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.
- 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.