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How does one appropriately navigate the situation of having a lot of prior knowledge before applying to university + courses not on main interests?
I am new here (but have been around MSE for some time). Generally on the mathematics stack exchange it's considered inappropriate to ask for personal advice but I see many such posts on this stack, so I hope this question is suitable.
I am currently an AS (Year  12) student in the UK, thinking about applying for a mathematics degree in two year's time (ideally at Oxford, if that's relevant). However, I am not really an AS student. I have been self studying rigorous and higher mathematics for quite a while now from textbooks, papers, MSE and whichever scraps I may find online. To give a brief idea of what I have studied:

An introductory term complex analysis course, finishing with rigorous treatments of the residue theorem, the polygamma functions and the contour work of Prof. Blagouchine
Much linear algebra, from proofs of existence of the Jordan Normal form, the spectral theorem (and Hilbert-Schmidt's infinite dimensional extension for compact operators), the entirety of the functional and operator analysis course from Royden's textbook, and analytic functions of matrices (matrix Taylor series via contour integration!)
Measure theory (all the major theorems, with full proof) as learnt from university lecture notes + Royden's real analysis
Introductory to mildly complicated topology (again as learnt from Royden's real analysis)
Applications of the above three points to ergodic theory and dynamics (up to chapter 12 of the text "Operator Theoretic Aspects of Ergodic Theory")

The things that come to the forefront of my mind when thinking what I'd like to learn more about at  university under the guidance (at long last) of a tutor:

More advanced contour integration (Hankel, Pochhammer, keyhole, etc.) and more on special functions and advanced complex analysis (e.g. Picard's theorem, Riemann surfaces, Weierstrass factorisation, zeta function and its uses)
More on (advanced?) ergodic/dynamic theory and its applications (e.g. physics and probability theory - I just read and understood the proof of the Strong Law of Large numbers, which was nice!)
Differential topology and geometry
Geometric measure theory, fractal analysis
This one's a bit vague, but fixed point theorems and other (mostly functional analytic) ideas that are useful for solving practical problems (e.g. I have studied the Peano and Picard existence theorems for ODEs but these are limited examples)
Integral transforms and their applications in differential equations, distribution theory, Sobolev spaces etc. and their applications in solving difficult definite integrals
Practical skills! I am good at absorbing theory but not as good as I’d like at doing exercises and research, as I have been self taught (with the exception of the wonderful volunteers in the MSE community).

EDIT 26/08/22: This post has gained a frightening number of views. Something recently reminded me about this post, so I looked back - and looking back, I cringe. I don't want to delete the post, since that would be discourteous to the nice answers and comments - in particular I am very grateful to Prof. Glueck. Since so many people have seen / are seeing this, I want to clarify a few things. At the time of writing, I (a) didn't have much perspective - I hadn't gone to any open days yet, for instance - and (b) was worried that the things I "really wanted" to learn weren't going to be taught to me. But I now realise that that's actually not a problem, at all. I've recently read some algebra textbooks and ended up getting quite into certain ideas from abstract algebra and number theory - before the summer, I had thought that I disliked these topics. It just goes to show, what you think are your main interests probably won't remain your main interests - as Prof. Glueck's answer points out. So, this question was initially written from a position of mild panic, and accordingly doesn't hold up to closer inspection (again, something Prof. Glueck helped me to realise!).
There is another thing I'd like to clarify. Some parts of what I wrote, from the original question, were very poorly phrased. Some responses on this post seemed to pick up on this and accordingly misunderstand me, which was upsetting at the time, but looking back, it makes a lot more sense to me why that happened. I don't want this internet record to represent me with that misunderstanding. Maybe I'm just being anxious, but I want to try to set it right. When I wrote that paragraph, and the whole post in general, my intentions were:

To communicate the concern in a respectful way
To do so without coming across as over-confident (I think this failed)
To communicate my love for mathematics and desire to learn more (that one hopefully came across properly :) )

Where I say: "this is not hubris, I have checked", I did check, and it wasn't hubris. What it was, was lacking in perspective. Although I may recognise the titles and chapters from a set of course notes, I would still benefit from going over it again. I knew that at the time too - I wanted to communicate the familiarity, but I fear some users interpreted it as my claiming to have mastery, which I don't and never thought I did. I am very much looking forward to tutorial sessions, lectures etc. and talking about maths with other people face-to-face, to learn and improve. I was just afraid of - and this is because of my experience in secondary school! - going over things I'd already learnt as if I'd no familiarity at all. But of course, university is much more independent, and as other answers have pointed out, there is not any risk of this.
When I said: "... not much overlap with what I'd like to know", that was born of the mild panic that was previously mentioned. Yes, I would love to study complex analysis (e.g.) in more detail in my undergraduate. But that is certainly not the be-all and end-all of interesting mathematics. Again, that is something I also knew at the time, but I didn't think to emphasise it in the question - which was a mistake. There are many topics on the Oxford, Warwick, UCL etc. courses that I've never studied at all, and which I'll certainly find interesting once I get there. When writing the question for the first time, I didn't quite appreciate how much having prior experience with a topic affected the perceived interest of said topic.
I always have been aware of the gaps in my knowledge and ability that come with the difficulties of self-learning. I thought that highlighting that, and the fact that I knew that, would shift the focus of the question away from what I wanted to ask. I should have highlighted it more anyway!

But when I check the course details and lecture notes of the maths courses at Oxford (but also at Cambridge and Warwick) I find to my dismay that a sizeable amount of first and second year content is familiar to me already (this is not hubris, I have checked) and that it seems as if only the last two bullet points will actually be taught. Perhaps I'm blind, but the first five on the list of "major things I'd like to learn" don't seem to be particularly present in the undergraduate courses.
So I have this dilemma: I don't know how to appropriately contact universities or make decisions about courses when there is overlap with what I do know and not much overlap with what I'd like to know. I would ideally want to email a professor to say something along the lines of "Hey, I was wondering if you teach [X subject] as it doesn't appear to be in the lecture notes. What leeway is there for me to direct tutorial time to [X subjects I'd like to study] rather than [Y subjects I am familiar with]?" because university is after all rather expensive, and wasting time would be tragic.
Again, this is not hubris: I am well aware there will be gaps in my knowledge, especially as I have been learning without a curriculum, but when it comes to the topics I have studied already filling in those gaps can be adequately done through the odd tutorial, advice-asking and usage of university libraries - to spend many lectures and months on them really would be a waste.
But that type of contact seems unprofessional, perhaps, or not the best way to do things, and as a student seeking to enrol at (e.g Oxford) I really don't want to offend any professors, come across as arrogant or make problems for myself otherwise, so I am coming to this stack for advice as I have never dealt with university application processes and contact before.
My question(s):

How should I talk to a university/department/professors about this problem (both before applying and (hopefully!) after having enrolled)? The main component of the problem being: how can I ask to get teaching in topics that aren't actually (so it appears) on the syllabus?
How should I communicate my prior knowledge to get constructive use of teaching time and boost an application without appearing arrogant or -insert potential issue here-?

I'm aware I could get textbooks on the above, but I've been doing that for the last year and a half and self-learning can get intensely frustrating at times - I would much prefer to have a university tutor/lecturer go through them with me, which is why it is so sad to see them not present in the course notes.

Comments are not for extended discussion; this conversation has been moved to chat.

You might try and see if the university would allow you to take some sort of placement tests that would allow you to skip some of the less advanced classes you are not interested in.  Passing such a test would at least confirm that you really know as much as you think you do.

@user4574 It's more that I think I've studied quite a lot; I don't claim to have retained or mastered all of it - I just would not want to consistently resit, especially under the theme of "independent study" (which I have already done to the death), topics I have independently studied

I think part of the problem at hand is that your perspective is a bit idiosyncratic. This is obviously not your fault, since it's very difficult to get a clear picture of higher mathematics if your knowledge as of now is mainly self-taught.
So let my try to help you to "clarify the picture" a bit:
Mind your pace!
The other answers have made good points, but I sense somekind of misunderstanding here. Many instructors of mathematics and related subjects have a fine-tuned automated alert that is triggered whenever somebody tries to absorb a lot of material and claims to understand it, but says they are struggeling with exercises. This is because many of us have seen too many students who said that they "understand everything in the lectures, but don't know how to go about the exercises."
Actually, the exercises are the most important (seriously!) and the most difficult (seriously!) part of studying mathematics.
I think that main point of the other answers is not that they might perceive your question as "arrogant" or something. I think they are mainly worried that you are going way too fast. Let me discuss this in a bit more detail:
(1) The most elemantary problem caused by going too fast is that some people who go fast think that they understand a lot of things, while they actually neglect the details (and details are key in mathematics). However, given the comments you wrote here and given your posts and Mathematics StachExchange I gather that this is not a problem of yours.
(2) A more advanced problem caused by going too fast is that you don't have enough opportunity too double-check, possibly rectify, and reinforce the understanding and insights that you have (or think you have) already gained.
(3) Another danger of going fast when learning mathematics is that you don't spend enough time on the basics and start to lose yourself in abstractions.
I am a bit worried about (2) and (3), mainly because you say that you have, within quite a short period of time, learned various things and read the first 12 chapters of the ergodic theory book by Eisner et. al. Please note that - as already indicated by the series in which the book appeared - this is a graduate level book!
So there is indeed the danger that you are simply proceeding too fast, thus exposing yourselve to the issues explained in (2) and (3).
Suggestion:
The things pointed out above are precisely the reason why you probably need somebody to give you some feedback.
Now you happened to hit precisely one of fields that I am very interested in (functional analysis, operator theory, and connections to dynamical systems). So here is an offer:
If you agree, you can send me an email, and I'll reply with some numbers of exercises from the book you mentioned, and we can then try precisely what Buffy suggested. You try to solve them - without using the internet, just books - and then send me your solutions, so that I can offer you some feedback.
I probably won't have time to do this more than once or twice - but it might still provide you some useful feedback.
Courses you are interested in.
Here we come back to my claim that your perspective needs to be "clarified". Your problem here seems to actually be a mixture of several problems:

