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Minimal supervision in Phd
I started my PhD three months ago in the UK. I am a biologist, but my project lies more in the synthetic chemistry field and all the experiments I have to do are completely new for me.
My problem is that at the moment, I am alone in my group because everyone else have finished their PhD/Masters/Postdoc and left, and I have no one to ask when I am stuck with something. My supervisor tells me to do experiments that I have never heard of, gives me a very general and brief description and that's it! No protocols, no one to show me at least for the first time. In the labs where I have worked in the past it was completely different - PhD students have someone to assist them (a Postdoc or a technician) and they don't feel lost searching on the internet for the right protocol or method to follow.
I want to ask you if this is normal for a PhD student? Or is it how PhDs are in the UK?  I know that we have to work with minimal supervision, but is this normal? How am I supposed to start a new protocol involving sensitive instrument usage (like HPLC) with no one to show me?
Thank you in advance for your answers!

"How am I supposed to start a new protocol involving sensitive instrument usage (like HPLC) with noone to show me?" The sensitive instrument will probably not just lie around there. There will be someone responsible for it. Maybe you could ask that person.

@Trilarion really?

@SSimon generally, yes, in UK universities I'd expect there to be technicians responsible for this stuff.

I can say that this experience is very familiar to me throughout most of my UK PhD. You are expected to be more or less totally independent, and you'll spend a lot of time feeling both overwhelmed and/or lost. However, the red flag is that you're only 3 months in. Your advisor should be more involved in guidance at this early stage, holding your hand a little until you are developed enough.

@EnergyNumbers we dont have technician for HPLC, I think it is waste of money. dont you think? I mean every student in field of chemistry, NEED to learn to utilise those intrument by themselve.

@SSimon I think a technician or two can be a fantastic investment for a department. I've seen how a good technician brings out the best work in staff and students, ensures that we get the best value out of our equipment, and can be at the centre of the department's informal communications network.

@EnergyNumbers yes I agree, I wish that is reality, that departments or faculties actually hire them

It is not normal in UK biology PhDs for a student to be completely abandoned with no help; particularly not at the beginning. As you become more confident you will be able to take on new areas and new techniques with minimal advice, but I would never expect a student to dive into complicated, expensive and potentially dangerous experiments without so much as a protocol.
I would do the following:

Ask your supervisor straight up if he knows where a protocol might be found for the experiment, or whether he sees designing the protocol as part of the thesis.
Find others in your institution, but outside your lab that might be able to help.
Find an online protocol or protcol in a text book and specifically ask your supervisor if he thinks it is a good protocol and if he knows of anyone who can go through it with you.
If all else fails, a UK institution will almost always assign you a second supervisor or advisor: go speak to them.

Leave and find another advisor! Although you are being trained to be an independent thinker, you cannot become that alone nor stay that alone. Find a lab that gives you a gut-feeling that you will have the support and challenge to better yourself.

Switching supervisor is a radical option. But sometimes it’s necessary.

I agree with @Phil ASAP switch advisor, evil root is better to cut on beginning

@KonradRudolph Yeah, I wish I had switched. I had an advisor that would like you build skyscrapers on air if you wanted... He did not challenge your thinking or teach you how to think and break down problems like a researcher. If you are not at the stage to work highly alone or do not have a solid-researcher-mentality just yet or have not done guided research; and now you are confused... It is a mistake to try to fight your way through. You will suffer for nothing.

@Phil omg, I heard this story several times, it is such a usual occurrence. why?

My guess is you have professors smart in terms of books knowledge and getting enough done to continuously get grants; but many not adept in th psychology of training people outside technical knowledge. Like, inspiring student, studying a student and helping him/her see and overcome weakness, inspiring confidence, motivating, adapting to the individual needs, growing student technically and non-technically... I see this in industry too. People with those skills are less promoted than people who can bring in money to institute/companies.

A PhD degree is to say that you have earned the recognition of being an independent researcher in the field. From what I understand, the whole point of the PhD journey is to train to become a fully independent researcher.  Edit: by “independent” I don’t means “alone” as one of the commenters have stated.
Part of that training should come from your supervisor. Of course he or she cannot teach you everything. But I would expect him/her to show you how to use specific equipment or what the best practices are when conducting experiments. Firstly this is to prevent accidents from happening which may damage expensive equipment (trust me, this is in the supervisor’s own interest) and more importantly to protect you from injury.
Secondly, a PhD student is expected to tackle a completely new problem. There may not even be any “standard experiments” that would help solve your problem. It is thus expected that new problems need innovative methods or experiments to solve.
At this point even your supervisor would not know exactly what to do because no one has even looked at the problem before. However, he or she has (hopefully) been in the field long enough to guide you if you get stuck attempting something that has never been done before.

Thank you all for your help. I will talk to my supervisor about my concerns.

I disagree: “independent” ≠ “alone”. Most (all?) of the successful PhDs that I know constantly got advice during their PhD, either from their supervisor or from peers. Not getting any of that is a huge problem. The same is true, incidentally, for postdocs (and, gasp, PIs), if to a lesser extent.

@KonradRudolph yes independent does not mean alone. I have edited my answer in the main text. Thanks for the clarification!

@HarishVallabhapurapu also it means that you will NOT be alone after you finish PhD. that is hard to grasp to a lot of students and researchers, they need strong network and support from the previous lab.

It sounds like you should go to your supervisor and ask for guidance.  I experienced something similar during my dissertation work.  Eventually, I got the guidance I needed.  Near the end I felt like I had to take more of a leadership role.  That was especially true during the planning stages of my final defense.  At that point in time I was having to work more like a project manager, corralling my committee members and finalizing my work.  Good luck!

In 25 years of being a post-doc' I can assure you that there is no 'normal'. The university, the department and the supervisors are all different and with them the level of useful support you will receive. It's not uncommon to find the supervisor has no hands-on experience with the techniques you need to learn, so your best resource is often technicians or fellow students. Sadly, you seem alone, and in this instance, as with the rest of your PhD, you need to dig deep and read the literature, search online, and even ask in other departments if they may have the expertise you need. It's normal to have 2 supervisors. what does the other have to say? Don't withdraw from the problem, as it will eat you up, instead, be vocal and plan a meeting with your supervisor. Send a list of issues you want to resolve at the meeting, but don't go expecting to be given the solutions on a plate. Do explain what you've done to try an progress and what is your current level of understanding so you can discuss the best way forward. Try this approach for a couple of months and see if you start to feel better.
Don't worry. A Phd will be the hardest thing you have ever done so far. No student knows what the heck is going on for the first year, but by the second year you will be bringing in useful data, and by the third you will know more about your subject than your supervisor, which in your case sounds like it won't be hard!
A Phd is meant to be extremely taxing otherwise everyone would have one. You will become exceptionally resourceful and self-teaching, two assets far more useful than the PhD itself in your later career, and indeed life in general.
It will get better, I promise, but it's up to you to make it happen, and to organise your supervisor so in the end you are controlling them.
I've known around 150 PhD students during my career, most get through, a few drop out, and a tiny minority fail. It's always the most independently minded that have the easier time as they make it work in spite of the supervisors.
Above all do not expect them to show you much, or help you directly; but they should point you in the right direction, and respond to requests for meetings.
It's a crazy time and a quantum leap up from your graduate level experience so to ease the stress make friends with other PhD's wherever they hang out and support each other as It'll make you feel less isolated. Dissing supervisors is very therapeutic!
Don't give up, it'll be OK, we all had a really hard time for the first 6 months or so.
My only caveat is if after 9 -12  months you are still lost, figure out of it's you, your supervisor, or both and look long and hard about if you can still make it work. Also, you must be interested in what you are studying to be happy. If it's a total chore, then you need to ask yourself a whole bunch of difficult questions.  If you are absolutely convinced its not for you, then bail out rather than leaving it for another year, as by then you may be as much as 50% of the way to a PhD and jacking it in at that stage will be something you may regret many, many years later when you as old as me.
Finally, enjoy the environment, people and opportunities. Don't just treat it as a means to a few letters, but as a period in life that very, very few people are lucky enough to experience.
Good luck to you, dig deep and be magnificent. :-)

25 year of postdoc? Really? What is your role title?

Retired! Ex Senior research fellow

Wow! I think only in the UK one can be a postdoc for so long. A postdoc is usually a temporary position, for 1/2 years before a permanent or tenure track position.

