non-degree program" (or similar) student, either part-time or full-time. This is commonly used for preparing towards a degree program in the future for not-fresh-from-high-school student enrollment cases.

I have a couple of graduate degrees in the area of Computer Science and Artificial Intelligence

(emphasis added)
This makes no sense when you previously said "I'm interested in getting a Ph.D in Statistical Learning." If you have a M.Sc. or Ph.D. in Computer Science (I don't know off-hand of any place that grants degrees in AI).
Do you mean you previously have taken graduate level courses in CS and AI?
If so, I would expect you to start your search from that university unless geographic reasons prevent it, at least speaking to them as a starting point for recommendations.


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Attend conference without a published paper
What are the advantages for a new grad student to attend a conference without having a published paper?

Actually, there are several things. One is you get exposed to a lot of ideas and current research by attending the talks. Don't be afraid to be visible. Ask a question or two if the opportunity arises. Smaller groups/talks/workshops are more useful for meeting people than plenary sessions.
Another is that you can meet a lot of people in your field, some of whom might give you more idea or be interested in your ideas. If the conference has lunch venues, be sure to spend some time there. Introduce yourself.
If there is a poster session, you can meet with some folks, many of whom are also grad students, and learn about their research ideas.
While you are there, take a lot of notes. Collect a lot of business cards if you can. Pass out a lot of business cards with contact information if you can.
Ideas. Networking.

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text books - studying
Is there some sort of source/website that states all the required textbooks for a particular subject? For instance, I'm intrigued too learn about certain subjects however I am no graduate or anything, so attempting to enrol would be pointless, but I do wish to study in my own time and with the correct textbooks. Does anyone have any pointers? If they could offer any advice, I'd be very grateful. Thanks!

What "the correct textbooks" are depends a lot on whom you ask to. There is a bazillion textbooks around for every undergraduate subject.

While the question can be better worded, I don't see why it should be closed. The question asks for general directions in finding appropriate textbooks, which is certainly on topic and not opinion based (i.e., it is not asking which textbook is better).

Thanks, mate. I'm just attempting to begin a navigation through a maelstrom, hoping I could collect some information before I charted my course. Some people can't grasp my 'simple folk' mindset.

Many Professors publish their syllabi online. You could try browsing courses on the subject you’re interested in, picking the course you like the most and look up the textbooks the Professor requires. That way you can have your suitable textbooks without any need to enroll, and you’re free to study in your own time.

How would one find these syllabi? Google the subject and navigate through which colleges offer them?

That would be a perfect way to do it. Say you’re interested in thermodynamics, you could google “thermodynamics syllabus”, or find a physics degree and look through its thermodynamics course, or find a college and look directly on their site. You can do whatever you’re most familiar with.

Many major universities offer open access to all, most or some of their courses content. Some of the prominent examples are MIT OpenCourseWare, Harvard Online Learning and Open Yale Courses. They all have courses in a variety of fields, and will list at least the reading requirements for the course, including the textbook and sometimes the specific chapters and sections. Many courses also offer specific lecture notes, video or audio lectures, assignments and exams.


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How to avoid the "need" to learn more before conducting research?
I am starting a PhD this Fall, and I'm absolutely terrified that I'm nowhere close enough to the "frontier of knowledge" to enable me to do quality research.
For background purposes, I have a BSc in Chemical Engineering and a BSc in Pure and Applied Mathematics. I finished both of these simultaneously in a 4 year span, as my philosophy was always that I need to be as efficient in my learning as possible and race to that mythical place where the edge of knowledge resides. The issue, however, is that it's now all too clear to me that I'll never be quite where I initially envisioned I would be before starting my PhD.
It is also clear that I have a lot to learn before being able to contribute significantly to the project I'm being placed on, and I worry that I'll spend too much time taking classes and learning what's already known, and not enough time exploring what is not known. This is all made worse by the fact that my PhD will be mostly theoretical in nature, so I feel that the lag time between when I am done "training" and when I can start publishing papers is significant. I did a lot of experimental research in my undergrad, and it was clear to me there that this problem does not exist in as dire a form as the path I chose to go down (and that I am actually excited to go down).
To be clear I am not expecting to make monumental revelations in my PhD, and I don't think anyone is really expecting me to do that either. I am just wondering how someone that carries out theoretically/computationally focused research decides when "enough is enough" and stops taking classes. In sorting out my class schedule for this Fall, I am worried that my addiction from undergrad of taking as many courses as possible will interfere with my research, and at the same time I am not confident that I know enough to conduct research of any sort at the moment.
So if your PhD was largely theoretical and computational in nature, I guess I'm wondering how you were able to balance learning and researching? Anything you wish you did differently? Is it normal to spend a couple years accruing knowledge and not publishing much? Do you have any books/resources you recommend that deal with this sort of dilemma?

Is the question about classes per se, or it is about the desirability of further study and enhancing scholarship?

@paulgarrett Great question. I don't think I'm asking about desirability so much as necessity. Really I'm looking for general guiding principles on how to best approach a PhD that's largely theoretical. I worry about this because I can envision a trap where I spend all my time "enhancing scholarship" - whether it's through classes, reading papers, or some other method - and not enough time contributing to new publications in my lab group.

Nobody expects you to be at the frontier of knowledge when you start your PhD :-)

@Flyto Thank you for your reassurance! My concern, however, is more so about what's a reasonable amount of time to spend on reaching that frontier? But perhaps that's a terrible question and something I'll need to figure out for myself.

Note that the structure of a PhD can be quite different depending on the country, so you might want to add a location tag to your question if you want more specific advice. Anyway I tried to give you the general idea, hope it helps.

Make use of your supervisor.  Ask him/her for directions to the frontiers of knowledge.  I for example give my new PhD students new problems or research questions.  They are then tasked with learning all that's required to be able to understand what is it I'm asking them to do.  This allows them to be more directed in their learning process.

I had this problem as well and found myself in lecture hall hell where I seemed to be eternally attending classes, basically you have to go through a psychological switch where you stop worrying that you 'don't know enough' and try to learn stuff on the hoof as you go along trying to solve problems, ultimately this approach makes it more likely that you will get publications as you will be working on problems and might find something new.

That similar question on math.SE had been highly upvoted and hints that you are not the only facing that problem: https://math.stackexchange.com/questions/617625/on-familiarity-or-how-to-avoid-going-down-the-math-rabbit-hole

I sympathise completely with your situation, as I went through exactly that thought process when I began my PhD. I am in my second year.
I did not begin my PhD with a tightly defined research question, and I would guess that you are not doing so either. I did have an idea of what general area of research interested me and spent a lot of time looking at  published papers. Sometimes, while reading a paper, I realised that I needed more general understanding of that topic, and then read text books, or attended classes.
I also attended seminars and such like, to get some idea about how other researchers in my general field go about their work. And I volunteered to give talks on quite general subjects in my broad field, on the basis the best way to learn a subject is to try and teach it to somebody else. I have been surprised by how much of that haphazardly gathered learning has turned out to be helpful in my own very specific research - sometimes directly relevant but more often by suggesting ideas to me about a way of looking at a problem.
I thought, and still think, that it would be a mistake to take classes and read textbooks until I was at the frontier of knowledge, because I would never reach the frontier that way.
What I did instead, after reading, more or less deeply,  a few hundred papers, was to realise that none of them really addressed a particular question that I was interested in. Having, almost inadvertently, identified a specific research question, I could then conduct a very intense literature review to confirm, or otherwise, that nobody else had researched that question.
Guess what! I had reached the frontier of knowledge. I am now in the process of seeing if I can push the frontier further out.