It's probably hard for you to find some of the topics your interested in within a curriculum, because you don't know precisely what you should look for. For instances, at Oxford there are courses on on "Multidimensional Analysis and Geometry" and on "Geometry of Surfaces" (link to courses in Oxford). The second one clearly belongs to the realm of differential geometry, and the first one is likely to be related to it, too.

For other topics it's simply difficult to predict whether they will be treated in an undergraduate course or not. One of Picard's theorems (there are actually several of them) might or might not be taught in an undergraduate course on complex analysis. It might even depend on the lecturer.

For some topics you mention it is just unreasonable to expect that they will be taught in an undergraduate program. For instance, with respect to ergodic theory, it is safe to say that the vast (really, really vast) majority of undergraduate programs throughout the world will not routinely teach material that is more advanced than the stuff you can find in the book by Eisner et. al. For most universities it would be pointless to do so, because none of their undergraduates would have sufficient preliminary knowledge to understand anything in such a course. (If you are looking for an exception, you might for instance check the website of Imperial College London and hope for the best - but I wouldn't bet on it.) Some universities might offer some ergodic theory courses on a level similar to (not more advanced than) the book by Eisner et. al., but with different contents - but maybe not on a regular base, and in any case it might be hard to predict what precisely the content will be.

Some of the topics you mention will just pop up as part of other topics. For instance, fixed point theorems play an important role in some parts of the theory of partial differential equations, but they also occur in some subfields of topology.

I thus believe that if you insist on having courses on precisely the set of topics that you listed, you are most likely to drive yourself crazy about it, with little benefit in the long run.
So here is an alternative suggestion: Instead of pointing out specific theorems and topics, try to identify certain fields of mathematics that you like. In fact, it seems that you have alrady done so: Most of what you write points quite clearly into the direction of analysis - more specifically, functional and geometrical analysis and dynamical systems.
If you would like to pursue this further, your best probably is to join a program with a strong focus on those subjects in general, and then choose your specific courses as you go. In my experience, it often happens that an individual person's interests are not so much tied to a specific topic, but rather to a specific taste or "flair" that a certain field offers. Once you like the flair of a topic or a theorem, it's quite likely that you will also enjoy other topics and results from the same field.

Thank you very much. This addressed everything quite respectfully but usefully, and the offer you made was very generous! I take your advice. After all, I have only been researching universities for one day :) I went in thinking "Ok, the really hard stuff I see people doing let's see if I can find a course on that" and I got quite set back when I realised this was not the case.

What email should I use Sir?

@FShrike: On my profile page, there's a link to my website at University of Wuppertal - there you'll find the email.

@AtticusStonestrom I have been doing that! It was doing this that partly prompted my question - it was disheartening to see that many lecture notes from A,B and a few even from C were broadly the same as content I've looked at before. I expect there'd be more detail in places but ... it prompted the question

I slightly edited the answer since a user kindly pointed me (in a now deleted comment) to website where a lot of courses materials are provided online by the University of Oxford.

I have not forgotten your kind offer Sir - I will email you, if you are still willing, but I must first handle the other academic concerns currently in my life (e.g. choosing a university... revising for AS)

@FShrike: Please take your time, I'm not in a hurry.

Terrific answer - and even offering to field a couple of exercises by the asker.

Thank you so much for the link to the courses at Oxford. They are valuable learning materials...

I worry that you think you know more than you really do. Or, maybe a better way to put it is that you may have only a superficial knowledge of many topics, rather than the required deep understanding that is needed to be a mathematician. I'm not an Oxford Don, but was pretty good in some parts of maths. But if you ask a professor from Oxford or Cambridge to work with you they will ask the same sorts of questions I did here and will, if they are satisfied with the answers, give you that exam.
Here is the problem.
There is a huge (monumental) difference between being able to read and follow (and even memorize) maths written by others. When you think you understand it it might actually be more of a function of the quality of the writing than your depth of understanding.
What you seek, ultimately in maths is insight. That doesn't come from reading and memorizing, but from applying the knowledge and problem solving. If you can solve similar problems to the ones stated in a book you are getting somewhere, but it takes practice.
And you will only have achieved insight into a topic when you can conceptualize things that "might" be true but aren't yet proven - i.e. problems worth spending research effort on. Many professional mathematicians don't reach that stage by the time they earn doctorates. Some might never do.
Learning comes from reinforcement and feedback. The feedback, which you seem to be missing (in part at least) assures that you don't go astray and reinforce the wrong message. You don't get true reinforcement from reading and following along. It is too passive. You have to bring active processes to bear in almost every subject, but especially, perhaps, maths.
I worked in analysis primarily and a bit in topology. As a young student I wanted some insight into how rational functions worked (quotients of polynomials) and what derivatives could tell me about that - very elementary stuff, actually. I graphed, by hand (no computers then, no fancy calculators) literally hundreds of rational functions using mostly first and second derivative information. From that I got insight, not only into rational functions, but into the essence of derivatives. Very hard.
My advice is to find a local source, someone you can talk to - a teacher. It doesn't need to be an OxBridge professor. Go back through all of what you already think you know but apply it to the solution of textbook problems. Ideally, you should be able to solve every problem in any textbook, and not with 65% probability. And, your "correctly" solved problems are questionable without feedback. Only then can you be assured that you've "grokked it".
You may be ready, and I can't really diagnose it until you present that exam. But professors at top universities are already pretty busy and don't have a lot of incentive to work with you unless you first show them that you are exceptional. That is very difficult to do and unlikely if you can't solve the problems they would surely want to throw your way.
Let me give another warning. If you don't do the work then you might reach the point where you can't understand anything anymore. There are students for whom everything seems easy to get along the way and so they don't try very hard (solving hard problems...). But they may reach a point where their natural ability isn't enough to carry them anymore and they then actually fail since they don't know how to learn, not having worked at it.
If you don't have a teacher to give you the feedback, and if your family has the means, you might hire a university student to work with you. But you don't want a "tutor" to feed you facts and information. You want someone who will look at your work and give you feedback.

One caveat: There are a few textbooks that hide research problems in their exercises and don't say so. There have been a few cases where a student solved one of these, not knowing that there was no prior proof.
One note: The reason I asked if you had a teacher you worked with was also to note that if you want to approach a professor who you don't know, the best way is to do it indirectly, having another professional introduce you. It is best if they already know the professor, but not essential. But the professor will have a better idea of your skills and abilities if someone knowledgeable can attest to them.

Thank you for the response. I think I’ve carried the wrong tone despite best intentions. I didn’t post to display some “wealth” of knowledge (I really am aware of the limitations of what I know) but it is still not to be neglected. I opened lecture notes for several first and second year courses and recognised the titles of every chapter to be already in my own notes. My biggest concern with researching universities recently has been the nature of the courses (where they don’t cover my main interests). However I surely will put in hard practical work and heed that advice.

It's good that you take notes. That is good for reinforcement. Summarizing those notes is also a good practice.

I take great care to make sure I understand every sentence I write before I write it, which can mean taking an hour to complete a paragraph of notes. I am passionate about rigour, so whenever a source references a result or logical step I don’t follow I make sure to research, understand and include the proof of that result in my notes, before proceeding (except when the cited results are very deep and beyond the scope of undergraduate). I’m also not claiming to be ready for university: I am claiming to be very well advantaged compared to fellow year 12s! I have a year to get better at exams

Oh yes - everything for me is notes. I have the majority of Royden’s real analysis in my notebooks, paraphrased and expanded upon / shortened where necessary! Anyway, I am grateful for your response and I do trust your advice (I’ve seen your posts elsewhere on Academia). Would you please be able to comment on how to deal with the nature of the courses not really covering the big things of interest?