It's very unusual, and now nearly impossible in the current climate. I was lucky, well supported by a good research group.  I enjoyed full time  hands-on experimentation, something that's not possible in the normal career route into lectureship and with it, effective removal from the lab'. The downside is of course insecurity,  since employment is a series of short-term contracts with absolutely no guarantee of continuity if proposals are unsuccessful. I do feel career paths for pure researchers in academia are important, since we amass valuable practical experience which is often ephemeral.

I want to ask you if this is normal for a PhD?

It's not too uncommon, but it's also not the norm.
Regardless, the real issue's whether or not it's right for you.

How am I supposed to start a new protocol involving sensitive instrument usage (like HPLC) with noone to show me?

This is one of those things that you should be able to do by yourself, but, yeah, it's often a lot more pleasant if someone gives you a walk-through first.
To answer this question as-asked:

You probably know the basics already.

Normally, you'd want to research how these devices work in greater detail before using them.  This can provide important perspective when designing experiments and interpreting data.

Once you're familiar with how a hypothetical HPLC device might operate, then it's time to figure out how the specific one on your lab bench works.  These tend to be pretty expensive devices, so they should have thorough documentation to read through.

If you know anyone else doing work with them, you can look over their shoulder or/and ask them questions.

Moving forward
I'd suggest thinking out your life goals and how you're planning on tackling them.
Getting a PhD is often described as learning to function independently.  But there's a whole universe of variance in that vague point.  At one extreme, many PhD's essentially serve as high-end lab techs in 9-to-5-type jobs.  At the other extreme, some PhD's go on to make weird stuff no one's ever heard of.  Exactly how independent you need to be really depends on the path that you plan to take.
Without looking up the figures (which I'd imagine would be hard to get), I'd speculate that most PhD's end up working in high-end-lab-tech jobs.  They're nice, popular jobs because:

They're predictably about what you've learned to do.

They're typically 9-to-5 jobs, and frequently pretty stable.

You don't have to stress so much about general concerns, but rather can focus specifically on some specific facet of research.

If that's the sort of position that you'd like to go into, then presumably you'd want a grad school experience that'd align with it.  So, you might want to seek an appropriate supervisor.
Sounds like a mismatch, rather than right-vs.-wrong
This sort of supervision is right for some students.  This is, your advisor's not necessarily wrong, nor are you necessarily wrong.
Rather, I'd suggest seeing this as a mismatch, for which you might have two general approaches to solving:

You might talk it out with your advisor to let them know your needs.  If this works, then awesome!  Easy fix.

If your advisor isn't a good match for your intended approach, then ideally they'll be mature enough to acknowledge the mismatch and help you find an advisor who'd be a better match for you.

I agree fully with all the answers pointing out that your supervisor ought to give you more guidance. A PhD studentship can only work if the supervisor is giving you the guidance you need.
We cannot fully judge your supervisor on the bits of information that you have given. I would only like to add that not taking students by the hand does not necessarily imply that he or she is not a good supervisor. It is up to you to demand the guidance you need and you certainly shouldn't feel bad about it! (Like I said above, providing guidance is probably the single most important part of a PhD studentship.) By asking for guidance, letting him or her know what you can do by yourself and independently and what not, you are making this professor–student relationship work.


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Overcoming a profound basis of underperformance as a freshman
Relevant background:
Freshman at large public research university, pursuing Engineering/Applied Mathematics dual major.
Labeled 'gifted' after neuropsych evaluation in Elementary school. Processing speed about 2 standard deviations lower than other subscores, this may have contributed to learning maladaptations that have become evident in University.
K-8: Programs where I got A's and B's without actually making the effort to learn the material. Never a straight-A student, usually got discouraged when my (also 'gifted') peers did stuff more quickly than I could, decided I didn't like what I wasn't good at.
I got a few A's but mostly B's in high school math, where I completed up to Multivariable Calculus, in which I received a B.
This semester I took Linear Algebra & Differential Equations (1 course) which comprised 2 midterms and a final. I got an A on the LA midterm, didn't have much trouble with that. However I got a D on the DE midterm; it turns out I can't integrate worth a damn anymore. My final grade is TBD, but I'm guessing I will earn a B in the class.
The cruel irony is that by the time I realized I enjoyed math I also handicapped myself to learning it.
Question:
Has anyone here been in or witnessed an applicable situation? It feels as if I've dug myself a hole that may not be possible to get out of regarding future math classes.

I don't understand your question. Plenty of people get D's in math classes if that's what you're asking.

google impostor syndrome, because it sounds to me like you have it. 2) go to the library, find a nice thick calculus textbook with plenty of questions and start working through them. IMO, practice really does make perfect in the case of rote maths.

From your question, I can't tell anything about your study habits. I think that is relevant here...

Further to @astronat's suggestion, I recommend Thomas's Calculus.

Wait, how do you get a B in Differential Equations?  I thought you described that class as pretty much of a disaster? // And what's your question, anyway?

The cruel irony is that by the time I realised I enjoyed math I also handicapped myself to learning it.

I've had a somewhat similar experience so I'm going to describe it in a few words. Perhaps it will be helpful for you.
When in school I was labelled "gifted" as well. I was a member of a club for children with math aptitude and participated in competitions but never put in any effort. Eventually, after entering university, I realised that research is what I really want to do. That is the most important thing, I believe. I would have been smarter if I studied seriously in school. Sometimes I regret not doing it. However, it's in the past, it doesn't really matter, it's not relevant to the present situation. You have a goal and your current skills as starting parameters. What happens next is up to you. From your background section I got the impression that you think about what you could've done a lot. My advice: think about what you can do instead. Right now you're in a perfect positions to study what you like! Some people face a lot more difficulties.
Now to address your actual question.

Frankly, I don't quite know where to start relearning. Where would a starting point be?

I don't think you actually need relearning. Judging by your grades, you have a pretty good grasp on the material. If you really need to sharpen your skills do some exercises from a textbook. What I think you should do (it worked for me) is pick up a book/ online course / seminar on the topic you are interested in and just start learning. Whenever you encounter something you don't remember from the classes go back and refresh this knowledge. Eventually things you really need from those courses will become second nature (and this way you can avoid deliberately solving boring exercises).
You edited the last part out but it did indeed sound like the impostor syndrome. There is a ton of people with imperfect knowledge in academia, both students and teachers. More importantly, you're only a freshman. Even if you don't get into that cool project next year there will be other opportunities. You have a lot more before you than behind you.

Recall that 1% or work is inspiration, and 99% is prespiration. That holds for you as well as for the others, A-graders. You called youself gifted. So, you have this 1%. Now, you have to work hard on the 99%, not relying on your gift. That's simple to say and hard to do, but, in principle, everyone can do it: it boils down to properly organizing your study process, which should become a major part of your life. Self-organize, plan, study, and practice. How to do it efficiently is a different question, which is probably more related to psychology, coaching, and habits rather to academia.
(That's all, folks, sorry for that.)

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How to get my independent mathematical research accepted?
Though I don't have a graduate degree, I have original (and rigorous!) mathematics and have been cold-emailing some professors. Unfortunately, none of them want to collaborate online with me.
I really would like to get my research accepted. What can I do?

The standard way to get your research accepted is to submit it to a peer reviewed journal and have it accepted there. That can be done without the help of a professor (although admittedly it’s not easy). And if you need significant assistance from a professor, the standard way to get that is to enroll in a graduate program. So those are the main two ways to “break into mathematics”.

original rigorous maths without a graduate degree in 2021?

How are you emailing them?  Do you have a paper/result you are sending them?  Also, see: https://academia.stackexchange.com/q/45616/19607

@BCLC Math research prior to a master's or PhD is not that uncommon.  However, it is true that successful undergrad/etc research usually involves a mentor.

Contacting present professors is very unwise. Contacting future professors (a small percentage of present grad students) would be a bit less unwise.  You may be underestimating the demands on people's time in academia. An assistant professor I once collaborated with was ignored by tenured professors at one of the "big name" research universities because he had not yet built a reputation on his own (by publishing highly cited papers without his doctoral advisor's supervision).  He had a PhD from MIT and his emails were not returned. Please consider being realistic.

