text:
I absolutely hate my PhD and I don’t understand the purpose of it
I am in the fifth year of my PhD and seriously depressed. In my first year of graduate school I was not funded. In my desperation for funding I chose an opportunity that came my way even though I did not like the topic. The research I would say is pretty mundane, I was providing experimental support for an FEM group. I did t have any freedom until the end of my 4th year. Also my advisor was the co-PI and the project was not in his area of expertise and he had no clue about the work being done. Whenever I get lost and ask him for advice, he would commonly uses the phrase “Do what the industry sponsor wants, keep him happy ”. I literally felt like a contract employee and not a PhD researcher. As expected the work did not yield any papers. My lab mates working on other topics (areas of my advisors expertise) get to present at academic conferences and are writing journals. When I point out this fact I get chided for comparing myself with others and  that my work is “unique”. At one point my advisor himself said that the project was ill conceived. You can understand my frustration.
Now I am depressed, lost, and hardly do anything. I wake up, go to the lab and pass time. I absolutely hate my “thesis”, I don’t have anything meaningful or useful for my future career. I don’t even know the purpose of my thesis topic. initially I was scared to leave, now I feel it is too late to leave.
please advise me on what to do? Thanks.

which are are you working in, FEM sounds like engineering? also, which country / how long is your program supposed to take?

Perhaps this comment is ill-conceived, but I'm wondering if there's likely to be any consequences to the supervisor/advisor if none of the solutions presented here manifest, and whether this could be used as bargaining power in case things turn sour. If you've done work or been committed, then you shouldn't be leaving with nothing. Especially not if someone made a mistake in the project design or neglected rather than nurtured you.

Well the purpose of a PhD is not to make sense, or even to have a useful result (a result is nice to have, but secondary). PhD is a title that states that you have delivered the proof of being able to work systematically, find literature, read, and write it up, again systematically. Much like passing university is not about knowing or learning and mastering anything, but merely a proof of being able to deliver according to a strict time schedule, pass tests, perform exactly as required when required, and move on. (Also, PhD exists to have people work for free.)

Offtopic, but it might help you a little. Keep in mind that being a PhD student was one of the hardest period in many researchers life. You are not alone with this situation, even if the others in the lab seemingly advance faster than you. You need to follow the suggestions in the answers, but firstly: do not be affraid, you are not alone. Unfortunately, this is way more common than it should be.

I find it hard to imagine aproject where the co-PI "had no clue about the work being done"; if you weren't already in your fifth year, I'd advise you to get out. But under the current circumstances, your best bet is probably to (1) get counseling to deal with your depression and (2) do whatever is needed to finish your Ph.D. as quickly as possible. Item (1) will probably be a necessary prerequisite for (2) and also for subsequently repairing the damage caused by your unfortunate Ph.D. experience.

Quick comment: Sorry about your situation, and it is a bit odd that there is an experimentalist providing support for an FE simulation team, while it is usually the other way around. If you can tell me, is your industrial sponsor a CAE / tools company who want some aspect of their FE codes / models validated by experiments?

Ahh, bad supervisor, that sucks... but is a common issue. As a PhD student you should be learning how to do independent research. With emphasis on learning: somebody has to teach you how to. However, in my experience professors are very busy people. They only have a fraction of time to support you. So it's good to have additional support. But as somebody that's going to quit he's PhD, I think it's more than that: you also have to have talent for the job. You have to be able to take charge of your own PhD. Make a plan, be pro-active in writing and communication. Not everyone can do that.

Now I am depressed, lost, and hardly do anything. I wake up go to the lab and pass time. I absolutely hate my “thesis”

This is the problem; you are stuck in a vicious cycle. You need to meet with your advisor and come up with a solid plan - including a timeline - for what will happen in the next year or two.
Given that it has already been 5 years, I suspect this plan should involve you finishing your current work, writing a dissertation, and graduating. But, you and your supervisor could also choose a different direction (i.e., starting a new topic or choosing a new advisor).

I don’t have anything meaningful or useful for my future career. I don’t even know the purpose of my thesis topic.

At this point it's probably time to decide what your career goals are and make sure your plans are aligned with reaching those goals.

This could mean spending 1/3 to 1/2 of your time learning skills that will be useful for an industry career, and spending the rest of your time preparing for graduation. Given your level of burn-out, I suspect this is the wiser course, but I'm just guessing.
Or, it could mean switching topics / advisors / institutions so that you will be competitive for academic positions in topics that interest you more.

Upvoted. The most important thing is to reflect and take a decision in a limited time. The situation is not ideal, but people get stuck at different points in their career and a PhD is a more likely phase than others to get stuck in. A decision will be liberating.

I am not a psychologist, and even if I was I woudn't try to diagnose over the internet, but this:

Now I am depressed, lost, and hardly do anything. I wake up go to the lab and pass time. I absolutely hate my “thesis”

Sounds like you should seek professional help from a counselor. Even if you are not clinically depressed, a good counselor and help you work through your feelings and difficult decisions you have to make. They might help you decide to leave your PhD, so they might help you decide you should finish it having come this far, and help you develop strategies to get it done as quickly and painlessly as possible. I can't speak for your school, but we have a university counselling service here that can provide up to 8 sessions to any student or staff member for free. Have a look if your school has anything similar.

I'm going to take a slightly different approach, but bear in mind that the other answers are very right.  A PHD program is challenging, and as with anything else in life taking care of your health (and mental health) should be a top priority.
That being said, my answer assumes you have evaluated that and are still intent on continuing.  The challenge of a PHD program is supposed to come from working on ground breaking problems, not from your advisor's lack of support. Without further details, it's hard to say, but it does sound like he's prioritizing his career (finishing an industry sponsored project) over yours (publishing papers, working on interesting topics).  And he's apparently far enough out of his areas of expertise that he can't effectively guide you.  That is a recipe for frustration.  Been there, done that, with an added helping of arrogance and failed commitments on the side.  So, what can you do?
If there are resources offered by your graduate school that would be a good place to start.  I'll list some things that helped me, although I may not recommend all of them for your situation.

Treat it like a job.  You seem to be giving yourself a hard time since the research isn't working out how you think it should.  Just put in the hours and emotional commitment that you would for a job and trust that eventually your advisor will hold up his or her end of the deal and help you defend.
Get a solid timeline.  Already mentioned in the other answers, but I'm repeating it here since suggestion 1 doesn't really work if it turns out the research you're working on won't lead to a dissertation.  At the very least you should have an approximate timeline, what problems and approaches will go into the dissertation, and what scope of work you are expected to do, and what will happen if the project continues to fail. And get it in writing.
Work with your committee.  As with suggestion 2, make sure your committee is clearly aware of what work you intend to do.  Make sure they consider it notable enough for a dissertation.  Then, give them regular (monthly) updates - what progress has been made, what issues you've encountered, what problems need to be solved, and follow up on the rare suggestions they may give.
Take responsibility - unfortunately it sounds as though your professor may be incapable of supporting this area of research so you will need to take the initiative on ideas and experiments.

Those are just some suggestions.  I would strongly encourage you to consider what exactly you want from the program, and whether you can complete it with the current level of support from your advisor  (edit: and take a week or two off to vacation, spend time with family, or whatever makes you relax before making any choices). Depending on your goals, you may very well be better off switching advisors and research areas.


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How to inspire people to learn things they don't understand
I have seen far too many people that just cannot or are unwilling to learn due to the subject being rather uninteresting or boring. They tend not to want to learn things they don't understand because it requires effort and dedication.
For example, people wanted me to teach them how to create robots, android apps, games, etc. but when it came to programming and using database, no one really wanted to pay attention or cared about it, and what was worse, they complained about how difficult the assignment was even it was just creating a simple calculator program. I gave them tools to solve the problem. They had everything needed to create the program on their own without hand-holding but it seems like it just wouldn't happen.
Is this due to a lack of experience in teaching based on the method used or is this the student's fault for not even trying to learn? If this looks like a problem with the method of teaching, how should I improve or inspire learning?

I think in the tech world it's definitely a misconception on how things are done. They don't understand it's not just a magic fairy wand wave and more sweat and tears (possibly lots of tears) that makes those really neat apps and toys that they like to play with. They possibly haven't been taught proper problem solving and how to apply it to real world situations and/or coding.

I would suggest that "robots, android apps, games, etc" are ends which are aimed for/desired, while programming and databases are means to those ends (therefore are less easy to get excited about!).

In my opinion, you shouldn't ever try to spend time educating someone without ensuring that the person actually wants to be educated. There's a difference between, say, "wanting to be a programmer" and "wanting to learn how to program". You could ascribe the first sentence to literally anybody. Who doesn't want a free skill? It's only the people who also fit the second description that you should educate.
The difficulty is that people usually don't know the difference between those two things. I often don't know it: I once decided I wanted to learn how to play the keyboard. Picked one up, played a few tunes, but could never find myself interested in sitting there for hours and playing the same tune over and over until I had mastered it. Turns out, I didn't really want to learn how to play the keyboard: I only wanted to play it.
So, it's up to the educator to help decide whether the person really wants to learn it. There are a few ways to do it. First of all, be honest: don't advertise your education with what the students will learn, but how they'll learn it. Because that's actually what they'll be doing. So when people want you to learn them to teach games, tell them from the get-go what they'll be doing, how terse and repetitive and basic it can be. But then, during the education, ensure that the end goal is always in sight. So if you're teaching somebody to make a game, instead of them applying their skills solving some math problem ("compute the nth prime with property y", e.g.), why not give them a piece of code from an actual game, and have them solve a small, intermediary step - and then show them the result, the before and after effect of their contribution?
TLDR:
1) Students should come to you for how you'll teach them, not for what you'll teach them,
2) During the study, always keep the end goal in sight. Don't let them drown in repetitive theoretic stuff ... let them .... swim in it, I guess, always taking in fresh air whenever they need it.