For context: I am currently one of the second year external examiners for Oxford, and have had the same role for Cambridge in the past.
At places like Oxford, Cambridge, Warwick and Imperial you will find that every year there are a few incoming students who already know a substantial chunk of the syllabus.  It is not typically useful for those students to officially do anything other than take courses in the usual way.  You may find that you can do that easily, in which case you will have plenty of time to do other things.  You can talk informally to other talented undergraduates, either individually or in events organised by student maths societies, which are often very active at the kind of universities we are talking about.  You can try going to graduate level seminars or working groups, and talking to PhD students or postdocs who are also attending.  You can just spend time reading in the library.  You will not have any shortage of intellectually stimulating ways to spend your time.  Having the right people to talk to is much more important than the contents of the curriculum.

That is good to know, thank you

Are things like summer research projects or independent study projects useful to do at university in the UK? In the US, my friends who were self-study types did a lot of this and found it useful. I would also add on campus jobs teaching other students onto the list of formal activities that were useful, since deteriorating basic knowledge is definitely a thing and teaching is one of the less-painful ways to keep up your practicing--and you might even get paid a little ( https://math.stackexchange.com/questions/1189597/forgotten-old-results-break-my-motivation ).

Many UK universities have arrangements where students can help with research projects over the summer (although the number of places available may not be very large).  This is certainly a good idea for students who are hoping to go on to do a PhD.  Opportunities for undergraduates to do teaching are less common, I think.

@NeilStrickland I believe user 3067860 was referring tutoring jobs rather than classroom teaching jobs.

@mkennedy To muddy the waters further ... in a Oxbridge context 'tutoring' is essentially what I think you are referring to by 'classroom teaching'.

One advantage you have, as a young student, is time. It is great that you are ambitious and want to tackle hard topics. But, let's say you did get into a good program, and did spend two years on courses that mostly covered material you know like the back of your hand. Well, perhaps you will not have covered as much ground as you wanted. But -- (a) since you are very familiar with the material, you will have built up a CV with good grades in your courses that you can use as proof that you are talented, which can help you get internships and get involved research and later get into a PhD program, (b) you can take advantage of the opportunity get to know the professors who teach your courses, who will ultimately be able to write you letters of recommendation and suggest opportunities for your career growth. Even in the worst case scenario, where you really are just reviewing material you know over the course of these two years, you will have a lot of chances to grow your mathematical career. And two years, at the early part of your career, is really nothing -- it is well worth the trade.
Having said that, I strongly suspect that there will be significant advantages for you to take these early level courses. First, you will get extremely valuable feedback on your problem solving skills, which will help you develop your skills as a mathematician. Second, by learning the subjects systematically rather than on your own, you will very likely find interesting connections between subjects you didn't realize and build a stronger foundation. Third, you will get to meet your peers, and at the schools you are talking about there are very strong candidates in mathematics -- in fact you can really learn a huge amount by having a group of "math friends" who you study with at your same level. Fourth, you will be exposed to many topics in a degree, and you may find you are interested in something that you currently do not realize that you are. Fifth, there is a lot of knowledge that is not written down in books, like informal ways of thinking or ideas that don't work or the shorthand for communicating in a given field, that are best communicated by attending lectures and talking to your peers and professors. Finally, if you really have mastered a given course, to the level where you could take the final exam and ace it, you may be able to simply ask the professor if you can take the exam up front to "opt out" of some introductory courses. At least in the US, it is common to offer exams at the beginning of the semester that advanced students can take for credit to skip introductory courses.
It's very unlikely that you know more mathematics than the entire department. By going there and showing you are a serious student, you will find doors tend to open for you to explore your interests in as much depth as you want.

Oh yes, I definitely do not know more mathematics than the entire department. I'd just like to learn a few things which don't seem to be there but hey, as you say that is not too bad. This was an encouraging post

@FShrike There are usually a lot of options for exploring your interests, especially at a place with a world class mathematics department. If you find a mentor who does research in a topic you are interested in who is willing to work with you, you can do a reading course that covers an area you want that isn't in the course catalog. Sometimes the material you want really is covered, but it might not be obvious from the course catalog. You can take graduate courses which are usually more specialized. In the absolute worst case, you can get a solid well-rounded degree, and focus more in a PhD.

(Caveat: This answer is based on experience with universities outside the UK.)

How should I talk to a university/department/professors about this problem (both before applying and (hopefully!) after having enrolled)?

University administration and admissions staff will mostly not care about this. However, many universities have "honors" or "excellent students" tracks, with some closer attention and mentoring by senior academic staff. Ask about those.

The main component of the problem being: how can I ask to get teaching in topics that aren't actually (so it appears) on the syllabus?

When you approach individual professors of courses, ask for a one-on-one session. Assuming you get one, tell them a little about your relevant background, and see what they say. They might tell you that you can skip some of the lectures; or that you can just attend the final exam; or that you can get some written assignment instead of actually taking the course; or they will insist you actually take the course and be skeptical of your background knowledge.
If they don't want to talk to you, perhaps try the teaching-assistant-in-charge, or a second professor of the same course etc.

How should I communicate my prior knowledge to get constructive use of teaching time and boost an application without appearing arrogant or -insert potential issue here-?

Remember that, generally, you don't get individual teaching at university - certainly not as an undergraduate. That kind of relationship happens almost exclusively with graduate students, and even they doesn't always have their advisor actually sitting down to teach them things individually.
So it may (or may not) be the case that you won't benefit as much from teaching time. If you were lucky to reach some kind of arrangement with the course staff, you may be able to use that time differently; otherwise, consider it a form of practice (and perhaps even take the textbook to class and do extra exercises during the boring parts).

"Remember that, generally, you don't get individual teaching at university - certainly not as an undergraduate." This is where Oxford and Cambridge are somewhat different to most British (and other) universities, because the tutorial system means that students do get near-individual teaching – at Oxford, first and second years will generally have tutorials (as well as lectures) which will typically involve one tutor (a professor or other fellow) and 1–3 students.

This answer is also misleading about course structure and exam arrangements at UK universities.  There is no way that an individual lecturer can make ad hoc decisions about requirements for an individual student.

@NeilStrickland On the other hand, in the UK it is much more likely that nobody checks or cares whether a student is attending the lectures and most courses are likely assessed by a single exam at the end of the year rather than ongoing work.

@NeilStrickland: Added a caveat about this. I take it you're saying that the academic liberties, when it comes to teaching, are strongly curtailed in the UK?

@einpoklum See https://matheducators.stackexchange.com/questions/7286/

@NeilStrickland: Interesting. I would have thought the academic staff unions, and university senates, would not allow something like this. But it does remind me of things I've heard from a couple of friends who had taught in the UK.

I was recently at Oxford studying maths (and philosophy, though I was never good at the latter). I'll try to address some of your points.
Based on what you've stated, you are probably right in thinking that the coursework will be rather easy. This will give you a lot of freedom, either to learn other mathematical subjects on your own (as a friend of mine did a lot of the time) or to pursue interests outside mathematics if you happen to have any (some of my acquaintances were math nerds to the teeth, whereas others weren't).
So the fact that the coursework is easy shouldn't deter you from applying to Oxford. (Perhaps there are other reasons? I don't know what your preferences are.)
Sitting here, it's hard for me to guess where the intense frustration of self-study comes from, in your particular case. Other answers do speculate that you're perhaps rushing through your material. If I go by your question about diophantine approximation, to which I gave the answer, you might not have had enough practice acquiring or coming up with non-rigorous ideas. These non-rigorous ideas are an important shortcut for understanding and proving things. Here's a blog post about this by the great Terry Tao: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/
In this day and age, many tutors are busy bees. (In case you're wondering, it's a result of the managerialisation of the universities. Two people who have explained how this reduces the freedom that professors used to enjoy are Terry Eagleton the literary critic who taught literature at Oxford, and Lord Sumption the former Supreme Court judge who taught history at Oxford.)
There was one year when one of the maths tutors at my college was so busy with lectures and marking problem sheets and helping his PhD students during Hilary term that the undergraduates who needed to meet him for various reasons could hardly get a word in with him.
On the other hand, I had another tutor who throughout my time at the college gave ample time to his students, myself included. He really liked the teaching side of it. (He's now at Warwick.) Ideally, by having more and more conversations about the maths that you're doing, you would organically get into the orbit of such a professor, who would then be able to give you more personal guidance. It's easier to do this than in America, where because there are "office hours" it would be more difficult to speak to a professor outside office hours --- not so at Oxford.
If research is up your street, there are plenty of researchers at the Maths Department at Oxford who want undergraduate assistants. My friend did one of these research internships during the summer, and even got his paper (now a couple of papers) published on Arxiv.
And of course, things become easier when you have peers as enthused about maths as you are.
So once again, when it comes to what you're going to experience at Oxford, my advice is not to worry too much.

Thank you for this post, and - good to see you again!