As @RodrigodeAzevedo comments, people with stellar credentials already have difficulty with cold-calls and publication and so on. That is, to be realistic, you are competing with better-credentialed people who already are having substantial difficulties getting what you're wanting... Not that you shouldn't try, but do be aware of the steep uphill here...

https://philosophy.stackexchange.com/questions/86667/large-cardinals-and-in-intellectu-existence

A solid option is to just put your work into a well formatted, well written pdf and slap it on arxiv. For example, this paper: https://arxiv.org/pdf/1809.03443.pdf as far as I can tell has no journal or academic affiliation, but is influential and widely cited because it's well written and relevant to a hot research area

You say you have no graduate degree. What subject is you undergraduate degree in and have you contacted your former institution (for advice, not collaboration) ? A lot of people involved with teaching students are quite happy to help out a student they remember positively. Cold emailing professors with a 'paper' will get you onto their spam filter quicker than you can say "EInstein was wrong !!!!!"

@Marianne013 I have talked to my institution but because of my history of mental illness some of the professors don't reply to my emails. I was the #1 math student at my university.

https://philosophy.stackexchange.com/questions/88557/continuum-hypothesis-true

Read enough math papers so that you have a sense about how they are written up. Then write your own, emulating the style. Then submit it to a journal. You don't need any degree for this.
You will get feedback. If the work is not very good, it will be immediate feedback and you need to think about what you are doing. If it is good, then, after a while you will get a reviewer report suggesting changes.
You don't need collaborators, but finding someone to read your work and give you some feedback is helpful. If you have an undergrad degree, or are an undergrad, one of your professors can probably help.

I think this answer is good, but it is really hard to believe that someone can learn enough about proper math writing without mentoring of some sort to be able to publish.

@DenisNardin I had a colleague who started publishing quality papers in his early twenties without any help — it took me a couple of years worth of mentorship to get to his level. Some are naturals, which I have witnessed a couple of times already.

@DenisNardin: I suppose there are two parts to that: writing scientific papers, and writing math papers specifically. I suppose the latter could be self-taught, but that comes from the ability to compare papers in your own field to that of papers in relevant math journals.

@DenisNardin i know people who have published their bacelors thesis as a paper already, and OP could've just dropped out right before getting his graduate degree. Its also unlikely but definitely not impossible to self-teach, it just takes a lot of effort

@MSalters The skill of writing math is actually quite specialized (how to structure the paper, what fixed expressions to use, how to reference things etc.) and it is quite different from a scientific paper from another discipline, so I don't know how many transferable skills there are. I guess one could try, but I expect the resulting paper to be rather hard to read (because it likely won't conform to the norms of the field).

@DenisNardin Srinivasa Ramanujan labored alone for most of his life and had no advanced degrees. It isn't common, but it isn't really hard to believe that someone can do it.

@Buffy That's actually a terrible example: Ramanujan was unable to get other mathematicians to take him seriously until Hardy and Littlewood gave him mentoring in how to write. Still, I do not disagree with you, it's just that it's a lot harder than you're making it look.

@Buffy That's one absolutely extraordinary example. Surely the topic of statistics is not foreign to the mathematical reader. One person's success, out of billions, over centuries, does not change the fact that the probability of OP having made a groundbreaking discovery is effectively almost zero. Every professor evaluating the prospects of engaging OP is acutely aware of this, which is why nobody wants to talk to them. Correcting bad ideas and guiding students is the job a professor is paid for.  OP wants that service for free - completely on the merits of their "amazing" research. Unlikely.

@J..., probability doesn't apply to individual cases. I don't claim it is easy to do math in a vacuum. The probability is probably near zero that the earth exists at all.

@Buffy That's still faulty generalization.

@DenisNardin, I'd like some evidence for your claim that Hardy et al. taught Ramanujan "how to write". The problem may well have been that his intuition was so deep and correct that he could take large steps in proofs that others weren't able to follow. Many thought he was a fraud because they couldn't follow his work and claims that he made lots of errors may be true or just a misjudgment by others. Maybe he needed to learn to write for an audience unready for him. I can't say, but would like to see evidence for the claim. Hardy recognized his genius pretty quickly.

@J..., I assume you mean that faulty generalization is true of those you think won't respond to the OP because there is a low probability, hence inferring that the probability is zero. Maybe I misunderstood the intent of your latest comment.

@Buffy Your last comment read like a logical argument that went something like "because the earth exists despite a low probability, therefore OP might be the next Ramanujan".  If you're a math professor and you're willing to pick up the phone on the basis of that argument, fine, but we shouldn't be surprised if you're alone in coming to that conclusion.

@J..., I only intended to say that low probability events happen. You seem to want to discount that.

@Buffy No, I'm just saying that we shouldn't expect events to be low probability events simply because they occasionally happen.  Nobody will be faulted for giving the benefit of the doubt to the more probable explanation - in this case, that OP is almost certainly not a mathematical genius and is, quite likely, sitting on a paper that is worth nothing.

Possibly you don't want to really ask for "collaboration" as much as "advice" or "feedback". Advice and feedback are "limited" responsibilities, while collaboration is an on-going thing. People might be willing to give advice but not make any long-term commitment.
Also, keep in mind that most professional mathematicians, by their own design, are already "fully-booked" for the foreseeable future, apart from possible large surprises which would justify changing their schedule abruptly.
That is, they already have many on-going projects, as many collaborators as they'd want (if not too many!), as well as other obligations. So, imagine that you are asking a person who is already extremely busy to do something extra. Not that that's impossible, but "it's a big ask". :)
EDIT: ... and, anyway, as in other comments and answers, you don't have to have collaboration of a professor to attempt to publish (in the old sense of "in a refereed journal") your work. Yes, the stylistic expectations, and judgement about level of innovation and so on, will be easier for an experienced person, but in that regard all you really need or want is "advice" rather than "collaboration".

Is there a latex template I can use for simple papers?

On arXiv, you can see the (La)TeX source as well as the typeset version... Choose a smallish paper and see what's involved. Yes, starting from scratch it'll seem intimidating, but it's just a one-time thing. There's also no necessity of understanding the inner workings of TeX, etc., any more than we reeeeally have to understand how networked computers work to use them. :)

@Wakem Depends on the journal you plan on submitting to. I know that there's a template for IEEE journals, for instance (though I doubt that the IEEE would be interested in publishing a paper on pure mathematics unless it was directly related to their areas of interest, like some computer science stuff might be). You might be able to ask the editors of a specific journal if they've got a LaTeX template that they'd prefer for their submissions to use.

@nick012000 In my experience there is no expectation that math authors use a journal's preferred template before the final stages of publication.

Against some of the other answers, I will give a more pessimistic view here.
It will be almost impossible for you to get published from outside academia. I'll outline a few reasons.

Nothing prevents you from submitting your work to a journal; an academic affilitation is not a requirement. That said, to get published you need to impress one or more experts (the editor(s) and the referee(s)). Unless there is some crazy exceptional idea in your work, badly written mathematics will fail to impress. Which brings me to the next point.

It usually takes more time than a full Ph.D. for a mathematician to start writing mathematics decently. That is, years of practice and mentorship; a big part of supervising graduate students has to do with getting them to write properly. So I would deem it impossible that whatever results you have are written in a way that a referee will find acceptable. Refereeing a paper is a lot of work, and referees get understandably upset when papers are written poorly.

Without mentorship and contacts in academia, it would be really hard for anyone (not just you) to assess the quality of your own research. Most non-trivial research benefits greatly from discussions with other experts. And it is a common experience for all (most?) mathematicians to have a "brilliant" idea, that immediatly becomes stupid when we explain it to another mathematician. We have all thrown some "earth shattering" paper to the trash when we finally realized it was worthless.

Which brings us to the issue of contacts. Even for an absolutely brilliant mathematician, it would be hard to be heard from outside the community. Most famous case being that of Ramanujan. His first attempts to get in touch with British mathematicians ended up in rejection precisely because his (brilliant) arguments were poorly written. And some were wrong, as most of his results on prime numbers were. Hardy's wisdom and mentorship were required to make Ramanujan a mathematician. Another famous example is Fourier, whose papers were (rightfully) rejected even though they had brilliant ideas.

The only chance you have to get an inside is to hopefully get in touch with someone who is a direct expert in the (sub) area your results are in. That said, we are all used to receiving papers from cranks; the only way I would (maybe!) take a closer look to a paper sent to me out of the blue, would be if it makes interesting claims in topics I have worked myself. Even then, most people are usually busy enough with their own projects, students, classes, and service work, that they won't have the time to pay attention to you.