I would say the particular details of this question is a bit vague and hard to assess. Because we will have a hard time knowing/understanding the actual details of some of your statements. (For example exactly how difficult the actual assignment was, what prior knowledge did the students have etc.)
However getting to the actual question: How can you be more inspiring to people learning what they don't understand.
In my opinion there is one major thing that you can recognize in any good teacher that I have had during my studying, both in mandatory school and later in university aswell.

As a teacher/lecturer, there is a difference between an interest in your study, and the interest to teach others.

The best teachers I have had always have had one thing in common. They LOVED their subject so much, that they really BURNED for others to realize the beauty in it aswell. As a student, there is such a difference getting an half-assed answer to a question from a lecturer just wanting to get out of there and 'do something important' instead to a lecturer that really takes his/hers time and explains stuff thoroughly to you.
Furthermore, trying to answer a bit of those later examples (as I personally have encountered situations where the teachers expectations clearly was out of reach for us students). I hope that you have some kind of evaluation of your course, why did they find the assignments hard? Were they unprepared? Did you assume them to have some basic knowledge they did not have? It has happened to me , where a teacher assumed us to have a lot of knowledge in programming while we only had taken an introductory course. That teacher was flexible though and added extra lectures in those subject what we had to know and that course turned out rather good in the end.
Still, at the end of the day, you can't magically change people into wanting to learn. You can make your best effort in that your teaching will be fun and inspiring, but at the end of the day, the willingness to learn has to come from the pupil.

First I would like to state that I do not have decades long experience in teaching other people, but I have decade long experience in making myself learn something.
This is what I often do to help myself stay focused: I read books "from back to front". Not literally, but what I mean is that I first look up what I really want to learn and then go back to study (sometimes not that interesting) things I need to understand to learn the the part I looked up before. This really sets a big sign in my head: you go through this to be able to do that later. And it helps.
I am a mathematician but let me try to propose a possible implementation: you said the students want to make apps/games but don't want to learn databases. Assuming they have some basics in some programing language (e.g. Python) you can tell them that you will explain them how to do one part of a game - inventory. You can ask them how to implement an inventory for a game and push the idea toward databases, gradually going from the example of game inventory to the abstract database stuff. In the end you can give them an assignment to make something like game inventory.
I am not really sure if my example makes 100% sense as I am not that good programmer (database for inventory is probably an overkill), but I hope I made a point.

I think Jin5's answer is quite good, however, I do think one of our responsibilities is to do our best to motivate our students. That said, not all students want to be motivated.
Many people would like to speak a foreign language, but they want to skip the boring work of learning vocabulary, grammar, and pronunciation. It's the same with programming. Students must learn the components before they become masters.
I agree with others here that the passion you feel for the subject will have a motivational impact on the students. I remember one student at her graduation come up to me and said she changed her field of work from her original major to something completely different. I asked why. She responded, "Because of that one class I took from you. You were so inspiring about it that I really saw how important it was and I really fell in love with that subject."
I never really thought I felt anything special about that subject. However, I have always thought it was really important and that came through in a way that has motivated more than one of my students to end up focusing on that area.
In short, show them how what they want (making robots, etc.) includes many areas, including database work. With luck, they will see how databases are just one more building block they can use to build their own masterpiece.

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I want to quit PhD after 4 months. Should I?
I am a first year PhD student. I am doing it from an institute in a remote part of the world where the temperature is -10 degrees most time of the year. I took this opportunity after doing my Masters in Europe and I was really happy with the time I spent during my masters thesis research and really liked my research and came up with good results and very good grades. Now, I have taken this scholarship position but I am already in dark. As said, this institute is in a very small town. I can't speak the language of local people. I can't go out for a walk because its too cold.
I don't think that the project is difficult. In fact it is easier for me. But my daily life is not perfect because to read a research paper or write an article I need a fresh and happy mind, which in this city and cold temperature is not possible. I have money just for food and living and no more for even traveling to another city.
Can someone please tell me if I should leave this program?

We can't make life decisions for you.

It is your decision to make, do not let anyone or anything change your future. However, doing groundbreaking research in Alaska on million year soil would be fun, no matter the cold.

change as soon as possible, PhD without social life and conection is hell

Thanks for the comment dear. You understand my situation correctly. Yes, I am leaving it and without having any other option in hand. It is that much worst situation.

@SSimon, I disagree. There are many excellent PhD students that succeeded without social life and connections. Of course, it's not for everyone. It depends on the personality of the student.

can you give one for example @Dilworth ? Not in life sciences!!! It is academic suicide, unless if you are not in top tier universities

@SSimon, there are so many. But I dare not mention their names. (When I said connections I meant non-academic ones of course, since the OP is not concerned with his academic circle of connections, but with his friendship and social life---at least this is what I understood).

@Dilworth there are many example of phd students studying in cold, foreign institute?

@SSimon. No. We are talking about the veracity of your statement: "change as soon as possible, PhD without social life and conection is hell". It has no mention of cold or foreign institutes.

@Dilworth I was referring there to your nonsense that there are students that can make it isolated. Really? how many percent and in what field? Minority in range of statistical mistake.

As I said earlier, if you're talking about social isolation (not an academic one), I would certainly argue that many very successful academics (predominantly in theoretical exact sciences, e.g. mathematics) have been isolated throughout their PhD study. (And please refrain from rude speech next time.)

@SSimon Why do you think that social isolation's such a barrier to making it through a Ph.D. program?

@ChemicalEngineer bcs if you not share ideas with people, how do you expect to develop connections and wide your reserch network without connecting to people?

@SSimon Agreed that social isolation is a barrier to socialization/networking.  How's that connect to Ph.D. program completion though?

@ChemicalEngineer completion without anything else, is lost time, you cannot find job only with publications.

It's impossible to advise you what to do. But I would consider to give it more time, e.g., a year, before quitting. You may still be effected psychologically by the change of place, weather, geography and location; you are still not adjusted to the new place. So you feel anxious and perhaps panic. But it may not be a good strategy to haphazardly decide on quitting the PhD while in this position. So I would advise you to wait more time and see if you adjust to your place and perhaps start enjoying your project.
(Also, PhD is never easy, and so mild depression while completing a phd is quite common.)

Thank You for the advice as it seems more logical. You used the correct word panic for my situation. In a comment before there came Alaska so exactly its a similar story. Today was the most difficult day of my life. So much depression. I don't want to discourage PhDs but I am concerned with my social life here. Me and my roommates call it a prison. I can take this pain if I am earning 5000 Dollars on an Oil rig far from world but only for food and living, No. I think I am going to do something wrong in my career but I am leaving it.

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Asking the primary professor of your target grad school to review your statement of purpose
I'm applying for a PhD in the US (I'm from Italy).
I was thinking that maybe I could ask the primary professor of the school I'm applying for to review and give me his opinion about the statement of purpose I'm writing.
I'm afraid though, that in the US academic environment, this could be seen like ethically not fine. In Italy this would not be a problem, but I want to avoid to hit a cultural difference.
Is it something appropriate?

Not sure what you mean by "primary".  It is definitely okay to ask, but do not be surprised if the professor does not have time to respond.

yes. just to ask him suggestions about which program I should apply for. He was very gentle in his response.

You may not ask for a review which would be very time consuming. See this way, the professor would be receiving numerous applications from all around the globe and you are probably one among many. Think of him getting review requests from all of them (which would be statistically less probable by the way). He would be probably busy with his own research work and its more or less an annoyance if you ask for a review.
Now, if you are called for an interview or a presentation by the selection committee and if that requires preparation of topics specific to your potential advisor's research area, then its perfectly fine to ask him questions or suggestions for improving your presentation. Even here, I do not recommend you to ask for a review rather you may request for suggestions.

In Italy this would not be a problem, but I want to avoid to hit a cultural difference.

I don't think this has anything to do with cultural difference, professors  are ought to be equally busy in both of the countries.

Your hunch is correct.  It would not be ethical.
However, if you have a mentor in your current institution, you can certainly send this person a link to the primary professor's home page, and your essay, asking for feedback.
But before you do, here's a pretty effective exercise.  Imagine that the trusted mentor is reading your essay and jotting down comments in the margin, slashing here, asking for more detail there.


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What if I made a mistake in a paper and now I got a minor revision?
I have received a minor revision after a major revision for a paper.
I discovered that I made a mistake in the calculations. Fixing it will result in changing a lot of the results, but the final result will not change much.
So should I mention these changes to the reviewers (they are noticable) or should I just correct the mistake without mentioning it to the reviewers in the response?