@FShrike cheers. P.S. when they realised I didn't need much assistance with first-year real analysis, I was then paired up with somebody who didn't need assistance either, so that the tutorials could get more interesting.

I find to my dismay that a sizeable amount of first and second year content is familiar to me already

Two key points :

Universities exist for the primary purpose of certifying your education.  Providing that education is secondary.

In most cases you cannot simply skip the courses and write the exam, unfortunately, but there are exceptions and if you're confident you don't need to sit the course it doesn't hurt to ask if you can take an exemption exam. Some institutions are more open to this than others.

You don't have to attend all of your lectures.

If exemption is not an option, and if you already know the material, you can simply skip most of your lectures and use the time to continue your own autodidactic learning.  Use the time to advance other topics so that you're better prepared for whatever you plan to do when you graduate.  Just don't forget to show up to the exam and turn in any coursework that is to be graded.
Critical, here, is that you have clearly demonstrated that you do not need a formal course, lectures, or a professor to teach you new material.  You do, however, need the university to certify that you have learned this material.  So use the institution for the things you require of them and forego the things you do not need them for.  This frees up your time to invest in your own continuing education at a schedule and with topics that keep your interest.
Don't feel bad about showing up just to take the grades and run.  As long as you're continuing to learn at a pace you enjoy and as long as you're not missing any key topics there's no reason why you can't follow your own course of study.

I  think that neither (1) nor (2)   is true of the mathematics courses at the elite UK universities that the OP is speaking about. The courses are highly structured and although you may skip lectures the same is not true of the twice-weekly "tutorials" where the real teaching happens.

@erstwhileeditor On (2), are you saying that Oxford will fail a student simply for not attending lectures or tutorials?  Even if they are autonomously capable of mastering the material, completing all coursework, and passing the exams? I'm suspicious of that claim. On (1), it's true simply by inspection - there is no alternative if you want to possess a degree of higher education. You must register for and pass courses at a certified institution to receive a degree - no other means of obtaining a degree exists. An education, however, can be earned anywhere, even without leaving your own house.

(2) It's the "complete the coursework" bit: that means producing the work for the weekly tutorials and and coming along to participate in the classes/tutorials. I am pretty sure that the student contract specifies this. Of course in a very extreme case tutors might just ignore the rules ;-)  (1) I don't think these universities see themselves as being primarily "certifying" institutions, they also have important educational aims - producing the  next generation of mathematicians etc.

@erstwhileeditor Yes, most students either cannot or will not learn autonomously, so yes, Universities naturally are replete with educational assistance.  But we don't build houses to give us a reason to make bricks.  Educators readily focus on themselves and their own role in the process - normal human egocentrism - so if there's a predisposition to not see the forest for the trees, this is where it has roots, I think.  How they see themselves doesn't really change anything. All students need the degree.  Not all students need their hand held to get there.

From personal experience I can tell you that over-confidence is what trips you up.
For a BSc I had to do a course in electronics.  I'd previously done a similar course at the same level with relative ease.  I then under-estimated how much of a refresh I needed and very nearly failed the exam.
Lesson learned.
Assume the worst, hope for the best.  Assume you don't know it and work as if that's the case.  I'd suggest you just enter university and aim for the best results you can all the time.  I have no doubt you'll find that more of a challenge than you think now.
Also note that you need to learn a skill you may not have done yet : concentrating the most effort on the things you dislike the most.  It's easy to get into the subjects you like. To get a good B.Sc you need to work really hard on the things you're least attracted to.  I also nearly made that mistake on a couple of subjects.
From your comment :

My biggest concern with researching universities recently has been the nature of the courses (where they don’t cover my main interests).

This is a concern for later - again from personal experience I found that my interests changed and previously uninteresting subjects because a lot more interesting when I became more familiar with them.  Keep an open mind.

Thank you for your response. I fear everyone has read my question and just focused on “year 12 thinks they know more than they should”. Not really. I am made painfully aware everyday about my limitations. I provide the context of what I have studied only to justify my concerns, not to assert any kind of brilliance. It’s just that I want to learn the hard things I cannot constructively learn by myself - which does not include a non-negligible part of the course’s content. I appreciate my interests may well change - just not in topics I’ve looked at (at the levels taught in first year)

First year (and to some extent second year) will be dipping your toes into the water.  You'll discover their are way more things to do with math than you may do now.  I also wonder if you've really given much thought to other aspects of math - applied math and maybe numerical studies.  Plenty to explore ahead of you - no rush.


text:
How to write about awards in CV
I am currently writing my first CV to apply for grad school. While doing so, I am confused about how to write about some competitions. I tried to find this on internet, but the answer seemed not really uniform. I am particularly looking for the usual case in US, and field of applied math / CS.
Say a competition awards '1st prize' for one person, '2nd prize' for 3 people (2-4) and I got "3rd prize" with 8th rank in performance (awarded to 5-9). In this case, am I supposed to write '3rd prize' or '8th place (3rd prize)' or something else? It would have been easier if the award itself had some distinct name (such as bronze medal) but it officially does not. In these cases, what will be the usual way to write about this?

Are the names of the prizes translations?

And in which country were they awarded?

@Azor Title of prize is (quite literally) '1st prize' and so on. Competition is in my home country (not US).

Which country was that?

@Azor Korea. I am bit confused that this can be relevant. Can you please explain to me if this is relevant?

Because it might be! If you tag it with the country, someone who knows Korean academia might click on it.

And because I'm not familiar with this kind of award system, so someone who knows Korean academia will have a better answer for you than someone who's only familiar with American.

Thanks for clarification, but I am trying to get into US for grad school so I thought US convention might be more relevant :) In Korea, the competition is bit known so I thought I am fine with that. Thanks for advice!

I'd generally prefer mentioning "TOP 8 out of NNN competitors". This gives some leeway in case you don't know if you've placed 8th or 7th, and makes clear whether or not you have been ahead of a lot of people.
Saying you've got a "Bronze medal" in a math olympiad might look for an outsider that you were placed 3rd, while you might have been 200th out of a thousand competitors (still not a minor feat).
Consider that most of these prizes are usually not that relevant in a resume, that nobody should hire you just because of this, and if the interviewer believes this is relevant, he/she should ask for more details during an interview, where you can provide all necessary clarifications.
Keep in mind that you should be honest and not misleading, both in the resume and on the interview, and that a two or one-page piece of paper is never going to be a faithful representation of a candidate.

Thanks for advice. I agree that it will not be relevant but still I was curious. As you mentioned, I was also wondering if it will be misleading if I wrote so. I think I will follow your advice :)

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text:
Career choice：postdoc or non-tenure assistant professor job with little teaching load
I am a PhD candidate and I am currently in the mist of choosing my career path.
A little about my current situation:
I have a family in a large city where my husband grew up in. We are not likely to move because of his family and job, so I have geographical restrictions. I enjoy research but hate the stress of publish or perish. I do not dislike teaching, but it’s not my passion.
The first position: a postdoc at a R1 medical school with a PI who does exactly what I am interested in (autism intervention). The PI has good reputation of supporting her postdocs and she is a rising star with lots of NIH grants. It’s a two year position, but the grant(NIH R01) has four more years. I think it’s possible the PI might extend my contract if I cannot secure a job within the first two years.
The second position: a non-tenure teaching position for a lower-ranked R1 medical school near us. The campus is a satalite campus of the larger R1 university. Lots of research resources are available for that campus as well. But it doesn’t seem like faculty there, at least similar positions, are publishing as much. The main focus is teaching. But they were trying really hard to impress me during the campus visit that they do support research. Promotion is not as stressful. 5 year assistant then associate. If in five years you do not get it, you can stay for longer to apply again. Teaching load is about 3 hours a week. I will be teaching a theme that run through the first two years of medical school. Although this campus is small, it is growing. Pro for this position is that I can work there and look for other positions that has more research focus, since teaching load is not as bad, I am hoping to continue my research. They do have an autism clinic and there are ways to connect with the local community to conduct my research.
I am torn between this two. My advisor would definitely tell me to go for the postdoc because of the NIH funding training is invaluable. But I have geographical limitations, I do not know if I can land a better position than the medical school teaching position after I complete my postdoc. I have been looking at social work tenure track positions( my background), even R1 universities ask their early career faculty to teach two courses per semester. That makes the teaching at medical school not bad at all to me. Also faculty positions do not come around easily especially given my specialty. The teaching position teaches the area I am researching by and large.
Any suggestions or your first hand experiences would be appreciated. Particularly from early career faculty in applied health’s science. I know I am so lucky. But the choice is hard. It’s a perfect dilemma. Thanks in advance!

Welcome to Academia.SE, and congrats on your offers! I suggest you try to "boil down" your post to the key question you would like us to advise you on. As it is, the question seems to be "which one should I pick," which will generate the unhelpful response "this depends on the specifics of your situation; only you can decide."

Thank you! I guess my main question is, is it possible to launch a research career with a teaching position. I am mostly interested in others’ experiences in these types of medical school teaching positions as clinical assistant professors.