"it is a common experience for all (most?) mathematicians to have a "brilliant" idea, that immediatly becomes stupid when we explain it to another mathematician. We have all thrown some "earth shattering" paper to the trash when we finally realized it was worthless."  QFT.  The biggest problem with working in isolation is that you don't have anyone to tell you when you're going way off the rails.  Unless OP is a one-in-a-billion genius the overwhelming probability is that they are most likely sitting on exactly one of these worthless bad ideas.

So, this depends to some extent on who you are asking and what you are asking, and what your results are.
First, let's be clear that people aren't likely to to be interested if you have a claimed proof of some famous open problem, like the Riemann Hypothesis, or P != NP, or the Collatz conjecture.
Second, asking people out of the blue to collaborate is in general a not at all small ask. If you are doing so, you need to at a minimum explain in your email a) what result you think you  have b) how it works and c) what part requires collaboration. Is there a specific technique or aspect that they are an expert on? For example, if someone sent me an email asking to collaborate and it was on something I don't know much about, I wouldn't respond positively.
If you have a result that doesn't need someone else, then write it up, and then talk to someone who is an expert on a closely related problem. After you have their feedback, then send it to a journal if it is good and worth looking at.

There is a current paper about whether P=NP circulating. I don't know the details, but it has caused a stir - very recent.

@Buffy Those things do circulate but the hurdle to get people to take it seriously is very high.

This one is being taken very seriously, or so I read (slashdot, I think).

@Buffy, a link, please? I'd not heard anything about this, and lazy-googling is not adequate to find it. :)

Googling and wikipediang myself, and can't find it again. But I read it within the past week or so. Sadly the blurb didn't give any details, but the place it came out of was pretty reputable IIRC. Wikipedia still says unsolved.

@Buffy Was is it the one mentioned here https://science.slashdot.org/story/21/10/27/2024232/the-50-year-old-problem-that-eludes-theoretical-computer-science ?

That's it, and I found it confusing. But I don't have access to Transactions on Computational Theory, so couldn't check the veracity.

Anyway, the advice to stay away from hard known problems is good advice. Such problems are much more subtle than they appear. And, of course, don't try to square the circle. I'd love a proof of the four color theorem that doesn't use a long long case breakout, though, one that isn't dependent on computers.

@Buffy: The paper was desk-rejected, but was published by mistake due to someone clicking the wrong button.  See here: https://twitter.com/rrwilliams/status/1417161397960646658?s=21

@AndyPutman, Thanks for that. I got punked, I guess. I'll keep working on my proof ;-)

@Buffy, every year,  people put forward about three serious papers (and any number of crackpot papers) claiming to prove or disprove P=NP.  I wouldn't get excited unless it's circulating in a major journal.

What you are asking for here is for someone to do the job of a PhD advisor but to do it for free and without getting any credit for it at their work.  If you want a PhD advisor you're going to have to apply to a PhD program.
The only likely exceptional circumstances here would be:

You are a very talented high school student.  People like the opportunity to work with someone stronger than their usual students, and like the idea of helping out someone like their younger self.
You are extraordinarily rich and are willing to fully employ a mathematician.  Given the current realities of the job market I expect you could hire a postdoc for something like $60K plus benefits as a personal PhD advisor (assuming that you make a good impression on them as someone reasonable during an interview, any whiff of crank and even that kind of money won't be enough).

It would be foolish of a recent PhD to allocate energy to a short-term thing like this, of course, considering that (unless, by a miracle, it does turn out to be a Big Thing), as opposed to working on more conventional ways to enhance their CV. So, ironically, if a person outside of conventional academe wants to have young people contracted to advise them, they'll probably only be able to engage people who have given up on success in academe. True, that's not the filter that we formerly pretended, but, still, ...

Probably, but it's hard to get a second postdoc, and you could finish a lot of papers if you didn't have to teach any classes...

True. An innovative way to survive? Sigh...

Another idea is to pay them. A university lecturer or professor normally gets paid to help students get published - and it's a full time job. Why would they do the same work for you for free? But lecturers also often offer for-a-fee tutorial services to students, mainly for those who need extra help to catch up. But a maths lecturer offering such services may be willing to tutor somebody through the process of writing up and submitting a paper instead. It might make an interesting change of pace for them. And if in the process they end up reading your work and having ideas of their own about it, that could conceivably turn into collaboration.
But don't get your hopes up too high. It's hard for outsiders to judge what's of sufficient interest to be worth publishing, and it's particularly hard for outsiders to overcome the credibility barriers raised as a result of the thousands of amateurs who seek instant fame for solving famous problems, who turn out to be fooling themselves. You should take seriously the possibility that you might end up paying someone a lot of money, only for them to tell you that the work is unpublishable, or to see your months of hard work being rejected by the journals. Such is the academic life. If that's going to upset you, don't start down this road.
As others have already said, the best way to get into mathematical academia (or any mathematical career) is by going through a postgraduate programme - this is exactly what they are for! And you are competing with everyone already on that path. Attempts to find a faster shortcut are always going to be more difficult and less reliable than following the main highway.

If you are a  student at a university then, I'd think, that ethical rules would prohibit outside payments to any instructor at that university. It has the stink of corruption.

@Buffy would it? If you aren't taking any classes with them I'm not sure I see any issue. And people do consulting work on the side sometimes. I would presume that the university as some ethics guidelines but that would be up to the professor to look over.

@JoshuaZ, A pretty serious violation for a professor to take "side money" from any student. It goes beyond any individual class. They can consult for others, perhaps, but not for students. A professor from a different university would be fine, though.

@Buffy Is this the case at all universities? I taught at one university where graduate students explicitly were allowed to take money on the side for tutoring as long as the person wasn't in any of their classes. Another one had a much stricter rule. My guess is that this really does vary from school to school. (But it is possible that the one I'm thinking of was a minority. Certainly you have a lot more experience than I do about what the norm is.)

Re "A university lecturer or professor normally gets paid to help students get published": This could be read as students literally paying university lecturers or professors (probably not what you meant). Can you rephrase and/or clarify? (But without "Edit:", "Update:", or similar - the answer should appear as if it was written right now).

@Buffy For a non-student and an academia outsider it is not that terrible of an idea though. OP seems to be convinced their research is good and they're cold emailing profs - I'd reckon people would be more willing to spend their time on potentially bad (with no imposed limits on how bad) research if they were paid for it...

I think one should be realistic about the level of pay that this would require.  Hourly consulting rates for faculty are typically on the order of $300/hour.  And I'd guess (from having advised graduate students) that you're talking about something on the order of 100 hours.

Now and then, I give informal comments and critiques on things people email me. Fortunately, I receive few enough of these, and have no binding commitments, so that it's not an overwhelming burden. If someone proposed to pay me, with the obligations that that could entail (as opposed to no on-going obligations...), the ticket would be several or many thousands of dollars... for that entanglement with potentially never-ending crankery. The very notion of proposing payment would be a "red flag".

@NoahSnyder Well, you're only talking about 100 hours in the (improbable) case that the paper is actually going somewhere! It might be worth paying a professor $300, or better yet, a grad student $100, to spend an hour writing up a couple of notes on why the paper is no good, if that is in fact the case.

There are some good answers already. Let me add

There are dozens of new works appearing every day. Even if your work is correct, it's just one of n. Don't expect academics to be especially interested in your work, or take lack of interest as an especially bad sign
Academics get so much email. Cold emails from non-academics low priority.
More generally, academics are busy, and might already have many projects and collaborations on the go
Starting genuine, fruitful collaborations is hard. You have to have common interests, understand each other (sometimes I mean the actual language you're speaking, sometimes just the way you think about things)

As to what to do,

Don't get discouraged by lack of interest or difficulty beginning collaborations, though do listen to feedback.
Read many papers on similar topics
Write in a similar style, don't make bombastic claims and show knowledge of relevant literature
Submit to journal that papers similar to yours are published in
Read and take on board any referee reports you receive, or feedback from the editor

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


text:
Where can I find statistics about international movement of students/academics?
I am wondering where I could find global statistics on the flow of students between countries. This would include numbers such as "how many students with US bachelor went to UK for a master" and "how many Spanish master students went to US for a PhD"
They always say that the academic world outside US suffers from brain drain towards the US, i want to find way to quantify this

I don't have the statistics at hand for the case of students. But there was recently a publication of mobility statistics for scientists on 16 countries:
The report is here
The IEEE also made an infochart that was a bit more explicit using some of the data in the report.

The IEEE graph is really cool! Thanks for sharing that.