This is a wonderful story, thank you for sharing Jeremy! I definitely don't have a tightly defined problem that I'm working on, but I do have a pretty specific domain I know I'll be working in so I suppose my situation could be worse. "A few hundred papers" is mega impressive for being in your second year. I think your method of reading papers and attending seminars addresses my concerns. There are many frontiers of knowledge, each requiring more/less knowledge to access, and I'll probably be working hard with my adviser/reading papers to find a suitable one to push out.

Unsurprisingly, there is no one-size-fits-all method for balancing the tasks of acquiring knowledge and producing knowledge. It is a very common concern for starting PhD students like you, but in the vast majority of cases things work out fine. I'll try to explain why and how:
Acquiring and producing knowledge are not mutually incompatible tasks. In fact, they are often fruitfully combined with each other. For example, one may read a paper which gives them an idea for a new approach in their work;  conversely, one may stumble upon a problem in their research and switch to discovering how existing works deal with the same kind of problem. The balance is achieved by keeping a flexible goal-oriented approach, by this I mean that it combines two aspects:

Nobody can know everything in their field, so researchers don't try to achieve that. Instead they focus on the topics which are susceptible of contributing to their own research. That's the in-depth aspect of acquiring new knowledge: choosing what one needs to know in order to progress in their own research goals and study every detail of it.
On the other hand, researchers need to keep up to date with the major discoveries in their field. But for that they don't need to know or even understand every detail of every single new paper, they can skim through abstracts/papers and select whether they want to dive deeper depending on their goals (or depending on how soon their next deadline is). That's the in-breadth aspect.

Naturally figuring out one's methodology, for example recognizing the papers of interest,  takes some learning. Well, that's exactly what a PhD is for: a PhD student is a researcher in training. It's understood that they are not immediately operational and that they need to acquire a lot of new knowledge, both in their specialized field and in general scientific methodology.
It's very common to spend at least the first year (often more!) of PhD learning and exploring around the research question, sometimes a bit randomly. During this time it's frequent not to actually produce anything, and even occasionally spending time exploring something which turns out to be irrelevant. It's fine, mistakes are entirely part of the research process.
What is important is to keep a main research objective in mind: at the beginning it's often a very general and vague objective. Then progressively, by getting a better understanding of the state of the art and discussing with your advisor, you will start zeroing on what will become your specific research contribution.

That PhD students are still viewed as researchers in training at the beginning is reassuring. Looks like I'll need to work on marrying the ideas of Acquiring and Producing of knowledge together as you've suggested. Thank you for the very helpful advice Erwan! In response to your comment above, I'll be doing grad school in the US (in CA) and what I've experienced thus far of the program meshes will with what you've laid out. So much appreciated again :)

I'd rather like to give you some practical tips. In my view a big part of a PhD is learning how to learn and to deal with the huge information overload that might exist in your field of research, especially nowadays. Reading a 100 papers is warning signal in my opinion. Most PhD thesis only contain 100-200 references. Overflying a 100 is ok. This points me to the suspicion you are analyzing and reading the literatur not efficiently and this is of course a matter of time.
So, if I would start doing a PhD again, I would like someone to tell me how to select and what to read. My suggestions now after PhD are:

Start with a review article published in the past 10 years instead of reading the most cited current papers, which will not help a lot, as those are very specialized research reports, often in letter format.

If you don't understand most of the content, read a text book covering the topic, if you still don't understand you have to switch to university courses, lectures, seminars

after reaching some understanding look for open questions -> search google scholar with

"intitle:opportunities/advances/roadmap" AND "your field"

as these articles look into future, list open questions or methodological gaps. Simultaneously you should visit conferences, workshops and summer schools if you can and talk especially with PhD students which are at the end of their odyssey and discuss open questions.

then you have to decide what you can and would like to do, where sits the expertise of your advisor, your colleagues and you and what facilities do you have (intellectual, experimental, computational ... you have at all). This will narrow down a lot what you actually can do and running into too many dead ends, which you have to risk in fundamental research.

The rest is the PhD. And when you are able to come up with new correct and interesting questions to your peers in your field your supervisor could not think of, this is a very good sign you have become an independent researcher, which is the main goal of a PhD.

This is all golden stuff, thank you very much for your insights! In a way, I guess my question was essentially on how to deal with "information overload", and everyone here seems to be saying that managing that is actually a skill that needs to be learnt and practiced (thinking out loud here to make sure I'm not interpreting this incorrectly). I've also heard others mention that reading papers is something of a skill, so I'll definitely make that a key focus. Your Google Scholar suggestion is amazing, I can't believe I haven't seen that before! Thank you again for all your help.

I'm wondering how you were able to balance learning and researching?

You seem to be under the impression that one day a switch will flip and you'll know everything there is to know about your field, thus allowing you to start pushing that boundary of knowledge.
The good thing is: there is no such switch! You will not stop learning at all during your PhD, and there will never come a point where you know everything. The key thing is to recognise when you know enough, and your supervisor (if they're good at their job) will help you in this regard.

Is it normal to spend a couple years accruing knowledge and not publishing much?

Yes. There's an old saying: a PhD is three years of training in how to do a PhD in six months. It sounds like you're in the USA, so that timescale is likely going to be a bit longer, but no one is expecting a new PhD student to be churning out papers. That's not a healthy way to do science. You need to get the lay of the land first; not know and understand everything there is to know, but to be able to identify the key open questions in your field and the ways in which they can be addressed.

My PhD will be mostly theoretical in nature, so I feel that the lag time between when I am done "training" and when I can start publishing papers is significant.

From my experience, theoreticians publish papers far more quickly than experimentalists. In my department, it's common for the theory people to finish with 3--4 papers; observers (I'm a cosmologist) with only one or even none. However, try not to fall prey to the "publish or perish" trap. Yes, it's important to publish your work, but don't rush it. Make sure you do good science and write it up really well.
for example, I started my PhD  having already done a Master's degree, i.e. I had some experience of research and the literature in my field. I still spent at least three months just reading papers when I started, and only began to feel like I really knew what I was doing about 18 months in, when my first paper was published. This is normal. Just be patient and be open with your supervisor about your worries from the start.

