How does one appropriately navigate the situation of having a lot of prior knowledge before applying to university + courses not on main interests?

Question

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).

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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!

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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.

Answer 1

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.

Answer 2

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.

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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.

Answer 3

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.

Answer 4

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.

Answer 5

(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).

Answer 6

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

Two key points :

1. 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.

2. 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.

Answer 7

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.

Answer 8

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.