The UNESCO Institute for Statistics offers some student mobility statistics by country in their online database (not by degree level, I'm afraid). It's apparently not possible to give a direct link to a particular item from this database, but you can find the figures you're interested in the section EDUCATION > Other policy relevant indicators > Number and rates of international mobile students (inbound and outbound).
From that, there are several indicators of interest, e.g. Mobility indicators > Net flow of internationally mobile students or Inbound students > Inbound internationally mobile students by country of origin, depending on what you're after exactly.
Generally, you'll have to "play" a while with the database interface to get an interesting view of the data. See the "customize > selection" and "customize > layout" options available in the menu on the top of the tables. In particular, you'll often find some interesting additional variables or filters hidden under the "customize > selection" menu. It may take some time to format the data the way you want.
They used to have a simple, nice interactive map of this data on the following page https://uis.unesco.org/en/uis-student-flow, but it doesn't seem to work anymore (the map seems to load forever without displaying anything). I gave the link anyway in case they eventually fix it.
Another option, more restricted in geographical scope, is the Organization for Economic Co-operation and Development (OECD) online database. They offer some data on mobile students by country of destination (but only for OECD countries), degree level, and field. You'll find several variables of interest in the section Education and Training > Education at a Glance > Students, access to education and participation (e.g. "International graduates by country of origin", "Share of international students and all students by field", etc.) Look also at other variables without "mobile" or "international" in their names, sometime they include filters relative to international students, so it might interest you. Here is an example for doctoral students: https://stats.oecd.org//Index.aspx?QueryId=118857 . Like the UNESCO database, it requires some time to get how the database works, so you'll have to invest some time to find what you're after. Hopefully the links I gave previously will help you.
Otherwise, if you're interested in some particular countries, it may be worth looking at the websites of national statistical institutes or higher education ministers, they often offer data and figures relative to international students.

all_licenses: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/3.0/

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text:
What does the typical workflow of a journal look like? How should I interpret a particular submission status?
What steps does a manuscript typically go through from submission to publication (or rejection) in a typical journal? How are these steps referred to, in particular by editorial systems, and how long do they each typically take?
Note that this question is about the typical situation and hence not about:

Journals with an atypical workflow, e.g. those that allow for an instantaneous reviewer–author interaction.
Exceptional steps or rare occurrences such as withdrawal or clerical errors.

This is a canonical question on this topic as per this Meta post. It is usually used as a duplicate target for questions of the form: “How to interpret submission status S at journal J?”, since taking all combinations S×J would be unwieldy. Due to its nature, this question is rather broad and not exemplary for a regular question on this site. Please feel free to improve this question.

"those that allow for an instantaneous reviewer–author interaction" -- is this a thing? I know of no example of this in my field, but was thinking it would be a good idea. Could you share examples?

Feel free to edit this answer to improve it, in particular to add other names you know to be used for the individual steps or to extend the maximum typical durations from your experience. The source for the diagram can be found here.
Overview

Initial Check
This step is usually performed by the journal's administrative staff. It may include for example:

Checking for missing or broken files.
Checking compliance with length requirements, if any.
Checking central formatting requirements, e.g., line numbers, if required by the journal.
A plagiarism check.
Excluding manuscripts of very low quality, such as automatic translations or manuscripts with very poor language.

Also known as: technical check, initial QC (AIP), admin checklist (IEEE), Awaiting Editorial Office Processing (ScholarOne), quality check (NPG)
Typical duration: A few workdays.
Editor assignment or invitation
Based on the topic of the manuscript and suggestions by the authors, an editor is assigned to handle the manuscript.
Depending on the journal, the assignment may be done by technical staff, the journal's chief editor, or automatic by submission category or author suggestion.
With some journals, editors are invited and not assigned. An editor who is invited may decline in some cases.
Also known as: with editors (APS), editor assigned (Editorial Manager, AIP), AE assignment (IEEE), assigned to the editor (NPG)
Typical duration: A few workdays to several weeks.
Editorial assessment
The editors decide whether the paper should enter the review process or should be rejected directly, e.g., because it does not fit the journal’s scope or requirements on importance or quality. A rejection at this (or the previous) stage is called desk reject. The paper may also be returned to the authors for reasons other than rejection, such as to request more data or clearer figures prior to formal review.
With revised manuscripts, the editors assess whether the existing reviews have been addressed adequately. If yes, they either proceed with another round of reviews or jump to editorial decision immediately – this mostly depends on the magnitude and nature of the revision.
Also known as: with editors (APS), waiting for potential reviewer assignment (AIP), under review (ScholarOne), assigned to the editor (NPG)
Typical duration: This strongly depends on the journal: With some journals, it is less than a week; with others it may take a month, in particular if several people are involved in the decision or the initial quality hurdle is high.
Peer review
The editor selects a number of potential referees to review the manuscript. Should a referee decline to review or not perform the review in a certain time (as given by the editor or journal), the editor usually has to select a new referee. The main exception to this is if the other referees already provided sufficient reviews at this point.
With revised manuscripts, usually the reviewers from the previous round are selected. The editor may also decide that certain or all reviewers need not see the manuscript again, as their comments have been adequately addressed, or take the opportunity to seek the opinion of one or several additional referees.
Also known as: with reviewers, with referees, under review, awaiting referee assignment, awaiting referee reports, awaiting reviewer scores, awaiting reviewer invitation (ScholarOne), reviewers assigned, manuscript assigned to peer-reviewer/s (NPG)
The initial selection of referees is usually comprised in the previous step. Some editorial systems give the status as with editors, awaiting reviewer assignment (or similar) if a new referee needs to be assigned and no other referee is currently assigned. Others will show under review regardless.
Typical duration: This strongly depends on the field and journal. It typically ranges from a few weeks to several months, but in some cases (particularly for highly theoretical work where intense proof-checking is expected), it may be as long as one to two years. Moreover, the key factors for the duration of an individual peer-review process are how soon the reviewers perform the review and how many reviewers decline or fail to review the manuscript. Thus, even for a given journal, there is a strong variation of review durations. Some journals give their statistics on this time (or a related one) on their webpage.
Editorial decision
Based on the reviews, the editors decide whether:

The manuscript shall be rejected.
The manuscript needs to be revised by the authors before it can possibly be accepted. If the authors submit a revised manuscript, the workflow is mostly the same as for the initial submission.
The manuscript shall be accepted as it is.
A decision requires further reviews.

Note that the editor might not always wait for all reviews to be returned before making a decision.
Also known as with editors (APS), review completed, required reviews completed (Elsevier Editorial System (EES)), awaiting AE recommendation, awaiting decision (ScholarOne), awaiting EiC decision (IEEE), Editor Decision Started (AIP), Decision Started (NPG), or pending decision (Bioinformatics Oxford journal).
This may be followed by a short stage denoted decision letter being prepared (or similar).
Typical duration: A few workdays to a week. This may take longer with some journals, in particular if several people are involved in the decision.
Copy editing and typesetting
The article is copy-edited and typeset by the publisher. Occasionally, requests to the authors may occur at this stage, e.g., due to low-quality figures.
For some journals, a pre-copy-editing version of the manuscript will be put online at this point under a category like Just Accepted, with a warning that the current version has not yet been copy-edited and may change further before publication.
Also known as: in production, in press
Typical duration: This mostly depends on the publisher’s backlog – between a few workdays to over a year, roughly correlated with the length of the publication delay (see below).
Final proofreading
The authors are sent the paper’s proofs, i.e., the paper as it is about to be published. If corrections are necessary, it goes back to copy editing and typesetting.
Also known as: proofs with authors, Galley proof
Typical duration: Most journals request proofs to be returned within a certain time, usually between 48 hours and a week (reasons).
Publication
For some journals, particularly newer ones with an online-centric publication model, an article will be published immediately after the previous step has been completed. If the article is subject to an Article Processing Charge (APC) or other publication fees, publication will typically only occur after the applicable fees have been paid or waived.
Other journals with a more traditional process will queue up the publication for collation into a journal issue with other articles.  The time before this issue is published depends on the size of the journal’s publication backlog and can range anywhere from a few weeks to several years.
Many journals with an issue-based delay provide “online early” access to articles so that they are available to the community before the final issue date.  Articles thus often acquire two publication dates: one for online and one for print publication.
Further reading

IOP Publishing: An introductory guide for authors
IEEE: Peer Review and Decision Process for Authors
The secret lives of manuscripts (American Naturalist)
List of events in Editorial Manager

It might be useful to update this to account for this question: https://academia.stackexchange.com/q/177378/75368


text:
What is an endowed chair exactly? How does it compare to a “normal” position?
US universities have a tendency to name chairs after people. Just to give a few examples from Harvard: Hollis Professor of Divinity (love that title), Steven and Maureen Klinsky Professorship of Practice for Leadership and Progress (too long to fit on business card), Carl H. Pforzheimer, Jr. Professor of Teaching and Learning, etc.
I understand that the people whom the chairs are named after are rich and donated money to the school in the past. What I do not understand is in what aspects it changes the nature of the position. Are these chairs more highly regarded that just being "Professor of Divinity" for example? Are they better paid?