You should always mention this! If you don't, then the entire manuscript becomes suspect. The reviewer has no way of telling what else you changed, and would be forced to recheck the entire manuscript. It's very time-consuming (and cruel of the authors).
Tell the reviewers exactly where your revisions are and why you're making them. You're already using their time, try to make their work as simple as possible.

Hijacking this great answer to add that something you can do to be kind to the referee (at least, in the journals in physics I am familiar with) is to make your revisions to the text in color, so the reviewer can quickly see what has changed. (The final page proofs will remove this color).

We usually also include a "changelog", addressing each of the referee's comments:
Referee comment 1 We added a new paragraph in section 1.2.3 to address this.
Referee comment 2 This is not valid because ....
Referee comment 3 We fixed this error, affecting equations 1, 2, and 3
Other changes Apart from the comments by the referee we also fixed errors in equations 7, 8, and 9. They do not affect the final result

Excuse the formatting above, apparently a lot of markdown is not supported in comments?

@Simon Yeah, each comment is forced to display as a single paragraph, so only inline formatting is supported; no line breaks, paragraphs, bullet lists, etc.

@Andrew I sometimes use latexdiff, so the reviewers get a final document and the exact list of what was changed.

Been there, done that. You need to redo the calculations and rewrite the paper. Make sure you communicate your needs to the handling editor. They will decide if it needs to go back to the reviewers.

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Fiancée has a position lined up, but none of the universities in the area are hiring. What else can I do to get an academic position in the area?
I find myself in an unfortunate two-body problem situation. My fiancée got two offers earlier this year, one a post doc and the other a tenure track position. She deferred the tenure track for a year and is currently halfway through her post doc. The university she got the tenure track position from was happy to do this for her so she could broaden her teaching and research skills.
I am a PhD student who is finishing up in May, but I find myself not having many good options for employment. The area in which she got the tenure track position is very small and there are not many options for people with a math PhD outside of working for one of the universities. Obviously I knew this already. However, none of them are hiring for this coming year.
The two body problem in academia is a classic problem and it's not one that is easily remedied. I'd really hate to be a lecturer because that's a really rough lifestyle and I have a lot of research I want to do. I have at least three or four papers lined up for the next year or so in addition to one I have published, two that I have submitted, and another I'm submitting in the next week or two. I'm starting to feel a lot of dread about this. What should I do (or even my fiancée do) to get an academic position for the fall (ideally in the area she will be in)?

It's not clear what you're asking us. ("What should I do" is of course a very personal decision)

Thanks @ff524. I fixed it up a bit. Hopefully that gets the point across better!

Thanks for fixing my title @ff524. That's a much better title!

Has your fiancée discussed this with the university where she has been offered the tenure-track job?  They might be able to create a position for you, or offer you a temporary position until a permanent one opens up.

Any postdoctoral possibilities?

What field is your fiancee in? You can be fairly general, but it makes a rather big difference if she's very employable (e.g. big data specialist) or if this is likely her only good shot at a TT job (e.g. some of the humanities).

You (or rather, your fiancee) should talk to the department where she will be taking the tenure-track position.  Many universities try to support the employment of faculty members' spouses/partners, as part of their family friendly policies.  This support can take a number of forms.
The simplest form that the support can take is for the university to hire you in an academic position.  You don't mention whether your fiancee is also in mathematics, which may be relevant.  It's often easier to get the school to hire a partner in the same department.  The reason is that the hiring has to be done with the approval of the department that is going to be employing the spouse, and that may be easier to get if the department has already made a commitment to one member of the couple.  Sometimes (and ideally, I would say), the trailing spouse can be hired in another tenure-track positions.
Other times, the trailing spouse may be hired in a non-tenure track position, but one which can nonetheless be effectively permanent.  You say you do not want to work as a teaching-oriented lecturer.  Unfortunately, you are unlikely to get a permanent, non-tenure-track position in math as anything else; academic positions in math almost invariably involve teaching.  The situation can be somewhat different in the laboratory sciences, where people can be hired as research scientists with little or no teaching component to their duties.  At my institution, I know of a case where a professor was hired in the physics department, then his partner was hired as a research professor in biology.
The third possibility is that, even if they cannot offer you a suitable academic position, your fiancee's university may be able to help you locate some other kind of employment.  Some schools have official policies that offer this kind of support; but some do not.  However, even if there is no official policy supporting this, you may still find people (especially in your fiancee's department) willing to assist you.  The help could come through personal connections, or making some of the resources of the university's career office available to you.  After she finished graduate school, my own wife got some help from my institution finding job leads, which were extremely helpful.

Note OP doesn't seem to have an objection to teaching, just being a full-time non-tenure track lecturer (i.e. teaching 5 courses/semester for $45k/year or so, which can be tough!).

To go along with AJK's comment, I would say a reasonable solution is to get hired as a part-time lecturer in the short term, until you find a permanent solution.

@Kimball: I disagree, somewhat strongly: if you know you don't want to be in a 100% teaching job, you should not take a lecturer position, even as a transient solution. Being a lecturer improves your teaching - but doesn't let you improve the skills/CV items you need if you want a job that is not as teaching-focused.

@AJK It depends on preferences, but I think a part-time lecturer position (1-2 classes/semester) is better than no position at all, at least in math.  You can get an office, library access, university affiliation, etc.  Plus when applying for tenure-tracks, teaching experience/recommendations are important and you don't have to explain away a gap in your CV.

I voted this down not because it's inaccurate exactly, but because I think it paints a much-too-rosy picture. These things can happen, with planning and coordination that started a while ago and a HEFTY portion of luck. But at this point it seems very optimistic to think they will. I think Ben Webster and Raghu Parthasarathy's answers are much more realistic.

@TomChurch In particular, this answer totally fails to address the fact that the best window of opportunity for this sort of arrangement (before the spouse accepted the job) is already past.  It's worth trying, but I doubt much will come of it now.  Of course, it would be nice if the OP actually clarified what had happened (whether they just didn't bring up the issue, or did and were refused).

@TomChurch Many universities will consider a spousal hire at any time.  Obviously, the earlier this is brought p, the better; however, the questioner has to deal with the situation as it exists for him now, not months in the past.

There's nothing particularly unusual about your situation, but I'm a bit confused by your post.  What have you done to try to obtain employment thus far?  It would pretty late to be just applying to TT or postdoc jobs now.  Have you applied for positions in other cities?  As other people have mentioned, the lack of any discussion of whether you tried to negotiate a job for you when your fiancée was offered her job (and what happened if you did) seems like a pretty big oversight.  I'm going to assume you have asked and were turned down (since if you haven't, why are you talking to us and not her chair?).
Unfortunately, you face some nasty choices here.  My personal recommendation is to try to get a job somewhere else.  Being separated from your partner sucks, but your other options are probably being a lecturer or being unemployed, both of which will be bad for your long-term prospects.
In the long term, you should expect that you and your fiancée will have to apply for TT jobs elsewhere (either to actually take the jobs, or use as negotiating leverage).  So, what you should focus on is being marketable when you do that in a year or two.  So, get the best job for your career in the long-term wherever that is, and get used to flying.
If you really can't stand to be apart, you can contact local universities, even if they aren't hiring.  It will be bureaucratically impossible to make you a TT offer at this point (it's way too late this year for that), but they may have lecturer-type positions for you.  Often in the spring, they'll realize that they have teaching needs (someone decides to take a job elsewhere or take a leave, and they end up with a hole in their teaching schedule).  But since you've said that's not what you want, that just circles us back to the long-distance option.

This is essentially an impossible question to answer, but I'll point out a few things that might be useful. (I'm a physics professor at an R1 university in the U.S.; parts of this may or may not be relevant to your field or area.)
The very best thing you can do to get a long-term academic position in the same geographic area as your fiancée is to be the best you can possibly be in your field. This may sound like a silly thing to point out, but I think it's under-appreciated. Creating positions, or even creatively making use of existing positions, is tough; a really good motivation for faculty to work on this is for them to really believe that you would be great to have as a colleague. Conversely, if they are not very excited by you, doors will not open. I've seen this happen many times. (And, it makes sense.) You might not like this, but if staying away from your fiancée for a year and working on your research at some other position makes you a stronger mathematician, it will probably help both of you in the long run.
Another route is for your fiancée to threaten to leave unless a position is found for you. Given that she hasn't actually started, though, this would be a bad idea -- it will generate a lot of ill will, which junior faculty should try to avoid. Of course, and I know this doesn't help you, the time to have negotiated and set this up would have been before she accepted the position, when it would have been very normal and understandable.
A third path is for you to take some sort of adjunct-like position, assuming such a thing exists or could be created. In my experience, people over-estimate the likelihood that this will morph into something "real." (See item 1.) I should also note that creating a job for a particular person is often difficult or sometimes even illegal (in contrast to running an open search.)
Good luck!

I might first try to decide which of the following is a bigger deal:

 Working in a job that is not centered on research and mathematics
  Having some significant uncertainty over the next 2-3 years about where your family will be living/working in the future.