I laughed pretty hard at "a PhD is three years of training on how to do a PhD in six months". I also definitely have zero expectations of ever knowing everything, and and actually look forward to always learning! I just wasn't convinced that one can arrive at a point where they know enough to do such theoretical research in just a few years, but the vibe I'm getting from this thread is that it's very achievable once you zero in on a specific problem. That theoreticians publish more quickly and frequently is an interesting observation. Thanks for all your helpful advice my guy :)

all_licenses: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
Asking a past professor to teach a workshop/short course
I would like to ask some of my former professors to come to my country and teach a small workshop, with undergraduate students, for a couple of days. I am doing this because I would like to raise the interest in my career, which is computer science, inside the faculty that I am currently working in. So the question that I have is: what would be the recommended things that I should offer to them, so that they are more likely to agree to pay a visit to my faculty? I ask this because I believe that I must submit like a report of expenses to my faculty dean or coordinator, so he or she could see if its feasible or not.
What expenses should my faculty cover? I was thinking of:

Plane tickets
Accommodation and meals
Material for the workshop

Any suggestions will be greatly appreciated.

What do you mean exactly by "workshop"? How many hours of lectures should the professor hold? What would be the audience? I'm asking because @aeismail and I seem to have understood different things...

I mean that the professor come and teach, in a basic way, some topic in which he/she is an expert. Only two hours per day and aimed to undergraduate students of my faculty.

@Layla: I think the term you're looking for then is a "short course," not a "workshop."

In general, the expenses you've listed above (travel costs, local accommodations and meals) are typically sufficient. Registration fees for the workshop would also usually be waived.
One other thing you might consider offering, if funds are available, is a small stipend to be used by the faculty member to sponsor a "young researcher"—a graduate student or postdoc—who can also attend the workshop and take part, perhaps by offering a poster or contributed talk. (Or, if it is not possible to offer a stipend, perhaps the registration fees could be waived for the young researcher instead.)


text:
Are there instances where citing Wikipedia is allowed?
For example, suppose you're writing an article about triangles, and you want to include a short proof of Pythagoras' theorem, but you can't quite remember how to derive it, so you pop over to: https://en.wikipedia.org/wiki/Pythagorean_theorem#Algebraic_proofs and check the proof there. It works mathematically, makes sense and is as far as you can tell, correct, so you include it. Now the proof on Wikipedia doesn't have a citation because probably someone has been taught it and copied it over to Wikipedia, but it's still correct. The only thing you could cite for the proof would be Wikipedia, would a citation be ok or should it just be copied down without a citation?
Edit: Ok, there have been many responses. Most with a firm "no", but some giving the reason: "Just get the proof from a book" since it's so common, but suppose then it wasn't a common proof of something. Maybe it's not very well known or something but suppose there is a small Wikipedia page with the correct mathematical proof. How much effort are people going to go to to find a proof in a book when there's a (correct (which we know, becuase it's maths and we can check it)) proof on Wikipedia? Why is it so bad to say: "I found this proof on Wikipedia, and the maths checks out so it doesn't matter who wrote it, it is correct, but that's where I found it." Or if you do find it somewhere else why not say: "Proof taken from Wikipedia and verified by the proof shown in "Triangles and their properties, Nature, 2014, p113 etc..."

What is the context? Is this a class paper or an academic paper?

I cited wikipedia once, and my advisor told me that the first thing referees look at is your references, and if they see wikipedia cited, they will reject the paper without reading it.

I think you've created a catch-22.  If the proof is so obvious that a wikipedia entry is reasonable, then you don't need to cite it - anyone with basic education will accept Pythagorean theorem without a reference.  If the information, however, is rare or cannot be found elsewhere other than wikipedia, then wikipedia itself cannot possibly be accepted as a valid citation. There is no reason to cite wikipedia for the trivial stuff, and your audience will not take a complex idea at face value with a wikipedia citation.

Whenever possible, cite primary sources -- or at least closer to primary. A good Wikipedia article will cite its sources, so you could follow those pointers and cite those. (It might be polite to add a footnote saying that you found Wikipedia a surprisingly useful tool in tracking down those sources, but I wouldn't cite it per se... not least because citing something that's subject to change and may show something different next week is bad practice independent of how authoritative it is or isn't.)

If it's not a well known proof, and wikipedia does not cite a source, then the proof does not belong on wikipedia and you should be sceptical of using it.

I can think of one very good example.
"It is commonly understood that …."
E.g. you are showing what anyone any easily find out about a topic.

I was recently tickled to find this citation of this article in Phys. Rev. A. 89, 032128 (2014), arXiv:1310.2687. Grumpy referee asks for a reference, sarcastic author cites Wikipedia?

It's a bit silly, but I know that people sometimes get information from Wikipedia (or from colleagues, lecture notes, etc.) which is perfectly correct, but they can't properly cite it, so they cite a book or review (that they didn't necessarily read) pro-forma. It goes against everything proper attribution is about, but its just one of those silly things you have to do in academic writing, because that's how it is...

I recently cited Wikipedia specifically to point out that a definition published there is incorrect. (I couldn't correct it on Wikipedia, because that would be original research.)

If you use text from Wikipedia you have to cite it as a source, otherwise it's clearly plagiarism and a tool like Copybot will detect it easily.

Are there instances where citing Wikipedia is allowed?

I'm not going to try to answer the question of "is allowed" or "isn't allowed" but instead I'm going to try and answer if citing Wikipedia is, in general, a good idea or not for research papers.
Here's five points I can think of for and five points against.
Good idea:

If Wikipedia is your source, and you learn something from what Wikipedia's editors have provided, then citing Wikipedia is the Nice Thing To Do™.
Wikipedia is a convenient, well-known, easily and freely accessible source (vs. final-copy published papers that are pay-per-view if not out of your wallet, then indirectly out of your university's).
Content aims to be written in an accessible manner targeted at a wide audience (vs. arguably the bulk of technical articles published in journals and conferences). Plus there's clickable links for jargon! Sweet!
Notable or Featured Articles will probably have undergone more scrutiny from experts (w.r.t. bias, technical correctness, sources cited, etc.) than your average peer-reviewed article on a similar topic.
At least with Wikipedia, readers should know where they stand with such a citation. Compare this with the artifice of citing papers for claims made in poor-quality journals or conferences that can be considered authoritative solely on the basis of being "peer-reviewed".

Bad idea:

A Wikipedia article is subject to frequent editing and the content you cite it for may not be there when the reader goes looking for it (adding a date-accessed only partly addresses the issue since it will be difficult for the reader to start going through versions edit but as per the comments below, you can use a citing service within Wikipedia to cite a specific version of a page).
Relatedly, the content may have been correct at the time of citing but errors may be edited into the content at a later point, making you look bad by association (edit, again, citing a specific version could work around that)
There is no minimal guarantee of the quality for most Wikipedia articles (which peer-review, at least in theory, should provide, depending perhaps on the venue).
Wikipedia is not a first-party source: in theory, no new knowledge is "created" in Wikipedia. Where feasible, you should try to attribute knowledge to those who create it (and optionally thereafter add a reference to those who report it or better describe it in a book or survey)
Attribution for the content of a Wikipedia article is difficult: multiple editors may be involved in a certain piece of content, many editors are anonymous, etc. This means that there is no culpability for information (unlike peer-reviewed papers where authors have something to lose by publishing crap). This opens up further possibilities, such as authors anonymously adding content to Wikipedia that they can cite to support the claims in their paper.