In the first sentence, "mist" was probably a typo, but I think it's well worth not correcting.

Teaching load is about 3 hours a week. --- Is this a typo? If not, this is about as minimal teaching as exists, at least in my experience and area, and thus it would seem to me to be a no-brainer in choosing this, since they want you and there is a reasonably good chance that you can stay indefinitely and you're still located in the vicinity of several people who research what you're interested in (both where you would be at, as well as those who are at the other university, which I would assume is within a reasonable driving distance).

Possibly worth considering is what is the chance of another opening for you at this teaching school being available when you exhaust the time spent in the postdoc position? If it's extremely high, then maybe do the postdoc, then reevaluate things when it's over (maybe your husband will desire, or have to, take a job somewhere else by this time), knowing that you still have a good chance for getting a position at the teaching school.

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text:
How can I handle group projects?
I'm a grad student and have been assigned to a group with several other students. This is a semester long group that will work on various projects. From the start, this group has been a disaster. I spent two days fixing grammar mistakes in another group member's paper. Some sentences didn't make sense at all and he obviously didn't proofread it because there were mistakes such as "for the for the". Another student in the group, an international student, told the student that wrote it, "great job". This was prior to me correcting it, but the fact that he complimented him made me think that he didn't read the paper at all. Absolutely no one would think it was a great paper, purely based on how many mistakes it had.
After we submitted the corrected paper, the professor sent it back to us with several notes about how to improve it. The original author told another group member that I should fix it since there were grammar mistakes and I wrote it (corrected it). In reality, his original paper wasn't even at high school level until I fixed it.
Another time we had to give a 20 minute presentation. The rest of the group wanted to go over who would say what in 5 minutes before class. However, with some coercing, I convinced them we should review it the prior day instead.
I have to maintain a B to keep my scholarship and I feel like this group doesn't care about their own grades. They seem like they just want to coast by into graduation. Should I approach the professor about this?

What field is this?

@Buffy I'm going for my MBA.

Generally speaking, a poor way to handle such situations is to divide up the work to individuals and then try to assemble the whole at the end.
If the group is reasonably small (say five or less) try to have regular meetings to brainstorm and outline things for the report. One person (probably the one with the best language skills) takes on the task of editor, or even writer.
Pass drafts around frequently and update them based on feedback.
If the group is larger, then do some division of responsibility, but to sub-groups, not individuals. Have each subgroup work as suggested above (brainstorming...editing...markups). But then, one or two people take on the task of getting everything together. Those two people need to monitor and shepherd the task throughout, making sure that the subgroups are doing their work productively.
It might even be advisable in a long project, to shuffle the groups a bit periodically. Not a radical reshuffle, but just some cross pollination.
You should also expect, and plan for, the fact that not every member will be able to contribute as much as others, or even at all, for various reasons, both legitimate and not. But just as important, don't set it up so that everyone needs to contribute in the same way. Try to draw on individual skills. Some doing research, some writing, some editing, etc.
A team should be a team, not a herd of individuals.
This might be different in a different field, but for an MBA it feels about right.


text:
How can I ask a professor for time-sensitive information when they are out for a family emergency?
I'm an undergraduate student, and at the start of the semester I was trying to organize something like undergraduate research (but more basic) with one of my professors. I have to fill out a form due at the end of the third week of the semester describing what the work would be, so I went to this professor during the first week of classes to discuss potential options.
About a week later, near the end of the second week of classes, I sent an email with additional research I had done on potential topics that looked promising based on what I found on their CV. They responded and told me that they would get back to me later with what they felt would work best.
The form is due at the end of this week, but the professor just sent an email out to the class that their office hours were cancelled and they would not be in class due to a family emergency. I obviously don't want them to feel any pressure, but at the same time, this is a time sensitive process for me.
What's the best way to ask for basic information about the project so I can begin working on the form?

Sometimes these things will have a faculty member who is in charge of the program overall even if they are not your advisor specifically (e.g. head of undergrad research in the department or something like that). Is there any identifiable point person you can explain your situation to / request an extension?

@TPM that's a good point, I'll make sure to do that.

@TPM, it would be best to make that an answer.

@TPM Yes I think that's the correct approach.

Keep in mind that the administration (or whoever is collecting the form) were probably expecting that you would have discussed the project with the professor well before the semester started.  So the fact that you only started these discussions in the first week of classes may seem to them like you left it until the last minute, and they may not be inclined to be lenient with the deadlines or requirements.

I'm an academic who has had to cancel class and office hours due to emergencies myself. We are all pretty tied to email these days wherever we happen to be, and regardless of what is going on in our personal lives, our professional responsibilities to continue to limp along. Depending on the specific nature of the emergency and the demands on their time, the professor may actually be able to help you nevertheless, but i) may well need reminding, and ii) may have higher priorities at this time.
I would send an email like the following (and if I received it, would respond as positively as I am able if the tone does not rub me the wrong way.)

Dear Professor X:
I understand you have had a family emergency. Sorry to bother you at
  such a time, and I fully understand if you are therefore unable to
  respond to this email. However, in case you happen to be working
  remotely, I would much appreciate your comments on the below, since I
  need to ... by .....  Or is there someone else you would recommend I
  contact instead given the timeline?
Thanks and apologies, ...


text:
Is it plagiarism and/or copyright infringement if research started from one person but another diverged onto a different path?
TLDR at the bottom.
Odd situational question for everyone (or maybe not) since this is out of scope for me. Hoping to get a clearer insight from a more experienced crowd about research, plagiarism and copyright. Unfortunately, I'll have to be vague about specifics and names. Apologies for the long post but I hope someone has some insight.
Just for background information but the theory has been around for almost 100 years but different concepts/ideas are used to prove or further research the theory. I'll start the timeline in the year 2000 just for convenience but information is easily accessible at this time.
Researcher A uses proven concepts that have been used in different fields of study but not widely used specifically in their field however it's not known whether or not they're the first to use. The concept used by Researcher A was discussed on a community forum in 1996 (4 years prior) and on another forum in 1998 (2 years prior) to researcher A starting their own initiatives in 2000. A also made their concept and findings publicly available on their website and started self-publishing a book in 2004 in relation to their findings, 4 years after they first started.
Researcher B is new to the field and comes across the concept from researcher A via their website around 2005 and becomes interested in the concept. During this time they're also reading other researchers' findings (100-year-old theory so lots to catch up on). The more B learns the more they find that while the concept may be feasible, researcher A's findings, definitions of the theory and methodology are flawed so B stops referring to A's research a few months later and starts to conduct their own research. In order to do this, B builds up the concept's methodology from what they believe should work after months of research. Because of the flaws that they found with A, B never read the book that A made either.
After B conducted their own research and findings they also release the information publicly and have changed the methodology of the concept. It's shown that both A and B while using the same theory and similar concept have different results and definitions for the theory. A also congratulates and supports B during this time and admits that their findings and methodology are different from B's. After 3 years of research (2008 now), B decides to publish their findings in a book. While writing the book they never referred to or looked at A's findings but did cite any work that they did look at when writing. A is now claiming that B has plagiarized and infringed on their copyright because of 2 of 6 the definitions written in the book have similar passages referring to the concept; the concept itself is somewhat limited as it's limited to 2 variations for each aspect (so if there are 3 aspects there are 6 definitions). However, these definitions which are similarily written are used for different aspects of the concept which results in people achieving different results for the theory more than half the time. Also since it's a limited scope there's only so much you can write that differs from others' writing.
As far as I'm aware you can't copyright an idea which is the concept in this case and even more so since it's been around even before A started. However, I'm not too sure about the plagiarism. A is demanding to be acknowledged for their work however they didn't contribute to B's research nor were they the originators of the core concept. They just happened to popularize it a little and were B's introduction to the concept.
Has B plagiarized A? Does A deserve the acknowledgement for the aspect definitions despite them being different and resulting in different results? Or just acknowledgement for getting B started on this concept? Or none at all?
TLDR: A accuses B of plagiarism and copyright infringement. B was introduced by A to the concept/idea but never used their information due to inaccuracy. Concept/idea has been around before A started just not as popularized.

Honestly, there's too less info provided here for strangers on the internet to guess and come to a conclusion. I'd recommend talking to your research guides to look into the content and sort out the matter. After-all, it's a few definitions, and you have branched off to create fresh research on your own anyway, so your research isn't in danger. "Pick and choose your battles". Some battles may not be worth the time and energy.
   Think about it this way: If you lived in a jungle and heard nothing about Newtons laws but you came up with those laws and definitions on your own and came to the city and tried publishing a book and then Newton accused you of plagiarism, what would be the right course of action? Unless you have hard evidence that you had not heard of Newtons laws earlier, nobody would believe you.

Unfortunately, the definitions are something like having a random person describe what an apple is from just looking at it. Without becoming too technical or abstract, there's a finite number of ways to describe/define an apple. Researcher A also tried to get B to abandoned their research and recruit them for A's research.