If the first is a bigger deal, you probably want to do your best to land in a TT or something close - which probably means you need to find a postdoc. If the second is a huge deal (for instance, if you want to have kids in the next year), you could consider administrative-focused alt-ac positions within your fiancee's university (https://www.historians.org/publications-and-directories/perspectives-on-history/november-2013/in-admin-four-history-phds-discuss-their-alt-ac-careers). These might be easier to get if your fiancee's department is trying to ensure she stays!
Are there multiple universities in this area (as suggested by your post)? One possibility would be to find a postdoctoral position in the same area, preferably in a different department than your fiancee. This is a temporary solution, but one that keeps you in the hunt for a tenure-track position. If you build a strong resume, it might be possible to be hired either at your wife's current university, or for both of you to together move to a new university. This is a difficult path, but can happen.
I suggest "preferably in a different department than your fiancee" because in many places, if you join a department as a postdoc, they may view you only as a trainee, and have difficulty seeing you as a potential equal/TT candidate. (Of course, other departments like to hire their own... YMMV.)
Finding a postdoc in a small town can, of course, be difficult - you may have to search in more fields and departments than you first planned. For instance, applied mathematicians could postdoc in some computational/theoretical groups in mechanical engineering, chemical engineering, physics, etc. You could also consider a long-distance relationship for a few years - though it is tough!
One last note: negotiating for a two-body accommodation usually requires leverage of some sort, e.g. "here is a great person we want to hire" or "we need to prevent this person from leaving." Right now, after your fiancee has already accepted the job and negotiated for a deferred start, but before she has started working, it sounds like your leverage is near its minimum.

I have at least three or four papers lined up for the next year or so....  What should I do to get an academic position for the fall (ideally in the area she will be in)?

First part, written by Buzz:

Many universities try to support the employment of faculty members' spouses/partners, as part of their family friendly policies. etc.

Second part:
If you don't succeed in getting the type of research position you would like, in the desired location, and you can get by without any income for a little while, you may want to consider asking a math department in your partner's town to host you as a visiting researcher in an unpaid position.  This would give you some sort of office space, perhaps shared; people to interact with; and an affiliation.
I've seen trailing spouses make this work.  Sometimes they apply for their own grant money.

There wouldn't be grant money like that in mathematics, but asking for an unpaid visiting professor position is indeed a good fallback option.

@TomChurch - I've seen unpaid trailing spouses get grants in applied math.  Bummer that it doesn't work in pure math.

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


text:
How does one research (pure) mathematics in a more inclusive manner?
Motivation
I don't want my mathematical research to be limited by me being a monoglot, Western, white, and a man; and I'm sure there are other limitations. Diversity is essential.
The Question
How does one become more widely read in mathematical research, beyond boundaries like languages one speaks/reads, one's culture, privileges, upbringing, etc.?
Thoughts
There doesn't seem to be much under my control. I could learn a new language or two. I could try reading from journals from places underrepresented so far in my research.
I don't know what to do.
Surely, we would all benefit from research that casts its net wider …

Comments (many discussing the premise of the question) have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on [meta], or in [chat]. Comments continuing discussion may be removed.

I’m voting to close this question because I think it should be on the Mathematics SE, but I didn't see that option in the "not within the scope" btton

Related.

I am a mathematician who has similarly been intentional about diversifying my research. My PhD training was in algebraic topology and category theory, very far on the "pure math" side of the spectrum. I can think of a few ways you can diversify your research, listed below. I'm going to ignore all the identity politics stuff, because I don't think it's relevant to the actual question of how to diversity your research and cast a wider net.

Do some applied math/stats research and choose application areas that stretch you.

I did research related to the opioid epidemic in Ohio (resulting in one publication and one preprint so far), and it connected me to a whole new world of Harm Reduction. As part of my statistical consulting for the organization Harm Reduction Ohio, I met many social workers, sociologists, epidemiologists, journalists, people who use(d) drugs and want to help others beat their addiction, etc. Now I give talks in Ohio and help connect other researchers into this topic.
Later, I did research related to policing and protests and have published two papers from that so far. Through this, I got better at time series models and learned about Hawkes processes from applied math. I also learned about different models of policing, the psychology of protests, collective trauma, etc. I've also done research about Ukraine (one paper in 2015 and one we started in 2021), to understand breakaway movements and civil wars. I learned a lot of game theory, and befriended Ukrainian journalists and sociologists.
Later, I coauthored a paper about wound-healing and diabetes, which taught me a lot about biology, pharmacology, and the lived experience of people with diabetes.
If you are interested in this kind of thing, you might enjoy the research coming out of QSIDE.

Teach courses that stretch you in the direction you want.

I developed and taught courses in statistical modeling, time series analysis, Bayesian stats, data mining, and big data. I also taught several computer science courses. In every course, I picked real-world datasets for my students to analyze. As a result, I learned about the world bank, racial disparities in median net worth in the USA, how climate change impacts vulnerable populations, racial disparities in criminal sentencing, and what factors predict for high salary, long life span, long-term happiness, etc. I remember learning about the John Henry effect from this. I'm working to develop a course that would teach statistics via social justice applications. It would be called "Statistics for Social Justice." There's already a book "Mathematics for Social Justice" and I look forward to reading it and teaching that material at some point.

Study the scholarship of teaching and learning.

After observing some trends in test scores in my early years teaching, I read up and learned about "stereotype threat." I then changed how I taught and reduced the impact of stereotype threat. Around the same time, I read up on fixed vs growth mindsets and again changed how I teach. These changes led to several publications in the scholarship of teaching and learning, e.g., a paper in PRIMUS, another in the Annual Review of Statistics, and a couple in the Journal of Humanistic Mathematics.

Intentionally read papers by authors from under-represented groups, that might be under-appreciated and under-cited.

I tend to believe that papers by mathematicians from developing countries tend to have a harder time getting published, achieve less visibility, and get less citations, than the same papers would get, if written by a mathematician in the USA. So, when I find such papers, I make sure to read them and to cite them if appropriate, because I feel many others in my field will have chosen not to read them.
Let me give you a concrete example. This past fall, I wrote a paper about coalgebras over comonads. As part of this, I wanted to do as full of a literature review as I could. I stumbled upon a paper by a mathematician from Burkina Faso. Despite being written around 2016, this paper had fewer than four citations, and the author had returned to Burkina Faso rather than getting a research professorship in Europe where he'd done his PhD. I might have said to myself "well, this paper is not highly cited, so it's probably not very important or everything in here was well-known." Instead, I read the paper fully and carefully, understood the main result, included a discussion of it in my paper (a bit of an advertisement to the reader that they should read his paper), and cited it.

Travel to research centers in developing countries, and establish collaborations with researchers you meet during such travels.

I travel every chance I get. Often, I reach out to universities as I go, and they invite me to give talks. For example, I taught a week-long course about model categories to a bunch of Moroccan PhD students in 2015. I gave talks in Argentina and Uruguay in 2018. I gave talks in the Middle East in 2022. My CV has a list. I've not counted but feel confident I've done this in at least 20 countries. It's led to some wonderful collaborations and lots of math I would have never learned about on my own. If you are interested in this kind of thing, you might consider visiting the AIMS centers in four African countries.

Go to seminars, colloquia, and conferences on topics you are interested in.

I go to the seminar in my main research area religiously, and every week I try to go to at least one other talk, whether in the colloquium, or a different seminar (e.g., topological data analysis applied to neuroscience data), or an interdisciplinary speaker series, of which we have several in central Ohio. Through these activities, I've found lots of new co-authors and expanded my research in lots of new directions. For example, that's how I met the professor who got me interested in the opioid epidemic. Some conferences that helped me a lot include a conference on big data at Wellesley in 2015 (where I met many statisticians and got my start in that direction), an applied math conference at ICERM in 2019 (which led to the policing and protests papers), and a QSIDE social justice math conference in 2021. There's a conference this summer on gerrymandering (specifically, the math you can use to detect unfair maps and to draw fair ones) that seems very interesting. Maybe you can attend.

The SOTL point seems weak to me. As someone who co-led a SOTL seminar at my school for a few years, I ultimately came to distrust that any of it made a valid difference. For example, Dweck's growth mindset theory keeps failing replications, but that shows no sign of impeding its prominence. Have read Steele's book on stereotype threat, that feels in a very similar category of too-good-to-believe effects.

@DanielR.Collins That's an argument in favor of more mathematicians getting involved with this kind of research, to clarify it and try to replicate it. For me, changing my teaching led to better outcomes for my students, plus it was interesting to read and think about.

I guess you are referring to Sore? I found his paper https://doi.org/10.1007/s40062-018-0210-x surprising and interesting, an explicit counterexample that reinforces just how big the gap is between algebras and coalgebras.

@GabrielC.Drummond-Cole Yes, he's the one! I ended up citing two or three of his papers, then randomly met him in Chicago last month.

Although there is little you can do about it, I did have to laugh a little bit regarding the irony of your last name! ;)

@Trunk I appreciate your comment. I'm not trying to boast. My issue is that I'm only aware of a small handful of people doing this kind of thing: Chad Topaz at QSIDE, my co-author Nancy Rodriguez, some applied mathematicians in California (all mentioned above). Expanding from pure math to applied stats during the pandemic meant I don't know that many people. If anyone reads this answer and is interested, I hope they email me, because I love to collaborate. Now, I know tons of people who have made their teaching more inclusive, but the question was about research. That's also why I (cont...)