Chaired positions are generally endowed, in that the money provided for the salary, and some other benefits (e.g., a dedicated assistant) or increased research funds comes from the interest on principal donated by the person or people the chair is named after, which is part of the school's overall endowment (UC San Diego has a more broad definition, as does Wikipedia). Being named to a chair is a prestigious award, and in many cases it is a lifetime (or at least, until retirement) position. Some chaired positions are given on a rotating basis (e.g., for five years), but nonetheless it is prestigious to be named to an endowed chair.
While it is not strictly true that professors in chaired positions are paid more (because in most cases pay is negotiable), because universities use a combination of pay and incentives such as an endowed chair to lure highly sought-after candidates, the pay is probably on average higher than the average non-chaired full professor would have at the same institution and department.
Finally, some schools use chairs at the university level (versus the departmental or school level) that they dole out throughout the university. The positions can therefore be used as an enabler for some departments with less money to attract particularly important faculty members that they might not have had an opportunity to attract otherwise.
For a specific example of the policy for endowed chairs at one university (Rensselaer Polytechnic Institute), see here.


text:
Proper procedure when promised funding not given to Ph.D. graduate student
I have a friend who was accepted into an interdisciplinary program at the University of Arizona a few years ago.
They are more than halfway to completing their Ph.D., however their advisor is indicating that they may not receive the full amount of funding this year.
Their offer letter and governing handbook all state that students will be funded at an annual rate of $25k for up to 5 years.
My friend has procured much of their own funding for many of the years through fellowships, so their salary was payed from outside money for over half of their time in the program.
The relevant handbook text is:

The basic graduate stipend is $25,000 (pro-rated the first year due to
  August start date). Depending on the source of funding, you may
  receive this stipend in the form of bi-weekly paychecks or in larger
  lump sum amounts... In addition to your stipend, tuition is waived and
  single only health insurance is paid. Miscellaneous fees incurred each
  semester are the student’s responsibility to pay.

and

All first-year students are funded from multiple university sources.
  By the middle of the second semester, students should have identified
  a dissertation advisor. From that point forward, financial
  responsibility for the student besides with the dissertation advisor.
  During dissertation research, students are typically paid as graduate
  research assistants (via payroll) for a period of up to five years,
  contingent on the availability of funds and continued satisfactory
  progress.

Now this was sold to them by the program recruitment as they would have complete coverage for 5 years at $25k salary, but the statement "contingent on the availability of funds," seems to be an escape clause here.
When a Ph.D. student is promised a certain level funding in the US, but legally the University can shirk that duty, what is the typical response, recourse, and avenues that the student should take when that funding is lowered or removed?
Note that my friend is still in negotiation about this with their advisor, and they may still obtain full funding, but I thought it would make a good question.

That "contingent on the availability of funds" is indeed an escape clause for the university, such that they technically have no responsibility to ensure funding.
However, it is entirely unreasonable for a Ph.D. student in a STEM field in the US to not be paid a living wage for their work.  Your friend needs to make the department aware of this situation, as departments often have resources that can be used to help to cover small gaps.  Now it's possible that neither the department nor the professor is able to obtain the necessary funds, in which case, I would recommend that your friend consider seeking paid employment in industry and finish their Ph.D. part-time.
Having a student do this would be quite embarrassing to most professors, so making it known that this is a possibility may help ensure that appropriate support can be obtained.
The situation, unfortunately, is much worse if your friend is not a U.S. citizen.  In that case, if they have the ability to seek employment through the training option on an F-1 visa, then they may be able to pursue the same strategy.  Some professors, however, will attempt to effectively enslave foreign students through visa-related threats, and this can result in very difficult problems.

Decent answer, but what if they want to pay them $17k, instead of the $25k promised? In this case the funding is not going to zero...I agree with your assessment, I just wanted to get some input from actual tenured professors and perhaps academic HR people who have encountered this kind of thing before.

The phrase "up to five years" is another escape clause for the university. This does not mean at least five years, it means at most five years.

This might need a lawyer's attention. My guess is that the university can get away with cutting funding by claiming funds aren't available, but that claim might be undercut if, for example, they're admitting new students and offering them funding.  They might also try using the "continued satisfactory progress" clause as an excuse and claim that your friend wasn't making satisfactory progress.  But then there should be some real evidence of that, and the claim could be undercut if they continued to fund other students who weren't doing as well as your friend.  All of this, however, is my opinion of how things should be --- the university should not be able to easily get out of a promise of funding --- but my opinion might not be the law's opinion; that's why I suggest a lawyer might be needed.

The statement "financial responsibility rests with the dissertation advisor" suggests that the system is that students are funded from their own advisor's grant.  I don't think there's a claim that the university or department as a whole has insufficient funds, just that this advisor does.  I wouldn't see any inconsistency in the department admitting new students at full pay if they are going to work with other advisors.

@NateEldredge, yes, and unfortunately the interdisciplinary programs here don't have a formal department, it is usually a collaboration among departments, so students who are members of these programs seem to have more difficulty with issues like this...and I understand the funding situation in the US has been particularly bad for the past few years, but RU/VH institutions shouldn't make promises that they cannot keep...

Also, according to http://www.cbpp.org/cms/?fa=view&id=3927, AZ has the highest level (as a percentage) of university funding cuts in the nation.

@daaxix: I understand what you're saying, but at the same time I would not go so far as to say the university made a "promise" - it seems to me that it was clearly conditional, and the students were made aware of the risk that funding could dry up.

@NateEldredge, ok, an ethically grey, almost promise during recruitment...I agree the handbook is fairly clear, however, I was under the impression that there is an informal understanding among RU/VH institutions and STEM graduate students that they will be fully funded for 5-6 years, every biological science, engineering, and physics Ph.D. graduate student and professor that I have interacted with seems to have this understanding in the US, but perhaps my perception is incorrect, which is one of the reasons for my question.

@draax: The problem is probably that when accepting a such offer, you either don't read the small print, or even hear feedback to ignore it as "it just works out." So those other students might have been lucky, or simply be unaware that next semester, they might find themselves in your situation. It's a dirty little power play some departments play which can hugely impact your success and happiness. But legally, you're out of luck, and the US is a country where lawyers rule.


text:
My PI is irresponsible and I worry about my career as a postdoc
I am a postdoc in life sciences in the US and have started my postdoc around 6 months ago.
I am at a great (top-10) university. I like the university, lots of smart people and friendly environment.
I love my work and plan to stay in academia. I am from europe and have left home just for the postdoc.
I am, however, very worried because my PI does not seem to bother too much about our career prospects.
Some points:
He is often away from the lab due to family-related problems. He spends a lot of time with his family (has 2 kids). Although that is great, he does not spend much time in the lab, forgets meetings, is not up to date with current research and often does not have time for supervision (this was the case even before the pandemic). He always comes up with some excuse but after a while it has become clear that his priority is just very much somewhere else than research.
Currently, the lab is supposed to write a grant. He has left that to us, he says that he is too depressed because of the current situation with COVID-19 to be able to focus on work.
This is very worrisome, because our future is dependent on the grant, yet at the same time
he wants us postdocs to go back to the lab as soon as possible (presumably to keep face and show the university our lab is active, while at the same time he does not seem to care about what happens to us postdocs).
In addition, we have currently submitted a paper (~2 years of work from the lab) which gets rejected from all top-tier journals.
He has a lot of good properties, he listens to our opinions, accepts criticism, leaves us a lot of freedom. As mentioned the university is great too. I am highly motivated and love research, but
worry extremly that I am just wasting my time. I have my own money for now but can't change labs easily and would prefer to stay at the uni.
What should I do?
Edit I do understand that as a postdoc it is important to be independent. I am. I have learned several relevant skills, have my own ideas, I am generally quite pro-active. But typically as a postdoc you can also join an environment which allows us to find your own goals within the work of the lab. My PI is just so out of touch with everything, that his ideas are completely outdated. He just does not take research seriously (even though he still wants me to publish asap again under the current circumstances). Hence he can't also give me any guidance (he suggests experiments which have been done many years ago already) and, worse, does not even bother to change that.
Edit2 The PI just fired the other postdoc in my lab. I am under a complete shock. Basically a mix of blaming the other postdoc for not advancing with the grant proposal (PI promised many times to finally get it together and work on the proposal) which was never really officially discussed as being our task to begin with. Yet he has the audacity to whine during the "crisis" meeting with me about how sad his family life is (his kids can't go to school which they don't like yada yada) and then to interrupt the meeting to go and play basketball with his kids. I have told him now that I want to have a co-supervisor and he has promised to reach out to people. He is so reckless, selfish, if he does not get it together I will change labs even if I lose my fellowship.