Thank you for the input as well.

Independently of whether you like the treatment of the theory by A (maybe it's incomplete, low-quality or even partially wrong), you have come across the theory through A. At this point, it is a matter of integrity to give credit to A for this.
Yes, the theory has been around before, and OP can develop their own derivatives, that is perfectly fine. It is not entirely clear now to me whether OP's definitions were derived/motivated by the definitions by A, or were developed completely independently. But, in any case, knowing about even a parallel (and not contributing) development, it is good form to cite it. People discovering things in parallel should give each other credit (check for a classic example for a case with a major dilemma the case of Darwin and Wallace).
I frankly find it difficult to understand why people insist on not citing others, despite the former clearly having delved into the work of the latter as a basis for their own, and insisting these others didn't contribute. Personally, I think it is at least appropriate to criticise them with open visor (but professionally) for lack of precision, lack of contribution or whatever one finds objectionable in the other person's work, than just silently drop them.
What is described here is not plagiarism in the strict sense, as far as I understand and assuming the description is accurate. Instead one could call this "diminishment or withholding credit". In this case this is credit for recognition of the importance of the problem, even if not dealing with it to the satisfaction of whoever cites them.
TL;DR: OP asks: "Or just acknowledgement for getting B started on this concept?" - Yes, at the very least.

I would agree that an acknowledgement for getting B started could be doable but A publicly said B plagiarized their work despite the divergence early on as A tried to stop B's research as well. A is claiming the research done after the fact is A's; because they introduced them and had some similar definitions. I added an example comment above regarding the definitions that they claim were stolen. An acknowledgement of a parallel study makes sense but how do you cite a book which you've never read or had access to? A does have information available online publicly but B never used.

Thank you for the input as well.

That A tries to recruit B is new information here. That puts an additional light on what's happening here, but, to be honest, it does not change the fact that acknowledgement of an existing source is the appropriate thing to do. Even if you had discovered it like after the fact, certainly, however, if you have been motivated by the person. Otherwise, people could just read an abstract, ignore the rest of the papers, develop a theory and say that they didn't use the work of the other person [it is polemic, I know, but is an extreme version of the situation]. Anyway, damage's done.

Best you can do, in my opinion, is, state you had been initially introduced to the topic by them, but then not further followed their development and start citing them from now on. I can only recommend not to play "who was first" games, this usually ends badly for both sides, but worse for the weaker party.


text:
What is the percentage of Latino/Hispanic females with a Ph.D. degree?
I am curious to know the percentage of Female Latino/Hispanic with Ph.D. degrees in the USA.
Does anyone know? What is the percentage of male/female out of the approx. 6% of Latino/Hispanic population with a Ph. D. (out of the of the 1.6% of all Americans with Ph. D. degrees)?
Thank you

Although the linked "duplicate" question doesn't directly address the breakdown by gender, the answers include links to data that I think does answer this.  I'd encourage you to see if that data answers your question, and if not, update this question to explain what's missing.

If I correctly read the Census data I linked in this answer, it appears that among Hispanic people (by Census definition) with a doctoral degree, 53% are female.


text:
Asking for a letter of recommendation after only some days of collaboration
I plan to apply for a Master's degree at a new university, and I require three letters of recommendation to do so. At my current university, there are two professors I did seminars/my bachelor thesis with, and both have agreed to write one - so there is one left. However, among the rest of the academic staff, there is none who knows me well (meaning I had one or two lectures with them, but not much personal contact). So I am wondering, if the following alternative is viable:
After finishing my thesis, my professor has offered me to continue research, which has now developed into some mathematical software. Our idea was to integrate it in an open source project, so I participated in an hackathon-style event of said project, where I have met another professor. The event was only 2 days, but he was interested in my project and supported me a lot. Therefore, I consider asking him for the last letter of recommendation, as he probably has a better idea of my work than most lecturers at my university.
Is this reasonable?

I am guessing you aren't applying soon? Since cycles are mostly over. Can you get more work in with this prof?

The deadline is in February, so this will be difficult.

The outcome of such things is unpredictable, but you might do well with this. But you can have a conversation first with this professor to see what they think of the idea.
Letters of recommendation are best when they make a positive prediction about your future success. It may be that this professor has already formed such an opinion and would be willing to express it formally.
You can ask, of course, but have a conversation first, getting their opinion on your future and how you can optimize it. A letter can flow from such a conversation.
Note that this is a US perspective and a personal one.


text:
NSF IIS status date changed, status still pending
It has been close to three months since my NSF IIS proposal submission. The status date changed today on NSF Fastlane but the status is still pending. Does it mean it is going to be declined?

That almost certainly means that proposal has gone through a review panel, and does not indicate what the recommendation from the panel is. There are several internal stages the proposal goes through after panel review.

Does it mean whatever the decision is, the reviews will be available soon?

It could easily take two or three more months.  They'll make the 6 month deadline but usually not by much.

OP, any news on this front?


text:
What to Do About Inaccurate Dates on Letters of Recommendation?
As part of the application for a fellowship at an university, I acquired two letters of recommendation.
Apart from a factual inaccuracy in the body of one letter, which I am going to mention in the motivational letter, both of the letters I've got are stamped to a wrong date. One is dated earlier than the application for the fellowship was released (January instead of February, though the day isn't accurate either, lol). The other one is dated to the day before the application is due, while I received it a couple of days before that.
Should I worry about these inaccuracies and mention them in the motivational letter? I don't think it's particularly important, especially for the latter one, which contains a sensible date (though the 'date created' definitely shows a different date from the one signed), but I want to be sure.

There are a lot of reasons for such inaccuracies. Copy and paste is one. Inattention is another. But none of them that I can think of are your fault, and none seem to be attempts at sabotage.
As such, I would ignore the issue, especially in written documents in which you have more important and positive things to write. If you are asked about them, your reply can be just that the writer likely made a mistake and if it needs correction, you can probably handle it then.
Use your motivation letter for precisely that. Don't use it to apologize for the errors of others.


text:
Quitting PhD 2 years into a 3-year program
I'm currently in a 3-year (European) PhD program in Computer Science/Machine Learning, but I feel like I'm very far behind schedule. I started on a very open-ended project proposal and grant, which I spent my first year reading up on, including taking other courses, TA'ing etc. After the first year, I realized it was a very difficult topic to get anywhere on, and my advisor is also by no means an expert.
For this reason I pivoted to a related subfield of ML that is more mainstream and that I am more comfortable with, and basically started over. Over the past year I've been reading papers, discussing them with people, and trying to get concrete ideas for research. However ideas have been sparse and the ones where I spent time on coding and experiments, they didn't pan out. Now, 2 years into my PhD, I have nothing to show for it. I have no research, no publications, and only some vague ideas on what to work on next. Do I quit? Is it even possible to finish by this point? I'm going to bring this up with my advisor soon, but I would like advice before I do so.

The advice, actually, is to ask your advisor. Can you finish? Can you extend? What are other options?

First, be aware that it's extremely common for a "3 years PhD" to actually last more than 3 years, it's very often 4 and sometimes 5 years. The point is that at this stage your case is far from desperate. Since you seem to be motivated to continue, I think the most important thing is to get all the help you can:

Talk to your advisor. Even if they are "passive", they certainly know how the institution works and what are your chances. You could also ask to have a meeting with the director of studies (or whoever is in charge of the PhD program), it's usually a good idea to let them know in advance that you might need an extension. You might also let them know that it could help to have an additional co-supervisor, if that's possible/desirable. In any case both of them can help you establish a plan for the rest of your PhD, listen to their suggestions carefully.
In the next 6 months you need to have a clear research path. Talk to any person who could help you, not only your advisor. Go to seminars related to your topic where people discuss ideas. Read papers in the topic you're interested in, not only the best papers from top academics in the field but also average-level publications by PhD students: as a PhD student your aim is not to make a great discovery, just to achieve a piece of decent research work. Try to play your strengths as much as possible: choose a specific topic which suits your interests, your skills and your knowledge.
Talk to the other PhD students, postdocs and any academic you're comfortable with in your institution. Based on their experience, you should try to know as much as possible about what can happen after the third year: do they know other PhD students who are or have been in a similar situation? What happened to them?
How strict is the school about the duration of the PhD, how easy is it to get an extension? Very often the official position is that PhDs must be finished within 3 years, but in practice there are many exceptions; you can only know about this by asking people.
I assume you have a PhD funding for 3 years right? Your PhD is likely to last longer, so you need to start planning for that: is there any kind of grant, teaching position that you could have after the 3rd year? In the worst case scenario, do you have savings and how long would you last on them?