(cont...) emphasized the resulting publications. The main issue is that many mathematicians viscerally oppose anything with the word "diversity." A comment to the question asserts that it's racist and sexist of the OP to identify himself as a white male and ask if he has research blindspots. An answer says "promotion of diversity based on sex or colour or other irrelevant factors is discriminatory." Another says "What is unimportant: What's currently known as DEI, that focuses on race, gender, sexual orientation, etc." A majority of mathematicians feel this way (cont...)

(cont...) and it's part of the reason so many black mathematicians feel pushed out of the field. In this ecosystem, I am lucky to be at a liberal arts college, where my efforts to broaden my horizons, make my pedagogy more inclusive, and diversify my research, are lauded. Plenty of mathematicians deride such efforts instead, though I hope to convey in the answer that interesting math and good publications can result from such efforts (as well as hoped-for benefits to society). I hope that, in time, I will learn of many others doing this kind of work, and can plug their efforts instead of mine.

@DavidWhite Replicate and clarify what exactly? It's not as if mathematicians who embrace a growth mindset in their lectures will be equipped to carry out a psychological study. As for anecdotes (and specifically your anecdote) I'd wager that a growth-mindset could make the class environment a kinder place. This is a good in its own right, but any effects on academic outcomes are residual or illusory.

@Enforce For one thing, I do not stipulate that there is any issue with Dweck's work. I think it's far more likely that mathematicians who viscerally disagree with growth mindsets and stereotype threat are suffering from confirmation bias when they see a claim they think they can latch onto in order to ignore this research. But, to answer your question, yes, I think math professors are equipped to carry out a math education study, e.g., one section of calc 2 gets an intervention about growth mindsets and the other gets more review time. Then we'd have more research on whether or not...

...growth mindsets are associated with better performance in math, and differentially so for learners from underrepresented groups. I think diligent teachers do need a working knowledge of psych and education, just like we need to be able to draw good pictures in calculus even if drawing was not part of our formal training. I'd love to see more mathematicians teaming up with researchers in psych and education to test what does and does not work in the classroom (statistically speaking). Mathematicians' clarity of thought could also help in the experimental design.

@DavidWhite There is a comment in this same thread pointing to several large scale studies across different cultures that all cast doubt on the efficacy of the growth mindset approach in improving academic results. This is not a rebuke of Dweck as a scientist, there could be a host of reasons why her theories have not been borne out. As far as confirmation bias is concerned, of course everyone is prone to it. Case in point, you want to see more studies on the efficacy of a growth mindset while ignoring the existing body of research which suggests that its effects...

are nil or minimal (even where Carol Dweck herself has been involved). I would be really interested to know what you hope to see out of these studies. Are you just curious to see whether it works or not?

@Enforce I'm not "ignoring the existing body of research." I'm saying I was completely unaware of ANY such research till the first comment above. And it's final exam season. I don't exactly have time to read a bunch of stuff right now. Hence, I do not stipulate that there is any issue, because I have not (yet) read these papers. But I will read them, when there's time. None of this has touched on stereotype threat, by the way. From Whistling Vivaldi, it seems plenty of studies reproduced the effect. I'm unaware of papers claiming Steele is wrong.

I don't know anything about stereotype threat, but from what you say it seems to be significant. By the way, it would take less time to read the linked blog article that the time you have taken in writing these replies.

As requested, I expand my comments to an answer. Much of what I will say applies to science in general, not just mathematics, but I will limit myself just to math in the answer.

Your main question is:

How does one become more widely read in mathematical research, beyond boundaries like languages one speaks/reads, one's culture, privileges, upbringing, etc.?

What limits your mathematical horizon has (almost) nothing to do with  "languages one speaks/reads, one's culture, privileges, upbringing" or "being a monoglot, Western, white, and a man."
Your main limitations come from your specialized mathematical education. In your case, as I know, your initial math specialty was combinatorial group theory and, later, theory of algebraic groups. This is your "zone of comfort," these are the areas where you have read and are able to read current research papers. At the same time, there is a good chance that you will be unable to read current research papers in, say, probability, PDEs, algebraic geometry, analytical number theory, etc. Very few mathematicians can currently move with ease from one area to another (very few of us can imitate Terry Tao in this regard). I will make suggestions based on what worked for me. The main rule is to move only to adjacent areas of math. In your case, moving from combinatorial to algebraic group theory is a perfect example. There are two ways to move (both learning and proving new results) from area A to the adjacent area B:
(a) In order to solve a problem in area A you realize that some tools from area B might be useful. Thus, you learn enough math from area B in order to achieve what you want.
(b) You learn (either from talking to somebody, or listening to a colloquium talk or reading a paper in Notices of AMS) of problems in area B that seem to be amenable to tools from area A that you are comfortable with. Then you either find a collaborator from area B or do some reading, or taking a graduate class if this is an option,  and, maybe, it will work out.
After that, maybe you move to area C adjacent to area B, etc. How much of math from the new area you have to learn in order to succeed, of course, depends on too many factors to give a more specific advice.
What never worked for me is learning a new area of math just for learning sake (and not for the lack of trying).

"Diversity is essential." Well, yes and no. What is important is diversity of mathematical knowledge and thinking. What is unimportant: What's currently known as DEI, that focuses on race, gender, sexual orientation, etc. With few exceptions, you do not need to go to another country to learn and do more math. (Up to a point: If you have a collaborator who works in a different country, this is another matter. If you want to learn, say, condensed math, you might benefit from going to Bonn.) What's detrimental when learning and doing math is thinking in terms of ethnic, socioeconomic, geographic background that you have.

Regarding

I could learn a new language or two. I could try reading from journals from places underrepresented so far in my research.

Learning languages is, of course, a nice thing to do, but it is primarily useful if you want to visit a country. Almost all first-rate math these days is written in English and, to  much lesser degree, French. (There are also some older first-rate papers written in Russian, they are all translated in English by now. Also, sometimes I have to read old papers written in German, but this happens very seldom.) As a native English-speaker, it should not be difficult for you to read math papers written in French, you may need to learn a bit of grammar and vocabulary, but that's all. Reading journals from other areas of math  will not be particularly useful for expanding your math horizon. Most likely, you will be simply unable to understand what and why the authors are doing. A better option is to go to colloquia, to read articles, for instance, from the Notices of AMS which are intended for general mathematical audience.

In case (which is what might be underlying your question) if you want to do some math related to various social issues. My suggestion is to proceed as in Part 1 of my answer. For instance, if you are familiar (from studying combinatorial group theory) with Rips complexes, you can consider reading papers on mathematics of gerrymandering, where Rips complexes appear as one of the tools.

If you have time and energy and are determined to do something socially good as a mathematician, the best thing, in my opinion you can do is guest lecturing in middle and/or high schools in the city you are in (no need to go far). Here is a story told by two of my Mexican mathematical friends: They decided to organize a "math circle" in one of the local high schools. To their great surprise, many students came, even ones who were not interested in math. When asked why, these students replied that it was the first time in their life, they had teachers who really cared about the subject they teach. Sigh...

Just wanted to add my support for this answer in addition to my upvote.

As someone who comes from an underrepresented minority group to this field, here is what helped me:

Keep lecture talks and seminars open to public on a known platform like youtube. Also promote it so that people know about it.

Publish notes online.

Make discord groups/ mathstodon channels were peopl can participate.

Make it so that people who are also not in the university you study at be able to involve themself and talk about research. It is a minority, but there are some people who are very very interested but unable to contribute to the public mathematics place due to not being affiliated with a university.

Keep a blog where ideas are discussed at a more informal level than books.

Those are some tips how to concentrate people who are interested into a single place. The next step would be bringing those who are interested actually into the physical places like the university you conduct research. That could be achieve very simply by offering scholarships. However, this is not an individual thing but rather an institutional thing.
In the case of Germany, I know very many students from countries like India and Nepal who are very interested in the subjects that they do, then came here cause this country funds them (about 1.3k euro per month).

@DanielHatton that sort of gate keeping policies is how you keep math unwelcoming to people who want to come new and outside of your culture

How is trying not to get sued into oblivion a "gate keeping polic[y]"?

An increasing number of research institutes and universities host their own content these days. If your university doesn't want its content on YouTube or similar for-profit platforms, you can push it in this direction.

@DanielHatton in mathematics, many many people post research-level notes on their websites, many mathematics research seminars are freely available on YouTube (etc), many people post their lecture notes online, many people use the arXiv. This is not posting university-own IP in the form of lecture materials, and then suing people who re-use it without permission. The other end of the spectrum is where people privately circulate documents for up to decades and results are cited and used by those in the inner circle, and mysteriously are never published or released.

If you wish to help the countries where math research is not as developed, single out a conference or two in such countries, and travel there to report your results. Talk to the people there, exchange ideas, facilitate collaboration.  Take note of talented students, advise them on how to get to graduate studies in the West.
My close friend, a prominent astrophysicist from MidWest, has long been cooperating with a couple of universities in the South-East Asia and a couple of universities in the Latin America.  There, he is co-supervising MS students.  Some of them later come to his department, to read for PhD or as visitors.  The guy is doing a great job, helping to create new scientific schools.   At the same time, he emphasises that his work has nothing to do with race, sex or other parameters irrelevant to science and knowledge.  The only kind of diversity that he respects is the diversity of ideas.  I gladly agree with him, because the promotion of diversity based on sex or colour or other irrelevant factors is discriminatory -- and therefore harmful.