Okay, back to academic convention and "isn't alloweds" and "is alloweds" ...

Maybe it's not very well known or something but suppose there is a small Wikipedia page with the correct mathematical proof. How much effort are people going to go to to find a proof in a book when there's a (correct (which we know, because it's maths and we can check it)) proof on Wikipedia?

According to Wikipedia policies, it should have a reference to an original source. Cite that.
If it doesn't have a reference, or if you copy how the proof is expressed directly, I would recommend to add a footnote (rather than a formal citation) stating the Wikipedia article you found the proof on and how you verified it (and the CC-BY-SA licence if you copy the expression).
Generally avoid citing web-pages (unless they are normative online standards or something) if only to avoid getting your hand bitten off by a reviewer. Using footnotes is much safer in this regards and serves a similar-ish purpose (though it won't be counted as a formal citation).
In research, if enough reviewers believe something isn't allowed, then de facto it isn't allowed.

Or if you do find it somewhere else why not say: "Proof taken from Wikipedia and verified by the proof shown in "Triangles and their properties, Nature, 2014, p113 etc..."

Why not say "Proof sketched for me by Fred, the guy who sits next to me, who had previously read it in 'Triangles and their properties, Nature, 2014, p113 etc...'"?
In other words, what's important is the reference, not how you found.

I agree and I think that's covered in the answer under "subject to frequent edits" and "lack of attribution". (I deliberately avoided the use of the word "vandalism" since I believe that this problem tends to be over-exaggerated based on a few famous anecdotes: in the general case, Wikipedia has plenty of tools to suppress/revert vandalism and I cannot remember the last time I inadvertently encountered a vandalised article on my travels through Wikipedia.)

I remember one of the first times I used Wikipedia. I wanted to look up the scientific (taxonomic) name of the House Cat. The article stated that science had found that cats were actually reptiles -- cold blooded, scaly tailed, etc. By the time I researched how to edit this misinformation, someone had reverted the article. The vandal then struck again, and I recall that there were literally dozens of updates/rollbacks logged until the vandal was either blocked or got tired of the game. So, even scientific material can be vandalized.

"adding a date-accessed only partly addresses the issue since it will be difficult for the reader to start going through versions" Isn't this addressed by permalinks/revision IDs that can be looked up without needing to search the history manually?

@hexafraction, if such a feature exists, that could help. (I wasn't aware of such a feature before.)

Personally I haven't seen any vandalism.

@badroit Yes, it exists.

@hexafraction Yes, if you go to Special:Cite or click "Cite this page" on any article it will give a link to a specific revision, along with example citations, and a lot of the warnings found in these answers too :)

Okay, thanks for the pointer! I edited the answer accordingly to mention that service. (On a side note, I'm curious about the downvote I got if someone cares to leave feedback.)

Other online sources may be sensible as background citations - for example a news article to indicate the importance of a topic, or if the first discovery was published online ahead of any academic work (which would also be cited) as happens sometimes in CS/security.  IMO these are both more valid as citations than as footnotes, especially the latter

@PhilPerry "Wikipedia articles are subject to constant vandalism" No it isn't

@Grashlok, Yes they are. Only constant vigilance by others keeps it at all useful.

@PhilPerry Is that any different from any academic paper or encyclopedia where only worthy topics/information are let through?

In an academic paper or an encyclopedia, hopefully any vandalism or false material is caught before it goes out to public view as a reputable source. With Wikipedia, it gets published, and false material must then be retracted. Just look at the edit logs on a wide selection of Wikipedia articles, particularly on (but not limited to) controversial topics.

@PhilPerry So, really, your observations are purely anecdotal. Do you have reputable sources that have done the work to prove that Wikipedia is constantly vandalized and that its information is lesser to that of other sources of information?

You shouldn't cite the original source in wikipedia without giving credit to wikipedia, in some way.

@TheDoctor Assuming you have tracked down the original source and are actually citing from that (not citing directly from Wikipedia and essentially stealing their citation) there is no need to mention Wikipedia anywhere. Just like you don't cite the librarian who helps you find a book when citing the book, or the class where you first learned about an author when citing that author's work.

You should never use something that isn't your own without a citation. "Copied down without a citation" is not an option.
However, you should also not cite Wikipedia as an authoritative source; you should track down a source that is authoritative and use that.
Why not use Wikipedia as a source?
One reason for not using Wikipedia as an authoritative source is that it is simply not reliable enough in many cases. Wikipedia's founder, Jimmy Wales, himself says

that he gets about 10 e-mail messages a week from students who complain that Wikipedia has gotten them into academic hot water. "They say, 'Please help me. I got an F on my paper because I cited Wikipedia'" and the information turned out to be wrong, he says. But he said he has no sympathy for their plight, noting that he thinks to himself: "For God sake, you’re in college; don’t cite the encyclopedia."

Even if you can verify that the Wikipedia article is correct, you should not use it as an authoritative source. You asked:

Why is it so bad to say: "I found this proof on Wikipedia, and the maths checks out so it doesn't matter who wrote it, it is correct, but that's where I found it."

Citing Wikipedia as a source for an idea that isn't original to Wikipedia does not give credit to the original author of the idea, which is a major reason for citing something in the first place. (The same applies to any tertiary source.)
The Wikipedia entry on Citing Wikipedia itself advises:

It will usually be more acceptable to cite those original sources rather than Wikipedia since it is, by nature, a secondary or tertiary source. At the same time, simple academic ethics require that you should actually read the work that you cite: if you do not actually have your hands on a book, you should not misleadingly cite it as your source.

When should you cite Wikipedia?
It is appropriate to cite Wikipedia if you are commenting on Wikipedia. For example: "The Wikipedia page on X shows that it is a controversial topic, with a large number of rollbacks indicating disagreement on the correct approach."
Similarly, Wikipedia can sometimes be a primary source on popular culture. In this case, you should  use it and cite it accordingly. Update: I gave an example of this in another answer.
If you use Wikipedia as a source for finding other sources, then the question, Is there a problem with citing the original source instead of the source where the information was first found? applies.
Further reading

Appropriate use of Wikipedia (highly recommend reading this one)
Should I use or cite Wikipedia? Probably not.
Why you can't cite Wikipedia in my class
A Stand Against Wikipedia

Yes, "copied without citation" is bad. But/and also, "used without citation" is not so good, in my opinion.

@paulgarrett I agree: I think you should not use Wikipedia in such a way that it requires citation.