As I said I'm assuming that you want to continue, but there's no shame leaving if you don't feel interested anymore. It's not too hard to find a good job with a CS/ML background nowadays. The one thing I really don't recommend is to quit in the hope to restart a new PhD program.

oh It's not a shame at all, and if the PhD student is encouraged to do so why not. However I think it's in general a bad option because: (1) it's not that simple to find a good lab and a a good advisor, and to get accepted (2) a PhD is long and often quite hard psychologically (sometimes also financially), it's hard to stay motivated many years and nobody gets any younger (3) there's a risk for some people to develop excessively high expectations about finding the ideal PhD topic and supervisor, usually ending in disappointment... As stupid as it sounds, PhD success is a lot about persistence!

@Erik there are all kinds of PhD student/advisor relationships. It's true that your situation is not the best but it's not the worst I've seen either. Given that your official advisor isn't useful for your research, maybe you could try to collaborate with other people who are more expert/interested in your topic? My main advice is to talk to other people, don't hesitate to ask for help/advice: in my experience the PhDs which don't end well are always people who stay isolated with their problems.

@Erik try to talk to people who know you and/or know the context, it's really important to make good decisions. But at the end of the day you'll have to trust your instinct and decide for yourself... Good luck!

Thanks Erwan, I appreciate the in-depth response. I will ask about an extension and also push my advisor for more concrete research.

You most probably got into a PhD program with a bad supervisor. This is not your fault. The first option is to talk with your supervisor and see if he thinks there is a chance that you can finish your PhD. Most of the time a PhD is more a social concept than a scientific one: science is hard - if people think that you did things to the best of your ability there is a high chance that they let you graduate with minor results. But be aware that this can set you into a situation of dependence and pressure that can be exploited and might cause mental harm to you. This does depend on your personal relationship to your supervisor. If you think you can go through this period and he is still positive about your projected progress, then stick with it. If not, then start looking for alternatives. There is no shame in starting a second PhD or finding a job in industry. If you openly explain your situation, people will understand. Additionally, doing a PhD is a highly idealistic approach and most people show signs of depression during a situation of hardship. You might want to seek out for additional help if you show signs of depression.

Thanks for the advice. I don't think he is a bad advisor per se, but perhaps a bit passive and not experienced enough in guiding and pushing PhD students to pursue practical research enough. I'll try to be more explicit with him in my needs for things to actually work on.


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How hard is it to get a good to great grades in undergraduate school?
I am currently studying undergraduate school in Europe and am toying with idea of applying to a U.S. collage for graduate school.
So I‘ve read a bit online about GPA requirements and there it says that one should have a score of at least 3.0. My current GPA would be way below that and I‘m not a crazy outlier at my school. All of my friends have grades similar to mine.
At the same time I know a few friends in America who all never have grades worse than a B.
So do I just coincidentally know two absolutely non-overlapping groups on either side of the Atlantic is it easier to get good grades in America?

Your title question doesn't really match the body

Of course admission committees for graduate schools in the US have experience with grading systems from other countries.  So there is no need to "compare" two grading systems.

I have received my undergrad degree in Eastern Europe, and my GPA was below 3 when I graduated. Of course it was a disadvantage while applying the grad school in the US because the admission offices usually check the GPA of an applicant first, and I was rejected directly. Yes, it was difficult to get accepted but not impossible. Here are some things that helped me:

Even though I had lower GPA than my US counterparts, I had a very good lab experience and internships from my undergrad years. I made sure that I highlighted those in my statement of purpose and in emails I sent professors.
Speaking of professors, I found and emailed professors whose research area was a good fit for me and told them that I was interested in joining their groups. It particularly helps because the professors can accept you as a grad student even though admission offices not. It happened in my case, and I was accepted for PhD even though I applied for MS.
In my country, we had to take 8 courses per semester whereas in US, most students take max 4 courses. It affects the performance, and I made sure to indicate it in my documents and conversations in a positive way.
Even though I am not a supporter of GRE, I made sure I get a good score to increase my chances for acceptance as it was required.
I also used transcript evaluation services to define my education in US terms. It is common to see GPA increase with those services. But of course it is not a guarantee.

Overall, I recommend being aware of your strengths and advertising them. Good luck on your applications.

It is hard to answer this question as the educational systems in the US and the EU are very different at the undergraduate level. A bachelor's degree, say in mathematics, in the US includes many other courses such as writing, history, philosophy, etc. Only about half of the program is mathematics. In the EU the degree is normally much more focused on maths. But for any given course in maths, I'd guess that the level of difficulty is about the same, though this varies widely from place to place.
So, a GPA (overall) in the US means something quite different from one in EU. The GPA in field is a more comparable number.
But, with more focus in the EU, some courses that would be taken in graduate school in the US show up earlier in the EU.
You have to work hard on both sides of the pond to achieve success. Don't expect lower standards in either place.

Is there a reason you focus on the EU instead of whole Europe?

@user111388, no, just that EU seems to have fairly unified structures. Bologna Process countries is a bit wider. But I don't know about everywhere.


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Do Conferences provide the facility of pre-submission inquiries?
This is my first post here, so kindly pardon any Beginner errors.
I noticed that it is common practice among authors to write pre-submission inquiries to different journals when it comes to journal publications.
I wanted to know if Conferences provide the same provision as well ?
My reason for this question is that i am working on a paper that i intend to submit to a conference whose deadline is due in just over a month's time.
I am apprehensive about submitting my paper straight away and instead would like some semblance of assurance from the conference reviewers/judges
Any help & guidance in this regard would be highly appreciated !

It would be helpful if you could specify what kind of conference it is. E.g., computer science, economics, art history. Large/small etc. However, I expect that in most cases the answer is no -- the only way to get feedback is to submit.

@ThomassupportsMonica It is a Computer Science conference. It is not so big apparently, as it is just their 2nd conference.

For some conferences the Program Chair might respond to an email enquiry. In others, perhaps not.
But there is no reason that you can't just submit your paper. If it is inappropriate in any way, you will be told as part of the review process. Don't waste people's time with something obviously off-topic, but otherwise just do it.
If a PC does respond to an enquiry it would likely be something like the above paragraph. Note that deadlines are pretty firm because of the limited time prior to the conference itself, though some conferences also extend deadlines when they don't get sufficient papers.


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Is it reasonable to do number theory without active/good prof in uni
I'm from the top math university in my country (it's not really world class but on average one guy from every other batch in my uni goes to Harvard/MIT/Princeton/UChicago/Stanford/UMichigan for PhD).
I'm in my second year and I'm broadly interested in number theory and geometric analysis, but there's no "good" faculty in number theory in my university (there's only one prof and he don't do any active research and don't have much reputation outside my country), so I'm almost on my own to study number theory.
It's pretty hard to learn everything by myself with simultaneously handling the coursework, so I'm wondering whether it's a good/sensible idea (wrt say grad school application where I need to compete with many people from top colleges in USA who are taking legit number theory courses by world class number theorists) to switch completely to geometric analysis and ditch number theory ? I think I can focus more on geometric analysis in that way too -- the only thing that is bugging me from switching completely is that I like number theory more than geometric analysis.
There were one guy from my college who went to Princeton for number theory who wrote a solo number theory paper as an undergraduate, but I'm nowhere close to being that smart. Also I can do reading courses/summer internship under some active number theorist researchers in my country, but the problem is that you need to know some nontrivial amount of number theory before you can do some internship under some reknowned number theorist !

You seem to want inconsistent things. A top professor in a top university, who is therefore very busy and likely has many students, but who can give you a lot of support, which takes time and effort. It may not be a realistic expectation.

@Buffy Thanks, but I'm not wanting a top prof in a top university (as I mentioned again, I'm a sophomore and I've no plan to change to another uni - so I'm stuck with the current prof if I want to do number theory)

you can study number theory in grad school in the US without having learned much as an undergrad. 2) if there's no one to teach you number theory, have you considered transferring schools, or at least participating in special programs (summer schools, budapest semesters, ...)?

Number theory is perhaps the historically deepest, most studied area of mathematics.  To even begin to understand research in most areas of number theory, you need two full years of graduate coursework over several subfields of mathematics.
What one can learn in an undergraduate course, or even a first graduate course, in number theory is so far from the frontiers of research that it makes absolutely no difference whether the person teaching is active in research or even has ever done research in number theory.  You probably won't get into the 20th century in an undergraduate course, and not much beyond the 1940s in a first graduate course.
Except for a few exceptional students who are taking several graduate courses while still an undergraduate, no one is doing serious, mainstream number theory research as an undergraduate.  Hence, it doesn't matter.  You're not competing with people who have somehow learned lots of number theory already.
I should add that, in the job market for permanent positions in pure mathematics that support significant amounts of research, the sky has fallen.  Only exceptional people are getting permanent research-oriented jobs now.  Given that your job after your PhD (and your postdoc if you get and do one) probably won't involve doing research in either number theory or geometric analysis, you might as well study what you're interested in.

Thanks a lot for your answer. Taking graduate courses is the norm rather than the exception in my uni... in fact the last year is solely for taking graduate courses. I won't most probably do a postdoc or stay in academia, but just curious, why what I do in my postdoc will not be related to my topics of interest ?