Publish as open access. Otherwise you might exclude non-academics and people at institutions that cannot afford to subscribe to all the expensive journals.

Yes, or use arXiv, and make sure the last upload on arXiv matches the version of the paper that was accepted for publiction.


text:
Referencing if a YouTube video is the sole source
I have information that I found on YouTube, and when I searched for it on Google, I did not find it in any book, not even in an article, and nor in Google Scholar. Can I mention this information without a reference?
I am afraid that it will have a reference without my knowledge.

Did you search for background on the presenter? One could hope that a serious presenter has more substance than just that video. Some kind of credentials? Does the video implicitly refer to other work of the presenter? It might be possible to find more-primary sources...

Why not email the YouTube video creator and ask them where they got the fact from? If they don’t answer, that’s a good sign the fact is not trustworthy.

For what it's worth, wikipedia considers youtube an unreliable source and does not allow it to be used for citations at all.

In searching did you use "quotes"? Did you include the name of the YouTube presenter among the search terms? Did you paraphrase the information? Did you stop at the 2nd page of the results? Did you check your spelling? Did you search using only English terms? Any one of these things might have led you to a blind alley.

Credibility issues aside, one caveat with citing such online content is that it may not exist in its present form at a later date. For this reason it is good practice to submit the page to the internet archive first, whenever you intend to cite it, and either link to the archived link directly, or to both the canonical link and the archive one. (with youtube videos, this may be problematic, in which case, it may not be a good candidate for a citation anyway).

When citing, give a timestamp. I would hope that goes without saying, but I keep seeing people post links to hour-long youtube videos with no timestamp as if that's a valid cite.

We cite "personal communication" when someone told us something and there is nothing concrete to link to, a youtube video already looks way better than that!

Not to be a crank, but -- define "information". You might as well wonder if you should cite a rumor you heard at the barber shop, especially if this is the only source of the "information". Personally, I might be a bit embarrassed to have cited a YouTube video as my only source of information.

@DanRomik Some of my own research was presented at a high-level conference at a prestigious institution, and the video is on YouTube. I haven't written the paper yet, due to time constraints. I very much hope people care enough to cite my talk and aren't waiting for me to write it up!

If you have gathered some information from a YouTube video, and if there is no better source for it; then the YouTube video is what you should cite.
Now whether a YouTube video is a reliable source or not depends on a lot of details.
If, for example, the YouTube video is a recording of a reputable academic giving a scientific talk, then you can treat it with the same level of trust as you would treat it had you attended the talk in person (which should include "remain sceptical until you see the actual publication").
If, for example, you are discussing a piece of art and you have come across a YouTube video by the artist discussing their process of creation; or even just a relevant critic commenting on the art, this could be a great primary source.
If, for example, the YouTube video is about a mathematical result, and includes a sufficiently detailed explanation of the proof to follow and verify the proof, it is a reasonable source to assert that the result is true.
If the YouTube video is of a random person (or worse a known crank, liar, etc.) spouting off random stuff, you're better off ignoring the "information".

If it's a crank, and OP needs the information, they can say, the only source for this claim is <known crank/doubtful source>, so at this stage the information is unreliable. You still should evaluate any information you get. You are not responsible for the source's content, but for what you tell the reader to extract from it.

@CaptainEmacs Unless one is doing a meta-analysis of some kind of crankery, I definitely think one ought to ignore whatever they are saying. But these comments are probably the wrong place to get into a discussion about that.

Youtube can be a valuable source, even better than what you might find in print. For example, in discussing a particular printing press, I cited youtube videos demonstrating the use of that press. Worth a thousand words as they say.

@Arno Yes, of course. But if you try to use "knowledge" coming from somewhere else, you need to cite, even if you later discover it was questionable. Citing a crank is not a good idea without a good reason, but plagiarism is definitely worse.

There's also the category of non-crank videos that I'd classify as "backyard scientists" that record themselves performing experiments like "what happens if we pour molten aluminium on some Orbeez?" If you wanted to cite their work in an actual scientific paper, you'd have to cite the youtube video.

"If, for example, the YouTube video is about a mathematical result, and includes a sufficiently detailed explanation of the proof to follow and verify the proof, it is a reasonable source to assert that the result is true." I would even write down the proof with all details in the manuscript and cite the source.

@CaptainEmacs If the idea came from a known crank, you probably wouldn’t include it anyway, so no plagiarism would occur. Whenever we write, record, or present it’s generally understood that we’re operating within a narrowed scope of ideas. Not discussing a ‘relevant’ crank idea on the basis of scope is an implied norm in pretty much any venue that demands citations as far as I’m aware. If that weren’t the case, I’d be obligated to talk about quantum crystal resonance enhancing your chakras far, far too often.

@Greenstick I understand what you say, but let's take a more interesting case. There are a number of people who claim that history timelines at 1 millenium BC and earlier are off, possibly by several hundred years which would - they claim - resolve a number of inconsistencies. Assume this turns out to be true by improved archeological methods. Some of the earliest of these people were considered history cranks and did definitely cranky stuff in physics. Do they deserve citation? There are other examples, unfortunately, I only remember a vague outline, not the details and would have to dig.

@CaptainEmacs That's a good example. In my (entirely subjective) view, the resolution of inconsistencies is what makes such a theory worthy of consideration; we often say that a theory should be predictive but, more broadly, a theory should be explanatory and (subtext) not easily refuted by existing evidence. It sounds like what you're referencing has these characteristics. In my mind, a central goal of academic training is learning to take a more critical/rigorous view; sometimes that means someone branded a crank in academia is to most people outside an entirely sensible person.

I second everything in this answer, but I would added that many creators respond to comments on a YouTube video and contact information is often provided. It may be worth contacting the creator or even asking in comments if a more traditional publication with the information exists.

@Greenstick That would seem so, indeed. There is an element of groupthink in some fields which is resistant to change; and for an outsider it is not always obvious whether this is just well-founded skepticism or simply due to prevailing fashion (think Alfred Wegener). But the case I think about was clearly beyond the crank-line in celestial physics; it then becomes harder to give them credit for the non-conventional timeline. I still agree with you about consistency, and suggest that for the same person, each topic should at least be probed separately for being worthy of consideration.

@CaptainEmacs I can get behind that : )

If you’re going to use the fact in your academic writing, then yes, you have to cite it.
The real question is, is the source credible enough that it would be acceptable for you to trust that the fact it claims is a true one? The answer to that does not depend on the fact that the source is a YouTube video: some YouTube videos (say, of a lecture by a Nobel prize winner) will have as much authority as any academic text, and conversely some academic texts (e.g., by a well-known Holocaust denier or anti-vaxxer) will have no credibility even though they are written sources.
When you rely on a fact that someone else claims, you are tying your reputation to theirs. If they are spouting nonsense, it will make you look bad, unless you explicitly warn your readers that the source you are using is potentially unreliable.
But, regardless, you have to cite your source.

Agreed but I debate one point. Whether someone is a horrible person - holocaust denier, anti-feminist, sexist, racist, whatelse - (ideally) has zero effect on their scientific reputability, unless it conflicts with their field. A lot of very renowned scientist did awful things and were awful in general, but we don't debate their scientific claims. Of course, I wouldn't trust an essay coming from Newton about how to treat my wife, but that doesn't mean that Newtonian mechanics is invalid.

@Neinstein The point is that a Holocaust denier's textbook about the Holocaust is not a reliable source of information, even thought it is (at least arguably) an "academic text". Likewise for an anti-vaxxer's textbook about vaccines.

@kaya3 sure, just wanted to point out that important implication, which was not that apparent in the asnwer.

@Neinstein yes, kaya3’s interpretation was what I meant. In practice though, the sort of horrible people I had in mind usually but not always tend to have horrible views that focus on their main research area or at least are adjacent to it.

@DanRomik Ok, agreed. I'll remove my comment as it branches out too much and clutters the space.

You still need to avoid a charge of plagiarism. If the information is "common knowledge" then you don't need a citation, but otherwise it needs to be attributed to a source. Perhaps the YouTube video had a presenter or something that identifies where the information comes from.
But, as a last resort, credit the video along with a date that you accessed it. The date is needed since things change. For video citations it is common to give the time mark within the video where the information occurs

You can cite Youtube videos, just include some explanation.  The first reference of this PRL article about rigid-body rotations is a Youtube video so it's certainly been done before in a serious scientific article.

You have a link through your local library resource; those at other institutions won't be able to use it.

This is true, but I'm not sure how comparable it is to the scenario in the question. This paper cites that particular video (Ref. [1]) as providing an intuitive understanding of an easily repeatable phenomenon, as well as other videos evidencing related phenomena, but also cites some more conventional sources. In addition, the Youtube channel hosting the first video is probably more respected than most.