Well, that might entail never looking at Wiki, then, to be honest? After all, if someone learns something from some of my expository writing, and uses my document to find the primary source, I'd expect that this would be acknowledged. Otherwise one is keeping secrets.

@paulgarrett Perhaps you could write as much on this question then. (The question asks whether you should cite the source which led you to other sources.)

possibly, at least as counterpoint...?

@paulgarrett Counterpoints are always a good thing :)

This answer gave me a twinge of irony, citing and linking to Wikipedia, and then pointing to "Why you can't cite Wikipedia in my class" which sits behind a paywall.

@Xantix I was at school when I posted, didn't notice that the article was behind a paywall - I'm updating with a non-paywalled link :) And of course citing Wikipedia editors as an authority on using Wikipedia is somewhat different from citing Wikipedia as a commentary on, say, the Pythagorean theorem in an academic paper...

That is no problem, I didn't think the twinge of irony was bad, just pointing it out.

+1 for ""Copied down without a citation" is not an option."

In my opinion a lot of the answers are too negative about wikipedia, at least when applied to the part of wikipedia that applies to mathematics (my academic field, and the field the OP asked about).
I am a little surprised to hear people describe wikipedia as "unreliable", including links to university websites which say rather snootily to avoid it.  This is how I felt about the mathematics on wikipedia circa 2006.  It has gotten so much better in the intervening years, for the obvious reason: a lot of very mathematically experienced people (including at least one Fields Medalist, and also including me, for a period from about 2006 to 2008) put in a lot of time writing and vetting the articles.  Where it stands now is that wikipedia is the best single repository of mathematical information in the world.  It has been several years since I have seen anything that was wrong on a wikipedia math article.  Some of these articles contain content that is difficult to find in other places, and some of the content is new: it is far from unheard of that someone just put in their own proof of a theorem.  Many feel in principle that this sort of thing should not be done (I think I do; it's been a while since I've really thought about it), but in practice when someone writes a nice self-contained proof of a mathematical result, why delete it?  So there is some really great stuff there: I think that most research mathematicians who are frequent users of the internet have by now learned mathematics from wikipedia.
As others have correctly pointed out, the question of "when to cite" is more complicated.  Let me consider several of the alternatives:

Should you refer to wikipedia for standard proofs?

I think I believe that sometimes you should but I have never actually done it in a "serious research paper", in part because of exactly the sort of internet-phobic practices that Paul Garrett refers to in his answer.  Recently I was writing a broad-audience article, and I wanted to say that a certain aspect of a classical construction -- the Galois connection between ideals of a polynomial ring over an algebraically closed field k and subsets of affine n-space over k -- worked verbatim with k replaced by an arbitrary integral domain.  I ended up referring to Lang's Algebra for this.  That is really not (ahem) ideal: this is one of the most "standard texts" in the sense that a large percentage of professional mathematicians have a copy in their office.  On the other hand it is not free and even more mathematicians and math students don't have it.  But billions of people have internet access, and surely wikipedia (for instance) does a perfectly good job of explaining the point.  I wussed out and didn't give an explicit electronic reference, and I rarely do in formal writing.  (In fact I have myself written many, many pages of mathematical writing -- as has Paul Garrett, by the way -- and I usually wuss out and do not refer to it in my formal writing either, even though I know exactly where I would like to point and a student would understand my research paper much more easily with that reference included.)  At this point, when I say that something is "well known" I assume that students will look for it on the internet, and as a code between me and myself, at least, I try never to say that in papers except in cases where a student who looked for it on the internet would quickly and easily find it (and when that happens I don't worry so much about tracking down a print reference).
In the above case, the big advantage of wikipedia is its ease and convenience: it has almost exactly what any text would have but is much quicker, easier and freer to access.

Should you refer to wikipedia for non-standard proofs?

In other words, if a wikipedia article has a proof which is different from the one that you would find in any expensive math text, should you refer to that?  If you want the reader to read that proof, I think you have to do refer to it or try to track down the source of the proof that made it into the wikipedia article.  However, the latter brings me to my biggest complaint about the math articles on wikipedia: they're great for mathematical content.  They can be really bad as references: e.g. they can be taken out of some standard source without referring back to that source.  Or an article on the X-Y Theorem will have a statement of the theorem, motivation for the statement, the proof of the theorem, and then talk about further work and generalizations.  That would make for a great lecture about the X-Y Theorem, but for an encyclopedia article there's a lot missing: who are X and Y?  (Sometimes they don't even try to tell you, even when there are wikipedia articles on X and Y.)  Where was the X-Y Theorem first published?  (I'm sorry to tell you that many mathematically rock-solid articles don't contain this kind of primary source material.)  Is the proof included in the article the original proof of X-Y?  If not, where does it come from?
When I was involved with it, the culture of mathematical wikipedians was not good at addressing the above issues: if I asked for this information about an article, someone would usually nicely tell me that I was more than welcome to add it myself. I would mention that unfortunately I didn't know the source material that led to most of what other people included in the article...and there the matter usually got dropped.
So it may very well be the case that wikipedia has a proof of something for which it is not trivial to discern where the proof comes from.  As an example, wikipedia has a really nice proof of the Schwartz-Zippel Lemma.  It is not the original proof, I think -- it's slicker.  Where does it come from?  I couldn't tell from the article itself.  This is not a hypothetical example: I wrote a brief expository note including this proof.  As you can see, I did refer to the wikipedia article.  However, I should say that this is an article in the informal sense of the term: I wrote it up for myself, spoke in a colleague's seminar about it, and kept the document for myself.  I have not tried to publish it anywhere, nor would I, since it is "just an exposition" of a proof of Zeev Dvir's resolution of the Finite Field Kakeya Problem.  This brings me to my last point:

When should you include proofs from wikipedia in your articles?

If you use a wikipedia proof in your article in a critical way, then you should include a reference to it (or where it comes from, if possible).  However, if you are using a wikipedia proof in a critical way in your article, is your article a research article or even a "serious expository" article?  Why would a journal want to republish something that is available in a standard source?
In the OP's example he mentions including a proof of the Pythagorean Theorem.  No math journal I know is going to allow you to include (any one of; I'm sure it gives several) wikipedia's proof of the Pythagorean Theorem, but not because it comes from wikipedia: they're just not going to want you to rehash such old-hat stuff.  To be honest, the introductory passage "For example, suppose you're writing an article about triangles..." raises some eyebrows in this regard: are you trying to formally publish an article about triangles?  Good luck with that: it's going to be tough.  Such articles are published, but for every one that is, probably a hundred are rejected.
I also think that in a formal article -- even, perhaps even especially, if it's an expository article -- the burden falls more highly on you to investigate primary source material.  If you're teaching a class or something, then it's helpful to say exactly where you got the material from.  But if you're writing an article, it becomes more important to track down the provenance of the intellectual content itself: that is a much more challenging thing to do.  Still though I think there are cases where the answer really will be that the argument appeared for the first time on wikipedia, in which case you should cite it there.