You parsed that differently from what I intended - I meant job after (your PhD and your postdoc ..), not (job after your PhD) and (your postdoc)

It is true, as @Mr Cabbage observes, that there probably are economic realities to be considered. Unless you're independently wealthy, you'll eventually need to have a paycheck. And probably a reliable one. If you're "on your own", with no responsibilities to anyone else, you can obviously take more risks than otherwise.
So.
One point is that you have just one (immediately apparent) life, and if you are fond of number theory and don't pursue it, it'd be a pity. But, yes, risky, especially in your situation of not having senior people to give you good advice about it.
Also, in my experience in the U.S. at a "research univ", and having been involved with grad admissions for 35+ years, sensible people on admissions committees are well aware that people applying to grad school are very much in formative, transitional parts of their lives. What I myself look for is enthusiasm and self-starting, as much as anything. So, for example, the possible fact that your affection for number theory might compel you to pursue it despite lack of support in your immediate environment is a plus... while I'd know that you'd have some disadvantages in terms of advice and mentoring.
That is, to first-order approximation, do what you want to do. Second-order: try to make a living at it. :)

I think you should do what your heart tells you to do. If you love number theory, do number theory. However, there are certain economic realities to take into consideration. Most people study with the intention of getting a job afterwards. Therefore, since you have a talent for mathematics, I would suggest that you become a statistician, since there are far more jobs in statistics than in any other area of mathematics. You can make a decent living as a statistician, and pursue number theory in your spare time. The fact that the distribution of prime numbers has properties akin to a probability distribution, in view of the prime number theorem, indicates that work as a statistician would not be inconsistent with doing number theory as well. Also, there are primality tests that involve probability arguments, such as the Miller-Rabin test. I would suggest statistics-cryptography-number theory is the way to go, in view of your talents.

Is it reasonable to do [SUBJECT X] without active/good prof in uni

I will answer generally, since whether number theory or something else, the advice is the same.
It will be harder, for sure. Having a good advisor, active, in your research area can be extremely important, especially early on in your PhD, to make sure you are studying the right areas, and learning things that are relevant to current academic research. Without such an advisor, you will have to be a lot more proactive in learning things on your own, reading textbooks and papers from your research area, studying problems, and digesting material (e.g. blog posts) from some well-known researchers at other universities.
However, by no means is it impossible. And there are three points to make here:

Having no professor in the subject X would be even harder, but here you merely have no active professor. That leaves you in a much stronger position. I would advise against being officially advised by a professor who is not active, but you can still have a close mentorship with them, and they can be your "real" advisor if they are interested and available. Just because they are not publishing anymore doesn't mean you can't learn a lot from them.
Your passion and interest in a subject will take you far further than any mentorship. If you choose a subject that you are not interested in, then why do a PhD at all? To spend 5-10 years on something, you must truly love it, and truly want to become an expert. If you do want those things, you can succeed, although it is a lot of hard work. (Your interests may evolve later -- I find that almost all students' interests do, e.g. evolving more towards what their advisors do -- but if you are right now interested mainly in X, then you should do X.)
By the end of your PhD, whether you have an expert in your field or not, you become an expert in your specific sub-problem who knows much more than your advisor. So, although choosing an area where you don't have an active advisor makes you "on your own" earlier, you would be on your own eventually anyway -- that is the whole goal of a PhD.


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Repository or university working paper series
I am from the field of economics and preprints are ok with econ journals. I just cannot understand what is the difference between repository (like ArXiv) and publishing results in some working paper series?
I read somewhere that repository assigns DOI and working paper does not. Also, it is possible to update the repository's version, and I am not sure about working papers. From another point of view, working papers might be registered to more databases and make one's research more visible to the public....
So what are the main pros/cons in publishing preliminary results in a repository and in working papers (or discussion papers series??)? What are the differences?

You might need to inquire with whoever handles your working paper series, but in my experience the two aren't mutually exclusive. At least in maths its quite common for preprints to be both put on arXiv, as well as in the local preprint series. Although that seems to lead to the latter slowly dying out.

What do you mean by working paper series?

for instance, Working Papers European University Institute, or Working Papers any university

People actively follow arXiv. They subscribe to the RSS feeds, mailing lists, Twitter bots, etc. that inform them about recent relevant preprints that appear on arXiv. Google Scholar finds new papers on arXiv quickly, and this way the authors will get quick notifications when something that cites their work appears on arXiv. ArXiv papers are also indexed in other widely-used databases (e.g. DBLP in computer science).
I don't see any reason to use a local working paper series. In terms of visibility, it doesn't sound much better than e.g. simply hiding the paper somewhere on your personal home page and hoping that Google will eventually index it and someone will accidentally find it that way.

(+1) Especially for your second paragraph. I was about to write a comment saying essentially the same thing before seeing your answer. I don't know about economics (but I wouldn't expect it to be much different), but with arXiv you have papers all grouped at one internet location and sorted by field. I'd have to look at hundreds of university sites in well over 30 countries to be on the lookout for papers dealing with topics of interest to me, and if I wanted others to see something of mine, I certainly wouldn't stick it in some local place that few would think to (or care to) look in.


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Applying simultaneously for faculty and research-in-industry jobs
I'm a last year PhD student; I love doing research and mentoring younger researchers. I'm looking to apply for faculty this Fall. In my field, there are also multiple companies where you can currently publish and do open-ended research. I'm strongly doubting between both paths: more money, and more time for life&individual research (i.e. my own ideas, letting my students mostly come up with their own) in industry vs. more mentoring opportunities, and research freedom being assured for all my career [modulo tenure&funding].
My questions are not about the doubts, but about applying with these doubts:

Do people often reject offers from top universities to go to research positions in industry? Is it badly regarded? (Professors in those universities would still be my research colleagues if I go to industry)
Is it ok to ask for significant help (letters of recommendation, feedback on statements, invited talks, emailing PIs of other universities ...) if I think the probability of me eventually wanting a faculty job is between 10% and 40%?

You are probably wrong about work-life balance. The people I know in top industry research positions don't just punch a clock and go home. Their lives are pretty much as intense as any university researcher. Lots of conferences, for example. Likewise probably wrong about more time for personal research. They have a boss and the boss has an agenda. A tenured prof is much more free to set their own path.

Please ask one question per post.

@Buffy Thanks for your comment. Re work-life balance: having done internships and studying in a university similar to those I'm applying, at least in my university, faculty do have much less time for personal life. Re personal research: I misspoke, I meant ideas coming from myself, assuming I want my students to have their own ideas. I edited the text to clarify.

@Buffy: Hmm, I believe without a doubt that what you write is true for certain "top" industry research positions - but from the question, it doesn't seem completely clear to me what precisely the OP is looking for. For instance, I worked for one and a half year in automotive development (but admittedly this was really development rather than research), and I indeed had much more spare time there than I do now in a postdoc position at a university.

I think there is a wider diversity of experience you can have in industry research, than what you see in academia. "Industry job" is really far to broad a category for what are really many different careers; there are jobs where you will punch a clock, and others where you will spend a lot of your time working. In academia, there's really essentially one job, and there is a standard set of expectations around what you do (teaching, research, service) -- the exact breakdown may vary, but the amount of variation is small, relatively speaking. This is not "good" or "bad," just an observation.

Do people often reject offers from top universities to go to research positions in industry?

It happens. I don't have any statistics about whether it happens "often," but I also don't think this is relevant. You should make the decision that makes the most sense for you.

Is it badly regarded? (Professors in those universities would still be my research colleagues if I go to industry)

Anecdotally: I recently left academia and the response I had from professors and other researchers in my field was almost uniformly supportive.
My general take: People generally want to see you succeed, and have are busy enough with their own problems that they aren't going to bother holding a grudge. If you are able to productively collaborate after you leave, most people will care much more about the work than your position. The few that do care, are not worth keeping as collaborators.

Is it ok to ask for significant help (letters of recommendation, feedback on statements, invited talks, emailing PIs of other universities ...) if I think the probability of me eventually wanting a faculty job is between 10% and 40%?

Yes. First, I don't agree with the framing that this is significant help, because every faculty person is fielding requests like this from many people every year, and not all of them will end up in academia. Second, you have to advocate for your own best interest. If you don't make these requests, you are 100% certain to not get a faculty job. If you have even a 10% interest in a faculty job, then not asking for letters, etc, is working against your own interest.

It's currently August.  At least in North America, if you want to start a new job in the next three months, you might get hired into an industry position but you are unlikely to get a faculty position because they do not hire in that time frame.  If you want to start a new job in 13 months (September), you might get a faculty position, but you are unlikely to get an industry position now because they do not hire that far ahead.

I know; I meant applying to start next September. I've already started working on my application for faculty, but I don't plan starting to apply to industry until Spring to be able to decide simultaneously.