Just as you wouldn't cite Wikipedia directly, you also shouldn't cite YouTube directly.  The issue is that both sites present user-generated content immediately, with no peer review or fact checking process (Wikipedia arguably can have these features after the fact via community editors, but it's difficult to know when/if information has been verified).  YouTube videos can't easily have inline citations, but academic videos often do cite sources in the description.  If they don't, and especially if the topic discussed is something novel, I would consider the video (and maybe the uploader) to be an unreliable source.  Consider the video to be like a clue or rumor -- there might be something to it, but you need to dig up more information; as others have said, contacting the uploader (or presenter if it was uploaded by proxy) would be a good first step.
Another thing to keep in mind is that anyone can create a YouTube channel with any unique name; if you're going to consider the video credible based on the credibility of the presenter alone (arguably reasonable but somewhat dubious as a credible presenter would have the same content published in a more rigorous context as well), be sure to verify that the uploader and presenter are actually the same entity.


text:
For the average pure math US PhD program, what are essential topics after basics topics of complex analysis, abstract algebra and topology?
Edit: Right so I forgot this really big thing. My question is actually under this framework: Why are US PhDs different from European PhDs?

After the basics of topology, linear algebra, abstract algebra, elementary analysis and complex analysis, what are essential topics for the average pure math US PhD program?

If there are none, then how do you know? If there are, what are they?

Guess if there are: They are the basics of the following topics:

Algebraic topology, such as Part II of Munkres Topology

Algebraic geometry and commutative algebra, such as the rest of the second half of Artin Algebra

More group theory, such as the rest of the first half of Artin Algebra

The algebra topics in Dummit Foote Abstract Algebra that are not in Artin Algebra

Elementary differential geometry, such as Tu Manifolds

Guess as to what isn't included:

basics of complex geometry

My context: (Feel free to just ignore this bottom part if it makes my post too long.)

I was recently rejected for a pure math PhD program in Country A, where I live, for not having a strong enough background in "essential topics".

1a. The professor said I was not ready for a PhD or even an MPhil in pure math. I asked if he meant for Country A (I was very careful to not use the words "only" or "just"). He claimed that it was for all "reasonable" universities.

1b. He claimed that the average first year Country A PhD student (before starting the programme) in topology or geometry would know the basics of algebraic topology, complex geometry, Riemannian geometry and more algebra than the elementary abstract algebra. Some of the specific concepts are Gauss-Bonnet, branched coverings, Kähler manifolds, Poincaré duality, Euler characteristic, etc. Also, there's stuff about Riemann surfaces required (eg Mittag-Leffler and Riemann-Roch. I'm guessing also Abel-Jacobi, Riemann-Hurwitz, Poincare-Hopf and the list of 2-name concepts goes on.)

I am wondering whether my background is more similar to an applicant to a US-style grad school than to a European-style grad school.

2a. Consider Johns Hopkins University. Its maths phd requirements are the same as the "straight phd" programs in the top 3 universities in Country B, where I'm from and where I got my bachelor's and master's degrees in (unfortunately applied) math. (These 3 universities have "regular phd" which require master's or equivalent and "straight phd" which require only bachelor's or equivalent.) Both JHU and top 3 universities in Country B are actually even less than what I put as my 'guess' above.

2b. But anyway going back to JHU, it even says 'Nevertheless, the department does admit very promising students whose preparation falls a little short of the above model.' In other words, JHU isn't even as strict about these elementary requirements, but this Country A university is extremely strict about these highly advanced requirements.

2c. I just find it very hard to believe that my rejection from this Country A university isn't related to these US vs European questions that I've asked before. I would like to think that my rejection is that European universities simply require more. I don't quite have a chance there without further studies, but I do have a chance in the US. (And worst comes to worst, there's always Country B.)

You can see the previous revisions for more details.

Gonna copy some comments into the post:

If you're fact-checking this professor, what he said is kind of stupid. If you're trying to assess your preparation for whatever an "average" program is, you need to check with them. – Elizabeth Henning Dec 17 '18 at 16:41

(I think this comment is about the "reasonable" thing.)

I honestly don't understand what you're after here. All programs will say they want an undergraduate major in math or a related field. (...) A mid-ranked school will probably expect basic abstract algebra and calculus with proofs. (...) – Elizabeth Henning Dec 19 '18 at 16:57

(...) It is very likely true that more than half require no more than the GRE topics, if that is useful information. – Elizabeth Henning Dec 20 '18 at 4:52

Why should this question stay open when your last one was closed for being opinion-based?

This question was voted to be closed as primarily opinion-based. I think (2) is opinion-based, but I think (1) and the original question is not. (This comment is not in response to Bryan Krause)

The Math Subject test is not at all a test of preparedness for graduate study.  It's a test of preparedness to TA calculus.  Part of the reason is that US graduate programs vary greatly by level.  The average person earning a PhD in my mathematics program knows less mathematics than the average first-year graduate student at Princeton.

@BryanKrause This question is not really about the GRE. This question concerns topics which are in the GRE. Using the term "GRE" therefore is shorthand the topics. Therefore, the question can be rephrased "For pure math PhD programs, what are essential topics after (insert the topics covered in the GRE without referencing the GRE)?" Thank you! Your comment has actually led me to think my question is off-topic here and should be asked on math instead. Should it?

The subject GRE is a standardized test that can be optionally used by some institutions to guide their decisions about admission for graduate school. It isn't marketed or intended to test whether people "know math" or not, and I think most university math departments would prefer to set their own guidelines for their students rather than rely on the ETS. This question, and your last one, both seem to be arguing that the Math GRE does not contain the assessments of mathematics that you find relevant. And then your question seems to be "is there a right answer and if so am I right or are they?"

I certainly think if you don't mean to talk about the GRE you should not mention it.

@AlexanderWoo Thank you! 1. When you say "US graduate programs", do you include or exclude master programs? That's an important distinction in a later edit in (2). 2. When you say math, do you mean pure math?

My statement about "US graduate programs" is true whether Masters programs are included or not.  I do mean pure math.

@BryanKrause By graduate school do you mean PhD only or both PhD and master? I think you mean both PhD and master and if so, I think references to the GRE end here. Update: I think AlexanderWoo has answered (2), so I think we can end references to the GRE here.

@AlexanderWoo I think you have answered (2). Thank you!

The question is becoming a moving target. Let it settle, please.

@AlexanderWoo More accurately, the Subject GRE is a test of unpreparedness for both grad school and TAing calculus, in the sense that high scores don't mean all that much but low scores do.

@BryanKrause I edited the question.

@ElizabethHenning I edited the question. By the way, in the previous edit, the question isn't exactly about the GRE. GRE is just a shortcut for "elementary complex analysis, elementary abstract algebra, elementary analysis and elementary topology"

To answer your question, you really need to check with the US programs you're considering. Many, if not most, will describe their preparation requirements on their website. Nearly all will require the General and Subject GRE.

@ElizabethHenning I know, but what more than the subject GRE topics are required is my question?

It varies. You need to check with the program.

@ElizabethHenning I didn't ask for every. I asked for the average. Same response?

If you're fact-checking this professor, what he said is kind of stupid. If you're trying to assess your preparation for whatever an "average" program is, you need to check with them.

@ElizabethHenning Oh, thank you hahaha. For the "kind of stupid", I made a mistake. I should have said topology or geometry. I edited the post. Now it's not kind of stupid? And what if I define "average" in terms of uniform probability? List all the universities offering pure math PhD in the US. Pick one of them uniformly randomly.

I honestly don't understand what you're after here. All programs will say they want an undergraduate major in math or a related field. But there are a few hundred pure math PhD programs in the US and their expectations are varied enough that even schools at roughly the same "ranking" will have different expectations. A mid-ranked school will probably expect basic abstract algebra and calculus with proofs. But it's ridiculous to make generalizations because each program decides what they expect and they often post this on their website.

They might also post their written qualifying exams, which give some indication of what they want you to know early on.

@ElizabethHenning The qualifying exams are for second year PhD students and therefore cover topics in first year PhD? My question is about required background of PhD applicants. My intention is to be properly informed, so I can make a decision about whether I should continue studying for Country A or apply to US style PhDs. If Country A is like the US, then I'll pick Country A because I live in Country A, and there's nothing to think about. If Country is unlike the US, then I'll think about whether or not I apply to US or US style pure math PhDs

@ElizabethHenning Of course, we are talking about percentages or probabilities. If more than half of US pure math PhDs require only the topics covered in the GRE, then I have a decision to make. If we change more than half to less than 10%, then I don't have a decision to make. About variation, I shouldn't allow for many choices. Let us be binary: the university requires either only the GRE topics (1) or requires more than that (0). Maybe, I was not so clear about that. In this binary choice, is the average closer to 1 or to 0, in your best estimate?

Yes written exams are taken early in the PhD program, but since a main goal of the admissions process is to identify students who will pass them, they are an indicator of what preparation is required. It is very likely true that more than half require no more than the GRE topics, if that is useful information.

I tried to clean up the text; it had drifted a lot and was becoming a small novel. Feel free to edit my changes if I botched anything, but I strongly suggest you try to keep it concise; all the details about "University A" made this straightforward question very confusing.

@cag51 thanks! i edited further just now

@ElizabethHenning thanks. i edited question more. i removed references to GRE. I included mention of john hopkins university and those US vs European questions i had asked years prior to OP. i can't believe i didn't include the US vs European thingy. i think my question just seems pretty confusing otherwise. anyway, please help further.