What about the "unreliability" of wikipedia articles?

This is really "weak sauce" for math articles, because unlike most encyclopedia articles, mathematics articles are self verifying by any sufficiently qualified reader.  So saying "Don't include this proof from wikipedia because wikipedia is full of errors" sounds silly to me: on the one hand lots of published books have a higher density of errors than wikipedia math articles; on the other hand, every proof you read you're supposed to check anyway.  So don't worry about whether it's correct: see whether it's correct.  Most proofs in wikipedia articles are no more than a page or so in length, so they can be checked in a relatively short time.  If it's not correct, fix it or tell someone about it!

@SteveJessop asked:

Since you mention not being able to work out where a proof comes from: suppose you found a proof spray-painted under a bridge, that was slicker than the best proof you can find published. Are the concerns over such proofs on Wikipedia essentially the same as for that found on a wall? That is, there's no problem with verifying it correct, but if you can't print it or refer to it without giving credit then it's difficult to use?

Wikipedia seems better than "bridge proofs" because any interested party can access wikipedia and see the proof there as well as whatever documentation or lack thereof exists.  In the bridge scenario, one can question whether this is really where I found the argument or whether I did my due diligence in trying to track down its provenance.
In practice: in the mathematical community, the scholarly task of giving the correct attribution -- either in the sense of where you found it or, still more so, the primary source -- is not taken all that seriously by many (compared to other branches of academia).  We agree that you shouldn't be passing off others' ideas as your own, and that if you know who the "other" is you should cite them, but "I learned this trick from somewhere, and now I can't remember where" is pretty common in mathematics.  In fact math papers tend to be written in logical sequence rather than psychological sequence, so that there is a key part of the mathematical writing process in which history is removed or rewritten, so to speak.  I can only hope you understand what I mean by that.  It's a subtle phenomenon and not an inherently negative one, but we do it in mathematics more than in almost any other field (and I do it more than the average amount for a mathematician: a big part of my research process is to take others' ideas and writing and rewriting it in one way or another).  In general I have grown "more scholarly" over the years, but usually with the suspicion that at best the referees will not really care one way or the other.  I ended one recent paper with a section surveying the history and the literature of a certain problem.  That is very rare in a math paper.  I got zero comments about this section, and if the journal was better I would have expected them to tell me to shorten or remove the section.  In my most recent paper, see Theorem 2.1 and Remark 2.2.  Remark 2.2 explains the history of Theorem 2.1.  It is almost as long as the proof of the theorem!

Indeed. Good points. In particular, the rapid improvement of quality since not-so-many years ago. In my answer I did forget to mention the point that Wiki articles seem less obliged to "make the reader pay a price", unlike many standard textbooks, which work hard to "do things the orthodox way, no short-cuts". And I do wonder whether the nearly-anonymous editor policy is optimal long-term...

I like this answer. Any answer that says "wikipedia is unreliable" isn't quite right. It is generally unreliable for fields where the research is highly political (i.e. climate change, nutrition, medicine, conservation etc.) or a subject that non scientists care a lot about (sports, entertainment etc.) but not for highly technical fields. The general public does not feel compelled to to edit the article titled "Enriched category". That said, it probably shouldn't be used in serious research papers for the practical reason of getting the paper accepted.

Thank you for your answer Pete, it was a very interesting read. I'd like to clarify that I'm not actually planning on writing an article about triangles, I just thought it was an example that anyone would understand, whether they were mathematicians or not.

My only experience with citing Wikipedia in a math research paper was positive (I had another reference challenged by a reviewer, but not the Wiki one).  I had a reason that I thought a special one, though.  I was investigating very weak set theories and I needed to reference "standard" formulations of basic set theoretic axioms.  Unfortunately, because of the variable absence of their sister axioms, the various "typical" formulations of the same axiom were no longer equivalent, or at least no longer obviously equivalent.  I needed a very accessible and generic source.  I chose Wikipedia.

There is one other aspect of mathematical research papers in particular (less relevant to expository papers). In a research paper in math, many basic results just go uncited - the author is usually able to just make a bare claim, or say "the following proof is well known", and as long as the referee agrees there will be no problem. I have never found what I thought was a good and accurate description of mathematical citation practices, but any such article would have to menton this common avoidance of citing "well known" results.

Probably in the same category as "I learned this trick from somewhere, and now I can't remember where" is "The following is undoubtedly well-known, but it's easier to prove it here than to find a reference" (followed by the result and its proof).

If you got your information from Wiki, or used the Wiki article to find a more authoritative source, it would be honest and proper to cite it. Cite what you use, at the very least! Acknowledge your sources.
I know this is not universally approved, and, as in one comment, many people will react negatively if Wiki appears in the references... and this could have in impact on a referee's reaction to your paper, yes.
And, yes, Wiki's context for technical matters is a roller coaster... Best to see what Wiki says, maybe use Wiki as a pointer to key-words, to external/traditional sources. Then, if you are being honest, cite Wiki and the/an-other more conventional/traditional/orthodox (authoritative?) source.
In mathematics, sometimes Wiki comes up with connections or sources or people I'd not been aware of previously, as well as often mentioning historical sources that many standard sources do not. Also, it is ... available!
If you find yourself in the (unfortunate) situation of needing to conform to a retro standard of "what's legal" to cite, not only can you not cite Wiki, but probably shouldn't give URLs in your bibliography, either? Can't cite arXiv, either? Can't cite anything that was not refereed and "published" (in a now-archaic sense) in one of the known journals? Probably...
But, looking forward, it is happier to be in a situation where you can be honest about sources! Obviously!
EDIT: It seems that people react variously to two aspects (at least) of Wiki, namely, issues about authoritativeness, and issues about primariness of sources, which segues into giving credit where credit is due.
The bad scenario is, as in @ff524's quote from Jimmy Wales, that someone uses Wiki uncritically, and gets burned because something's wrong. This potential is arguably greater in Wiki than in other sources, but it's only a matter of degree. Everything needs corroboration, if it really matters.
Similarly, pretending that Wiki is somehow a primary source probably is never correct, just as no old-timey paper encyclopedia was every primary. But I recall, quite honestly, being exposed to ideas accidentally in encyclopedias that I would not have known to look for at all otherwise, ... and a good encyclopedia, and the better articles in Wiki, do give external pointers of various sorts.
For that matter, how many mathematics textbooks manage to come anywhere close to citing primary sources? :)  It's not easy, I agree! Not a good argument for not trying. I was amazed to observe, some years ago, that L. Ahlfors' "Complex Analsis" has no bibliography whatsoever, although there're many named-after theorems.
The near-universal accessibility of Wiki means that many, many people will turn to it first, rightly or wrongly. Again, I use it often to get clues about aspects of mathematics distant from my prior experience, although I certainly do seek corroboration! To pretend that Wiki wasn't used when it was used is a bit dishonest, as well as not giving credit to its utility, if not to its primariness as source.
There can be different reasons for acknowledgements, I think.

