I won’t have time for a substantive update for about a week, so here are some bullet points.
- Peter Cameron has a blog, which I somehow didn’t know before. Terence Tao and Tim Gowers both have links to it, but the Secret Blogging Seminar doesn’t. Anyway, it’s excellent.
- I’ve been looking over the course listings at Cambridge for Part II. Conclusion: by Cambridge standards, I ought to know more physics.
- Andrea Ferretti recently started a new math website, MathOnline. It’s for posting links to free mathematics resources on the internet. I think all of my favorite online resources are already on it, so I haven’t contributed anything yet, but I’m happy that they’re all gathered in one place.
- I’ve also been reading Kassel’s Quantum Groups and Turaev’s Quantum Invariants of Knots and 3-Manifolds, in part to better understand Scott Morrison’s answer to my question about the Catalan numbers, but mostly because what little I know about the subject is fascinating. But I feel guilty about doing this; I can’t help but feel that as an undergraduate I should be paying more attention to my fundamentals and learning about fancy stuff later. What do you think?
Annoying Precision 
If you’re statement that “by Cambridge standards, I ought to know more physics” is based on the Part II courses, you should note that noone has taken all the courses in Part II – a good student probably takes 10 or 11. Most people think of themselves as either Pure or Applied mathematicians (plus some statisticians) by Part II, and it is rare to take more than two courses from the other category (many people take zero). Of course, if you wanted to take some applied courses in Part III then it could be an issue, but not otherwise.
I suspected that, but it’s good to hear from somebody else. The Cambridge system is sort of frustrating; ideally I’d like to take applied courses in Part IB and pure courses in Part III, but I’m told I should focus on Part II for the sake of the Tripos. Maybe that’ll just be part of the change of pace from MIT.
by Cambridge standards, I ought to know more physics.
I am guessing (although this is based on very broad impressions and not attending either) that if you attended Cambridge, by MIT standards you ought to know more computer science.
Everything’s got their focus. It’s not something to get hung up about (unless of course you want to study mathematical physics!)
I think that – as long as you are not neglecting your basic studies to the point where you are failing exams – you shouldn’t feel guilty about wanting to learn fancy stuff! I have had similar thoughts in the past, and my experience is, firstly, that there are always more “fundamentals” to be learnt; and secondly, that the process of trying to learn about more advanced subjects often involves filling in any gaps in your knowledge of important basic material.
I don’t think it is a good idea to spend all your time thinking about one specialised subject, but it is good to have something to aim for!
I think it’s important to focus on fundamentals, but don’t you already have that covered? I didn’t really learn that much math as an undergraduate, and looking back I wish I took more broad courses. From my own personal experience, the deeper you get into a given subject and start to enjoy it, the less incentive you have to learn completely new things.
But then again, learning fancy stuff and learning fundamentals don’t have to be mutually exclusive activities, so don’t worry about it.
Face it, Qiaochu. Even by Cambridge standards, I think you’ve got your “fundamentals” pretty well covered already 😛