These questions are essentially unanswerable, because universities in the US are quite diverse in their expectations.
The average person earning a PhD from the department I work at knows less mathematics (and is certainly less capable of doing research) than the average first year graduate student at Princeton.
The average person earning a BA/BS with a major from the department I work at knows less mathematics than the average junior major where I was an undergraduate.
The average person entering my university as an undergraduate knows less mathematics than I did when I was 14 (and starting high school).
Of course these are averages and there are exceptions.
Do you want the answers for my university, or for, say, the University of Minnesota, or for Princeton?
(And, as for the ETS, they mostly are catering to be accurate for the middle of the normal distribution, which means universities like mine, because that's where the people are.)

Alexander Woo, do you disagree with Buffy? Thank you!

I edited my question. Is your answer mostly the same? Also, how about non-ivy leagues?

Yes; my answer is still the same.  There is no such thing as an "average" US university.  By the standards of your Professor A, my university is probably not a "reasonable university".  But "unreasonable universities" exist and are indeed quite common.

Bu the standards of Professor A is just about me specifically. Does that mean you think I know enough pure math to apply for a pure math PhD in your university?

What if I define "average" in terms of uniform probability? List all the universities offering pure math PhD in the US. Pick one of them uniformly randomly.

@JackBauer: How am I supposed to give you an answer then?  Say that 19.6% of the programs expect you to know commutative algebra?  And what do you mean by "expect"?  Certainly at a highly selective program, those who know more mathematics will have an advantage in admissions, but someone who shows exceptional promise in other ways might be treated differently.

@JackBauer: Ed Witten's undergraduate degree was in history.  Presumably, the Princeton physics faculty saw some reason to admit him to their PhD program despite his having studied far less physics than usual for a physics PhD student.  Clearly they were right.

Alexander Woo: Comment 1: okay I mean average as in most. How about now? Why don't we have the answer as "just the topics in the GRE" simply because the GRE is meant for applicants to average pure math PhD programs in the US, or is it not? if more than half of said PhD programs required algebraic geometry, then we expect the GRE to include algebraic geometry, don't we? COmment 2: Thank you!

The only thing more than half (measured by pure count of programs, not weighted by size, which varies a LOT) of US PhD programs in pure maths require of almost every admitted student is demonstrated ability to write a complete and correct proof of a non-trivial fact.  Almost every program will expect more than this, but the expectations are likely so varied that fewer than half the programs require any specific topic (and some programs will simply expect all of their students to have more mathematical background without specifically expecting any specific topics).

Alexander Woo, i edited question more. i removed references to GRE. I included mention of john hopkins university and those US vs European questions i had asked years prior to OP. i can't believe i didn't include the US vs European thingy. i think my question just seems pretty confusing otherwise. anyway, please help further.

If you want to know why ETS does something, ask them. The test needs to be broad enough so that most UG curricula are well covered, but not much broader than that. It tries not to disadvantage anyone in its coverage, though most students will find questions there that are about things they haven't studied. And you can still do extremely well even leaving some questions unanswered.
On the other hand, anything that you study will give you some additional mathematical background before you start to dive deep into a research area. You probably don't have time for all of them, so just choose something that seems interesting. If you already have a research interest, you could start there.
I'll note that the last time that it was possible for a single person to know all of mathematics was early in the 20th century. It has expanded too much since for it to still be possible as it once was.

Hello Buffy. Thank you for your wisdom and empathy! I had edited and further edited to hopefully clarify. Some edits include my assumption that the GRE is for PhD only and not for both master and PhD and the phrase "first courses". Does your answer change? (The professor I spoke to 12 hours ago suggested some textbooks.. My question is supposed to answer my real question on whether or not the average undergraduate or the average pure math master graduate would know about these textbooks.)

Most UG programs cover the basics and one or two additional things. I studied PDE, for example, as an UG (half a century ago). But measure theory was an early grad course. I'm not sure that ETS makes a big distinction between MS and PhD. Many students see the MS as just a stepping stone in any case.

On the GRE (my first edit): But ETS indeed expects some students to enroll in master (which is why its topics are what they are)? If ETS expects only PhD, then this sub-question might be opinion-based, but I think they should include algebraic topology instead of a lot of calculus? On the first courses (my second edit): Following up please! Every pure math PhD program would expect students to know, for example, algebraic topology right? I think this question is different from "Does every bachelor or master program in pure math cover algebraic topology?"

I studied algebraic topology as a grad student. In fact my first serious topology course was in grad school. The UG version was a pale shadow. Nice historical emphasis (Koenigsberg Bridges, etc) but not very deep. But Analysis (real and maybe complex) is pretty fundamental and many UG programs are, I suspect, pretty heavy in that.

Hello Buffy, I already suspected that not all bachelor programs in pure math will include algebraic topology (I am surprised about general topology), but my question is if pure math PhD programs in the US will expect its applicants to know algebraic topology. Probably your answer is no because US PhDs expect only bachelor, but please confirm if I guessed correctly.

Correct. Note that a doctorate in the US normally includes quite a lot of coursework. It is probably not like UK where the UG is more concentrated, so more maths are covered.

Thank you, Buffy! I just realized my question might be wrong. New question: Is algebraic topology expected of the average pure math US PhD program under the supervision of a professor researching in topology or geometry? I hope I included all the relevant adjectives, and I hope this may be the last follow up.

Buffy to clarify, your answer to "After the basics of topology, abstract algebra, linear algebra, analysis and number theory, what are essential topics for any pure math PhD program?" for the average US pure math PhD program is "None" ?

"Is algebraic topology expected of the average pure math US PhD program under the supervision of a professor researching in topology or geometry?" Probably not if the department is weak in algebraic topology and the professor is in set-theoretic topology. I think you're putting too much focus on specific subjects. If you're simply interested in admission (and not trying to conduct a survey of existing or exiting students from graduate programs), then future potential (can you pass the qualifying exams and complete a dissertation?) and past achievement (e.g. top 20 Putnam?) play a huge role.

@DaveLRenfro Thank you. It is not me. Country A seems to be leaning more towards European than US based on my interviews with professors even though the university websites look like they should be kind of US. One was surprised I couldn't describe what a manifold is, though I gave the correct definition of both manifolds and fundamental groups (I was unfortunately NOT asked to the describe a fundamental group).

@DaveLRenfro Gee, if manifolds are so important, you'd think the GRE would include it, so we wouldn't waste time. He said first year PhD students in geometry would know what a manifold is, and this applies to all "reasonable" universities on earth. Would this be strange if this was the average US math PhD program but completely normal for the average European math PhD program?

Buffy, I edited my question. Is your answer mostly the same?

@JackBauer stop changing the goal posts... You should have read about how to ask good questions....

@SolarMike How else can I get my question reopened, or do you mean I should instead ask a new question? That would risk a duplicate. So either way I'm screwed?

Buffy and @SolarMike , i edited question more. i removed references to GRE. I included mention of john hopkins university and those US vs European questions i had asked years prior to OP. i can't believe i didn't include the US vs European thingy. i think my question just seems pretty confusing otherwise. anyway, please help further.


text:
Disadvantages to publishing outside field of projected study?
As a recent B.S. recipient interested in pursuing a PhD in molecular biology in the future, are there any disadvantages to publishing outside of that field? I have a strong background in computer science and have been approached separately by a public health professor and a linguistics professor who are interested in collaborating on their respective projects and perhaps working towards a paper over the summer.
While I enjoy solving computational problems, I'm not sure whether this is a good idea professionally. Does publishing papers in other fields take away from the research I've already done?  I've so far only contributed to two papers in my "field of expertise".
What are the downsides to collaborating on a paper outside of one's field? Is this frowned upon by the academic community/graduate admission programs?
Related question -- advantages to publishing outside of primary research field

Although I cannot speak directly to the field of computer sciences, I can say that in social sciences any peer reviewed publications would be viewed positively on an applicant's CV. At this stage in your career this shows you're motivated, capable of contributing to academic work, and have some familiarity with the writing process. The fact is that many undergraduates may not have access to opportunities to publish in the field they hope to enter, but instead seek to gain research experience and publication opportunities with faculty who are willing to work with and train them. Many academic skills, such as writing articles, are transferable between disciplines, and working with these professors may provide you with strong reference letters when you do apply to graduate school. So long as this opportunity is not coming at the expense of something more directly related to your major, I think it could be a very valuable experience which could positively impact (or at least will not hurt) your future applications.
As a final note, be sure that you are able to explain the project and your role in the project in future application materials and interviews. If this is outside your field of study, don't be shy about asking the professors questions so that you have a solid grasp of the project's purpose. The only time I have seen undergraduate publications backfire is when the person is unable to explain the work, and it becomes clear they did nothing but data entry.

I completely agree with this advice.  The only thing I would add is to never something that might later embarrass you.  There is home for every paper, but some papers are just bad and should remain homeless.

Although I cannot speak directly to the field of computer sciences, I can say that in social sciences any peer reviewed publications would be viewed positively on an applicant's CV. — This is also true in computer science.

... and also in mathematics. They probably won't help as much as a paper in the field would, but they certainly shouldn't hurt.

Interdisciplinarity is a positive benefit in the sciences, collaborations have the capacity to be more than just the sum of their parts.  The thing that is required is the ability for e.g. molecular biologists and computer scientists to have a common frame of reference to communicate effectively, so as a molecular biologist with some computational experience, that is extremely valuable in facilitating collaborations.  I'd say there is no disadvantage to this, you are demonstrating a really useful skill for molecular biology.