"Don't use Wikipedia as an authoritative source" does not at all imply "Can't give URLs in bibliography" or "Can't cite arXiv" or "Can't cite anything that isn't peer-reviewed."

@ff524 I think Paul is trying to say that blanket statements such as "Can't cite URLs" are usually not very well thought out now that the internet is so widely used. (And by extension, a blanket "Can't cite Wiki" is also poorly thought out)

@ff524, I didn't mean to suggest that "not using Wiki" literally implies "don't cite unrefereed things", but, in fact, the same reasons given to support the one support the other, to my perception.

@paulgarrett My argument against Wikipedia is not "Don't cite unrefereed things." It's find and cite the original source of an idea (thus giving credit to the actual author), not a tertiary source.

@ff524, I agree that one should often give credit to the original innovator, yes, but I think also to a source that one used to find the original thing. (And, of course, many people do criticize the "lack of authoritativeness" of Wiki, and of non-refereed documents, as a fatal flaw.)

It's not very common in mathematics (the subject of the question) to cite intermediate sources and the original sources. Usually it's one or the other (or neither). Even the American Mathematical Society ethics statement focuses on original sources, and makes no mention of "cite what you use".

@OswaldVeblen, and, I sincerely think, "more's the pity", because then one fails to admit how one learned the thing, or became aware of it, and a certain mythology is perpetuated. It seems to me patently dishonest to not admit what one looked at, while also requiring acknowledgement of historical facts and priorities to provide another type of information.

One issue the other answers don't mention is the possibility of circular reporting: http://en.wikipedia.org/wiki/Circular_reporting
As other answers have said, if there's a citation then you should follow it. If there's not a citation, and you use the result in your published paper, the article may then be edited to use your paper as its citation.
This is dangerous, since referenced results will not be scrutinised as much as novel results. In this situation, it would be prudent to cite Wikipedia explicitly, so that such a circular arguments would become clear by looking at both sets of references.
The examples in the link above are false information about the real-world, which are difficult to verify and hence should be avoided outright. The situation isn't as bad for mathematical proofs, since you can verify them before including them, but you should make sure to present the proof in a way that shows it's novel (and hence needs review), but also cites Wikipedia as the source (or else it's plagiarism). A footnote like those mentioned above would work for this.

What is "allowed" and "not allowed"?
But there's almost no reason to ever cite Wikipedia. If Wikipedia states something without citing a source, then you're basically citing "Some random guy on the internet told me that..."  If Wikipedia states something and does cite a source, check the source yourself, to make sure it says what Wikipedia says it says; then cite that.

Some random guy on the internet answered a math question of mine, and we wrote a paper together: http://math.uga.edu/~pete/Clark-Diepeveen14.pdf.  (It was accepted by the American Mathematical Monthly.)  Other random guys (and gals) on the internet have directly helped with my mathematical research as well.  Of course some random guys tell me incorrect things as well, so naturally I check what people tell me, probably especially so if I don't know them.  If what they say pans out, of course one should cite accordingly.

@PeteL.Clark Sure but mathematics is verifiable essentially just by looking at it hard enough. For just about anything else, verifying something on Wikipedia involves finding a reliable, independent source. If you found such a source, you'd cite that, instead of Wikipedia. (And add the source to Wikipedia, please!)

@DavidRicherby but that was exactly the point of the question. Is it ok if what you've found is "verifiable essentially just by looking at it hard enough"?

I will not speak to the direct topic of your questions, which is proofs, because, to be blunt, that's not my field. I will say that there are some edge case where it's acceptable, in my mind, to cite Wikipedia - namely, where said Wiki has become the de facto repository of information for a given topic. These are rare, and usually pop culture focused, but they do exist.
For example, I have a paper which cites the Wiki for a game (which is even less authoritative as a source than Wikipedia), and it was cited without issue from reviewers, etc.

For the title question, it is almost always No!
Although you may be able to find a venue in which you could 'get away' with citing Wikipedia for something like that, it is almost never a good idea. If I see Wikipedia cited as a source (for anything except a comment on Wikipedia, as @ff524 mentioned), it will raise a huge red flag. My immediate thought is that this writer is too lazy to search for and find a better source! Don't do this to yourself. For something very common like this, you should be able to go to your local physical university library and -- without much effort -- find a book that references the needed theorem.
For your second question in the body, copying without citation, even from a Wikipedia page, is plagiarism. Again, don't do this to yourself!
Find a better source if at all possible, and no matter what, cite your source!

-1 for an authoritative "No!". In some cases Wikipedia is the most comprehensive source on a given subject (see a Wikipedia reference in my paper). Surely, I used it only to provide social context and motivation, not to support any scientific claims. But it's not binary - one should always be aware of trustworthiness of a given source in a given context.

@PiotrMigdal, I did qualify my statement by saying "almost always". You are right that in some cases Wikipedia may be the best source available, but (and I may be old-school here) to me a Wiki citation is still a red flag.

Are there instances where citing Wikipedia is allowed? Yes, when appropriate.
It is appropriate to cite Wikipedia in a case study or literary scrutiny of Wikipedia.
For scholarly articles, jurored sources are often mandated.
For literary articles, source citation is based upon the origin.
For example, a literary examination of Xenophon's Cyropaedia or Wikipedia's page on Cyrus the Great, may cite the origin text to draw conclusions regarding (inclusive of, but not limited to) erroneous statistics or accurate accounting.

There is already so many detailed answers are provided in this thread. I agree and disagree with some part of answers. But being a professor of computer science, reviewer and editor of various journals and conferences I always recommend.
1) DO NOT cite trivial things from Wikipedia which are commonly known to the research community.
2) DO NOT cite an established research work (algorithm, equation, findings etc.) from Wikipedia. Wikipedia is not the original work, rather it is a reflection and understanding of 100s of editors. If you learned something from Wikipedia, then refer to the original source and verify the correctness from original work before using it in your research/project work, and always cite the original work.
The reviewers and Editors usually don't like the citation from Wikipedia, because it may change during or after the review process, and we don't have enough time to search the original source to verify the correctness of referred idea/equation/algorithm etc.
So the bottom line is: Wikipedia is a good source of learning, but it is not the original work and subject to change. If you learn something new always refer to the original source to verify the correctness, and always cite the original work.

In many cases, an equation published on Wikipedia is a reflection of the understanding of one editor -- the one who uploaded it. The idea that all of Wikipedia's content is extensively reviewed is widespread, but inaccurate in many cases.