You are a postdoc. This means, that you are expected to have some degree of independence. What input do you need from your PI? Why was the paper rejected? Is it salvageable? Do you have other papers in the pipeline?

I am generally really very independent. Right now I have taken on organizing our online meetings bec. he can't do any of it, but as mentioned below even before the pandemic he was often away doing other things. I do need input though on general research direction, on the grant proposal. In general, knowing him, he is the type who tries to squeeze out of you as much as he can, but then forgets all appointments and promises on his side. Rejection: Desk-rejection. Do have other papers, but without some direction from someone with experience not sure how well they will turn out.

You seem to be unhappy. Have you considered applying somewhere else?

@lighthousekeeper Yes, but I am here on a visa and my fellowship is tied to the lab. I don't know if maybe finding collaborators would help? Would they care enough about me to give me the supervision I need? And the uni I am at is really really great

@Pugl You could certainly apply somewhere else. Not an expert, but I assume that your visa would probably be extended if you find another job. To me, it sounds like a great university isn't worth the misery of dealing with a poor advisor. Collaborators would be an option to improve the situation, of course.

@Pugl You have two choices: you really take over as effective PI, i.e. you proceed as you think fit, make your own decisions on the scientific direction and then just get his formal approval for the last step (assuming he is not the type that throws everything down the table on the last day before the deadline); or you seek another group where you find a more assertive/present PI. Keep in mind that if you compensate your PI's deficits well, this may make you more valuable to them. Whether they will be "grateful", it's hard to know, but it trains you for the future.

@ssquidd I think there's three separate issues here: unavailability, unreliability, exploitation. Unavailability (e.g., having no time for meetings) would be fine and give the postdoc the opportunity to develop their leadership skills. Unreliability (e.g. flaking on arranged meetings) is unprofessional, but might be acceptable, depending on your mileage. Exploitation (e.g., letting his postdocs write a major grant proposal and claiming the credit for it; sending them to the lab during a pandemic, while ignoring safety recommendations) is hardly acceptable.

Of all you say, the only indication I see of irresponsibility is his desire, you say, to send you back to the lab before it is safe. The rest is understandable.
We live in perilous times. He may be entirely correct in focusing on his parental duties and may have no other option with child care essentially non existent in many places. Many people are depressed now, but it isn't a sign of lack of responsibility. "Forcing" you to write a grant can be a boost to your career and he might have wanted to do that in the best of times.
Not getting accepted into the top journals can be pretty common and may not be his fault, or at least not entirely.
And yet you say he has many good qualities.
As to what you should do, I recommend that you maintain social distancing for your own safety. But beyond that, just carry on as best you can and hope that the horrors of the current situation don't increase or become cyclic. Make sure that you don't burn bridges with your professor, as that will make your next career steps (after the pandemic) much harder.
Encourage the professor to deal effectively with depression if you can, and deal with your own if you feel those twinges.
You are in a bad situation. A worldwide pandemic tends to cause that. It is no consolation, but in other times and places it was much worse.

Another consideration you might make as to the level of his responsibility is how helpful he is at getting postdocs their next position. If he gives good support and they wind up in good positions, then you can put up with some things that wouldn't make sense otherwise.

No, he has been always like this. Even before the current situation he was forgetting meetings and making up reasons which were not true, doing work on personal projects not related to research and coming up with excuses. Expecting us to push the lab forward while he was not taking on his responsibilities.

"to go back to the lab as soon as possible" doesn't seem to imply the PI is pressuring them "to [go] back to the lab before it is safe", at least in my opinion.

'"Forcing" you to write a grant can be a boost to your career' If OP gets to be PI on the grant, yes. If the grant is submitted under the current PI's name and with some subordinate role for OP, the accepted grant is a positive entry on OP's CV, but not a career boost. Arguably, in the latter situation (which seems to apply: "the lab is supposed to write a grant"), the current PI is exploiting his post-docs.

@lighthousekeeper Exactly. He is going to be the author, not us. He is pushing all the work and responsibility on us.

Actually, I was more concerned with the experience you get from it. That alone is a boost.

OK, that's an important clarification. Of course, OP stands in direct competition to other postdocs who will in the same time boost their careers by winning their own grants.

A PI is responsible for their group's safety and career even during a pandemic. Especially, it seems that his irresponsibilities are not tied to the recent events but run deeper. I also had a PI who did not keep up to research, always used his family as an excuse (we were sympathetic for a long time, but eventually you realize that there is a limit to exploit other people's sympathy with this excuse - they are not your family). The difference was that he was not even nice.. OP, your best possibility is to find an other research group. Do not try to change his behavior, change your environment.


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No summer funding, what to do during the summer?
If a STEM field PhD student is not able to be funded for the summer, what do they typically do?  I'd imagine they can't afford to take an entire summer off from their research - or they would probably risk not graduating in the standard 5 years in a U.S. program - but at the same time it would be hard for them to support themselves, eg, cover rent and living expenses, if they aren't supported by their parents / family.

See my answer here. They can also take the time to relax and maybe take up a new hobby. Grad school isn't just about research.

My question asks specifically for the case where a PhD student fails to secure summer funding from their school @seanroberson.

Road trip!!!!!!

Five years is not "standard."  It may be a common goal, but it is not commonly achieved in the US.

Depending on your specific field, and when you've found out summer funding isn't guaranteed or available, I present main options in no particular order:

Research internship in 'industry' ($ to $$$$$)
Apply for summer grants to work in another University in different, but hopefully related, research ($ to $$)
Get a summer job or non-research internship - ideal if it uses some set of related skills to your PhD, but if not, oh well, bills to pay ($ to $$$+)
As Dawn suggested, a sub-set of summer job is to seek the option to be an instructor, summer teaching assistant, or grader in your University (or a nearby institution), or even being an adjunct for the summer if that option is available to you ($ to $$)
Vacation, travel, relax rejuvenate, try some hobbies, spend time with friends and family (-$ to -$$$$)
Focus on your own personal research, free of official obligations (at least -$, due to living expenses)
Have an existential crisis due to the lack of driving structure which otherwise distracts you from your morality/mortality and the uncertainty inherent in life itself (free, and can stack with other options)
Choose your own adventure

Above all, I suggest planning ahead - if not for this summer, then for the next one. Most outcomes are uncertain, so judicious planning for multiple applications, budgeting (so you have options on how to spend your summer), and trying to develop a number of options will be important.
Note also that many of the options don't need to - and perhaps should not - take up the entire summer, so you should have the option to try out multiple things. Remember: academic life is a marathon, which extends off into the unknown future until you fade away (go Emeritus) or leave ("go into industry"). That makes the longest ultra-marathon on earth seem short by comparison. No matter what you do, try to change things up a bit, work in some rest and contemplation, friends and family, and a bit of fun for yourself too. If academic life starts to resemble a 24/7/365.25 prison sentence, you might want to reconsider your worldview.

A subset of "Get a summer job" would be: Work as a summer instructor for the department or a grader.

@Dawn Good suggestion - added it!

Have an existential crisis due to the lack of driving structure which otherwise distracts you from your morality and the uncertainty inherent in life itself -- Who says this is just for the summer months? ;-)

For the -$ research option, which I am most interested in so that the student can finish in 5 years, what do these students typically do?  Take out a student / personal loan to hold them over for the summer months?

@user73946 The advanced planner will have saved up some money during their undergrad days.  After that, saving money for the summer months from your paid months is the easiest solution, though how easy that is depends on what the cost of living where you're living is (relative to your income), and how indulgent you normally are.  It's cheaper than a loan in the long run.  If you've got a convenient lease that ends at the summer (plausible with university housing), you could just move home with mom and dad!

@user73946 It certainly varies by person, but the short version is either have saved up money from previous work, take loans, or best: build a 12-month cost budget that includes summer, using only the 9 months of pay you get, automatically saving each "paycheck" what you need and spending only according to the budget (not according to what's in the account). In building this, you might find there isn't enough money - so you increase income (jobs, gigs, grants/fellowships) or you cut expenses (live cheaper, get roommates, etc.), ask for help, or accept it isn't an option and work around it.

@zibadawatimmy - I take it you are not in the U.S.  Undergraduate study in the U.S. does not leave one with a positive bank balance.

Is the school trying to tell you something? Like go away? If so leave.
Assuming not Change Advisors
If your advisor is so poorly funded that he can't afford a summer stipend, there is no chance he can afford to equip his lab, and this really does not bode well for his future, and you're probably going down in flames with him if you stay

-1, as this doesn't really answer the question (regardless of whether or not it may be true).

And includes many unreasonable assumptions.


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What is a "professional student"?
What is a professional student, in the context of a "Graduate and Professional Student Organization" in the United States (cf. this one)? How are they different than or similar to post-docs?

Sometimes this refers to a graduate student in a program leading to a "professional" degree, such as law, dentistry, etc.

@NateEldredge Yes, that could be what they meant.

One of my former bosses (who resented educated people), referred to professional students to those who enroll in continuing education, and get degree after degree, even while remaining employed. Turns out, the head of our government organization had FOUR Masters degrees :-)

As in other comments and answers, there are two distinct uses, distinguished by emphasis, described in standard-ish U.S. English as: "professional student" (informal pejorative), and "professional student" (professional(=non-academic-goal)-program student).

@paulgarrett Now that the question has been clarified, it's clear that it has nothing whatsoever to do with the pejorative meaning (after all, such people wouldn't have an association).

@DavidRicherby, indeed so, and for those previously unacquainted with the two senses, I intended to give some information about how to distinguish, at least in spoken communications in a U.S. English context, to avoid similar confusion in the future.

Which country? Please edit your answer.

While Bill Barth's description of "professional student" as someone whose sheltering in academia and trying not to graduate is one I have heard, there's also a more formal designation for the term.
Mainly, post-graduate students who are in professional degree programs. For example, medical and dental students, law students, many students pursuing Masters of Public Health degrees, or MBAs, etc. are all pursuing degrees that are intended to be applied to a profession, rather than going further into academia. They often have somewhat different concerns than graduate students, hence the different term.

Could you clarify, please? It seems you are listing a somewhat restricted class of fields (law, medicine and health-related), and I somehow see a pattern there (in that those are typically fields where you need some official (read: national) permit to work in the field, so there usually is some kind of a nationally standardized exam), but I don't understand yet what is meant by being "applied to a profession". I take it someone whose profession is designing car technology would apply his or her degree of a Master in Engineering (or whatever degree they have) to their profession just as well?

+1. Now that the context has been clarified, I'm certain this is what is meant here.

The important distinction here is that these professional degres are not research oriented but are oriented towards training in the practice of a particular profession.  Students in professional programs in business, law, medicine, pharmacy, etc.  take courses and participate in various kinds of internships, but seldom are involved in academic research.  Compare this with students in research oriented MS and PhD programs, where the research component of the degree is substantial.

@O.R.Mapper  Yes, the degree of Master in Engineering is typically also considered as a professional degree, I believe.  This link might be relevant.

@O.R.Mapper I included the last one (MPH students) because that doesn't require an exam or certification to work, but where these degrees are still focused on practical application and the workforce, rather than research. I don't know very much about Engineering - while possible, Engineers are not generally who I think of when I think of professional students, but from some other comments, it seems that they may very well be.

An MBA student would be another good example of a professional student whose field has no professional licensing system.

@cpast Good call, I'll add that to the list.

See also: https://en.wikipedia.org/wiki/Professional_student

Ah, so the sense of "professional student" isn't that his being a student is his profession. It seems "vocational student" would be a better term.

@Geremia "Vocational Student" in the U.S. often implies a non-degree (or at least non-Masters/Doctoral degree) student pursuing things like automotive repair, etc.

My experience with the term is frequently derogatory. It is usually applied to a person who either doesn't know what they want to study, switches their major every year or two, refuses to graduate, and spends many more years than the average in school before graduating. Or, slightly less bad, is someone who likes school so much that they keep finding ways to get degree after degree after degree, never getting a job related to one of their majors, and typically working menial jobs to keep funding their education habit.

Given the clarification of the OP, I would suggest you may wish to delete this answer, since it is answering a different question.

@jakebeal I think that this answer needs an explanation that this meaning belongs to another context, but this is still valuable for future searches on the site.

Without knowing the exact details (e.g. country, university, link to that organization's website), I would wager the following presumption:
This is an organization of those students who are more experienced, mature or advanced relative to the regular/typical undergraduate students. One category of these are graduate students, another is students who are established professionals outside academia - either due to work experience or a non-academic qualification - and are engaged in some kind of supplementary or advanced study programs.
Examples of potential such "professional students":

Electricians
Woodworkers
Mechanics
Performance artists
School teachers
Accountants
Nurses

in some countries these might fall under the "graduate student" category, but in others the qualifications for these professions are not undergraduate academic degrees.


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Better organization of knowledge for courses?
I am currently thinking about organizing a course for next semester. The most important aspect that I worry about is: How to sustainably organize knowledge? This should be a new "hands-on" course where students research and work on novel solutions to real-world problems (Note: How to get the students to innovate is not the question I am asking).
One key aspect of this course is to keep track of knowledge from previous semesters, so students of later semesters can build on top of what was previously achieved, and can find good (over the years better and better) answers to commonly (and less commonly) asked questions. They should also be able to get a good idea of what students worked on previously, and even decide whether to build on top of what was previously done, or just roll their own thing.
The knowledge that needs to be tracked, in short:

Q&A's regarding the course contents
Interesting and useful material relevant to the course
Project approaches and results by students who have previously taken the course

I am currently thinking of using a Stackoverflow clone in combination with a Wiki. Does anyone have good suggestions/experiences? Maybe more professional solutions?

TL;DR First build a knowledge base, then a knowledge community (more than 1 person), then a Q&A site...
Apart from the practical questions (software, structure) I think you miss like many trying to start a community site/wiki that there really needs to be an incentive to participate continuously in such a project and answer and document things when there is a reader count of a few dozen (like it will be for a course). You see what trouble some specialized beta sites here on stackexchange have to get 1-2 questions at all a day to attract audience. Gamification will not work for a such a small audience. I participated in some research group wikis, the will of the members to participate is defined mainly by "how easy to use" and "what's my profit from this compared to my current personal note taking system (redundancy)"
Wikis work much better for such a small community for the purpose of documenting things and sharing knowledge:

easier to share and embed PDF's
many skins to get a layout suited for good overview
often plugins to export Wiki syntax to Tex/Docbook/PDF so you can write basically your bachelor/master thesis with the wiki
easier software to handle, many wikis can be file-based and run without server (more future-proof), I would only put my knowledge into a wiki if it is simple to copy and install at another place

If you put such criterions now here you get some candidates:
Dokuwiki, MoinMoin, PHPwiki, PMwiki I can recommend
MoinMoin runs without server (even on USB stick), has WYSIWYG editor (absolutely necessary or only few will use your wiki) and is pretty future-proof as Debian and other known communities are based on it. There is also a nice sidebar theme to keep overview and allow fast navigation, don't underestimate overview when people browse a new site!
So this is my recommendation, but the bigger question is how to motivate students to participate. Already putting some knowledge and stuff up yourself is mandatory, why participate in an empty website, that's why stackexchange invented the Area51 launching system, but even this is not reliable...

I tend to agree with Gamification will not work for a such a small audience. However, do you happen to have source/reference for it? Thanks.

@Hauser Thank you for your input! I absolutely agree with many of your suggestions, especially the ease-of-use factor. However; why participate? Because it is part of the participation grade and because it is a chance to help your fellow classmates and a chance to share. Building a community in the classroom is my main goal. This is only a small part of the MUCH bigger question: "How to get students to be a team/community?" I did not start that discussion here yet, but I will soon. I am still researching.

@Hauser another thing: A sense of community is created by team building exercises, a well organized stream of constant feedback in all directions.