Any time you cite something, you should consider whether or not the reader will accept it as an authority for the point you're trying to justify. That's just a basic principle of persuasion.
A well-known journal in the field you're writing for probably doesn't need much consideration. But just about everything else is possibly suspect. The New York Times is probably reliable for most facts, but for a fact that can't be sourced to a more authoritative scientific source, a New York Times citation could be questioned. A  blog post from a known expert? Maybe the expert isn't as "known" to your audience as you think. And maybe they made a mistake. Wikipedia is just one of many publications in this general category.
If you find it necessary to cite such a source, find a way to concisely make the case that your reader should accept its validity for the point it justifies. There are ways to do that in the text, or in a footnote. If you can't find a way to do it, your paper might be better off without it.


text:
How to avoid disappointing a professor interested in supervising me?
I am a PhD student in the social sciences and I already have a first advisor.
A few years ago, eyeing the possibility of a PhD, I had informally asked a professor of mine if she would be possibly interested/have the capacity to be my advisor. I had already worked with her and liked the collaboration very much. She is very competent on the data analysis techniques I will use, knows my specific subfield really well, and has a good network. She has invited me twice to discuss my project ideas informally in her team meetings with her other PhD and postdocs, despite me not being an official member of the team. All in all, she has been really encouraging and supportive.
However, she has some ongoing conflicts with different members of the department (I do not know exactly why, who initiated them, etc.). I am not sure about the relationship between her and my first advisor. For sure, the first thing I will do is talk with my first advisor about it. I would like to avoid any forms of conflict in my committee. Also, the techniques she is an expert on might have a relatively minor/subordinate role in the whole architecture of the dissertation.
In case my first advisor expresses preferences for other professors, how should I deal with turning down her offer to supervise my work? Have I behaved unethically by expressing interest, profiting from her feedback, and now having to retract, and if yes, how can I make up for it?
I would especially like to further the collaboration in the future, even if it is not in the form of advisor-PhD student role. I am thinking about telling her that my first advisors suggested other co-advisors and that the project will use, for the most part, a different methodology, so other advisors might be a better fit. I am also thinking of asking her if we can further collaborate and suggest co-authoring a paper about that specific subsection of the PhD dissertation. However, I will also have to ask my first advisor if she agrees to that.
Do you have any thoughts on how to approach this situation and how to best communicate with this professor?

Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.

You seem to know her fairly well; enough for a serious conversation. I suggest that you initiate one with her, first, rather than your formal advisor. Some departments can be very political with several factions. I'm guessing that you suspect that but don't have details. She will know about that better than you.
Tell her you have a current advisor and would also enjoy working with her as well. Ask if there are any issues around that concept that you might not be aware of. She will probably give you enough information, though not in detail to guide you now. If she suggests that it would be a bad idea, don't press it, but you can say you hope for a future opportunity to work with her.
You can also mention the other faculty members suggested by your formal advisor.
There is nothing wrong with any of that.
I suspect that she has a pretty good idea about how a formal relationship would work out.
You may wind up suggesting her participation to your current advisor or not, depending on what you learn. But don't get caught up in squabbles, whether rational or not, that might be surging around the department. Nothing good will come of it.
Also, if your advisor was assigned to you and you aren't happy with the choice, don't be shy about requesting a change. You don't hint at that, but it happens. Failure to speak up for yourself can set you back years.
I was a student in a very political department. There were people that wouldn't talk to other people or work with them. It mostly didn't affect students but there was a lot of sniping.

Note: "Political" just means factions. It needn't be about "politics" as in elections. Some of it can be racist and such.

It is fairly common that students change advisors or switch groups. For the professor running the group this is often not good news, since they might have invested time and resources on the students. However, professors are supposed to be adults and accept decisions of others that they do not like. You have not even yet established a formal relationship with your professor, so she is expected to be fine with your decision of not having her supervise your thesis.
It is fairly common for graduate students to collaborate with professors that are not their supervisors, though of course if this collaboration is to the detriment of their thesis then the advisor is supposed to intervene. Thus, some collaboration with her would be acceptable in academia.
Usually, professors should not "steal" other professor's student, meaning actively pursue them in order to leave a group or an advisor. As a student you can prevent the impression of stealing by being more open with your advisor and potential collaborators.
Finally, she might not be interested in working with you if you do not look for her as an advisor, since she might be spending a lot of effort on you without getting much back in return.
Overall, exploring opportunities for having someone as an advisor and trying out the relationship does not oblige you for the future. It would be nice if you can get up the courage to talk to her directly and thank her for what she did for you. You can ask her then directly what are the possibilities for a smaller project together, if that is what you desire. In principle, graduate students are to be protected by conflicts among the faculty, but this rule is not often kept.


text:
Pros and cons of length of research stay
I am a Ph.D. student in the Social Sciences at a German university. I want to apply for an exchange program at a research institute in France, which is very relevant to my research. It is a very big research institute, with ca. 40 senior researchers and ca. 70 Ph.D. students (half of them visiting). The university would provide funds that should cover most (possibly not all) costs related to the stay. I also have some savings that I could use for this. The research stay can last between six and ten months.
I am reflecting on the pros and cons of a shorter vs. longer research stay. I plan to return to Germany for my postdoc phase. The research stay will be in my third year. I would be willing to extend the length of my Ph.D. for a meaningful research stay abroad, but I will have to look for other sources of funding after the end of my third year.
Do you have some advice related to the length of the research stay? Do you think the advantages of a longer research stay would be worth spending a part of my savings or taking longer to graduate?

I voted to close because this seems like a personal decision that depends details of your situation and individual factors.  That said, if the place you will be visiting has a much more vibrant research community in your area that would allow you to get involved in more collaborations, longer is better all other things equal.

In my personal experience, six months is a minimum of constructive outcome, considering that issuing you a wifi pass, an entry badge, a weird-paper-you-need-but-not-understand, and finding some housing will take around 2 months.

In academia (being a part of life), everything is unpredictable. I would ask which is harder, lengthening a short stay or shortening a long stay, and then apply for whichever is easier to adjust.
If they are the same, then I would go for the short stay. After you have been there a few months, review this decision, and apply for the longer stay and look for a few months more accommodation if that seems to be wise.

A longer stay would, presumably, give higher probability of reaching some useful, publishable, result. But this might depend on the field. A longer stay might also let you make some stronger connections/collaborations with people there for the future. But that depends on the nature of the interactions generally in your field.
Otherwise, I see no significant difference for a small difference in time.

Your added question is too personal for me to give advice. I can't know all of your options.


text:
Mistake in abstract
I have submitted an abstract to a conference that has been accepted and will be published for the conference. There is a minor error that changes the data I have included in the abstract. What should I do?

The correct answer will depend on the usage of the field and the exact circumstances. I cannot tell whether only the abstract will be published or whether the complete paper is published or a short note made out of the paper such is done for poster sessions in some disciplines.

There is probably nothing to be done at the moment, though publishing a correction on your personal web site is appropriate. Even withdrawing the abstract is probably impossible now.
Make the correction in the final paper and, when you cite the abstract (as you should), note that it contains an error corrected here.
Lots of incorrect stuff gets published for various reasons. But we hope that statistically we wiggle toward the truth.