I’ve been writing a bunch of notes to myself about algebraic geometry. I posted them earlier, when they were smaller, covering just basic ideas of coherence. They’ve expanded (and, I think, improved) quite a bit since. Here they are. (Edited 8/30 to remove a few embarrassing mistakes.) My goal is for them to continue expanding significantly as I continue to learn more about the subject over the next year; ditto for my notes on algebraic number theory. For now, however, they are incomplete, inconsistent, and incorrect in places; please use only with caution.
I recently moved in to Harvard, so blogging will likely be light in the near future. I am toying with the idea of posting my notes from each of my (math) classes each day (and dumping a full PDF here at the end), but it remains to be seen how that will work.
Climbing Mount Bourbaki 
August 29, 2010 at 11:21 am
Exciting. What are the classes you plan to take by the way?
August 29, 2010 at 2:25 pm
I’m not really sure yet (Harvard has a “shopping” period to try out different courses). But Jacob Lurie is teaching commutative algebra, which should be fun; I’m also looking at potentially algebraic topology or algebraic geometry.
August 29, 2010 at 5:01 pm
Are you going to be live tex-ing, scanning, or just tex-ing after class. If it is the third I would advise against it. It is a lot of time, and while others would appreciate it, I suspect your time would be better spent otherwise. This is however, my opinion. I have done it a time or two myself.
August 29, 2010 at 5:11 pm
Live-TeXing. Typing notes up afterward would, indeed, be quite time-consuming.
August 30, 2010 at 12:08 am
Actually, I’ve found it to be a good way to review. You have to be a bit choosy about what to include in order to be efficient with time, and this forces you to think about the subject broadly… at least for those of us who are less capable of absorbing theories!
Anyway, I’m a sophomore concentrating in math, at least also shopping Lurie’s commutative algebra. I hope I get a chance to meet you!
August 30, 2010 at 5:56 am
Cool, see you there! I’m not decided yet, but I meet with my advisor today, so maybe I’ll have a better idea of my courses this afternoon.
August 30, 2010 at 11:13 am
Okay, nice (I don’t want to hit the reply button for fear of making this thread too skinny!). Although I certainly have no claim to wisdom, my advice would be to take no more than two math courses this term, perhaps even just one. When you go to graduate school and beyond, you’ll have all the time you need to think about mathematics anyway. There are so many interesting other subjects and only four short years to dabble in them! In particular, it’s useful to look for interesting classes that fulfill general education requirements, as you really don’t want to be stuck doing these in your upperclassmen years when all the really awesome stuff becomes available. This is something I wish I had been more mindful about last year.
August 30, 2010 at 3:58 pm
Thanks for the advice! I just had a chat with my advisor, and it indeed looks like I’ll probably take two math courses and sit in on a couple more. The main reason is that, yes, taking only math/physics/CS will be a questionable approach when there are general requirements.
August 30, 2010 at 11:12 pm
I’d personally recommend you to take language courses. You’ll need to learn several languages if you want to do research in math because sometimes papers are written in other languages (though that’s becoming less common these days, it still happens). So definitely take language courses!
It’d be interesting to try the scheme of taking one math course, one physics course, one CS course and one language course! It’s anyway your first year (if I’m right). You’ll have plenty of time to take math courses from your 3rd year onwards (up till the end of your PhD). And anyway the CS, physics and languages will play a big part in your math career (I’m a close friend of a very well-known mathematician though he doesn’t work in algebraic geometry (not properly anyway, sometimes he needs it though), he’s an expert in several other areas. He’s told me he always regrets he never did CS and physics and languages when he was young) Some of the areas he works in makes heavy use of these topics (like diff. geometry)).
I know it sounds sort of strange to think about other subjects especially since you look like you’ll become a mathematician (so you may ask, why am I doing other stuff?) but trust me, it’ll help you enormously later on (and I told my friend about you, he says that you have a *broader* background in math already, as an 18-year old, than many, many professional mathematicians will have in their lifetime. Do math of course but work on other things as well! Then you’ll certainly be the most complete mathematician by far (BTW, you mentioned comm. algebra – CS makes heavy use of that and conversly).
BTW, what are the requirements at Harvard? Are you supposed to take like a certain number of courses in other subjects or can you take what you like? I don’t know Harvard though I’d imagine that it wouldn’t be too different to other places where they ask you take lots of subjects in many areas.
August 31, 2010 at 12:25 am
Though I’m not your advisor or anything, I’d like to second all of the above comments.
Harvard is an excellent place for academic pursuits. As an undergraduate there, you’ve probably got access to the best possible courses available worldwide in various subjects. The math is excellent, and it’s great that you’re taking advantage of it, but you should probably look to broaden your interests. (Math is broad, I know! But they’re so many beautiful ways you can connect math to other areas. Speaking personally, I have found the connections between CS and math very interesting.)
I don’t know if taking only one math course per semester would be suitable for you (hence I don’t agree with Henry and Tony completely) especially since that’s your interest and that seems to be what you like to do. On the other hand, taking 3 or more mathematics courses might be too skewed toward one area. Two courses in math seem reasonable. It’s up to you, of course, to decide what you want to do.
Even with math, while you’ve already done quite a lot, have you considered taking Math 55? It’ll probably be material with which you’re already familiar, but it’s always good to solidify your knowledge, especially since you’ve learnt quite a lot of math on your own. Also, Math 55 is no walkover by any means, even for students who have done university level math while still in school. I’m sure you’re different, certainly much more experienced than most, if not all, of the students taking Math 55, but it’s just a recommendation that you might like to consider …
But I think your advisor has put you on the right path. Doing 2 math courses is definitely reasonable (I guess you’ll increase this as you progress through your studies) but consider learning other subjects as well, even if you do 2 math courses. Perhaps one semester, you might like to stick to one math course, just to see how you find other subjects, but I wouldn’t recommend going too far off the math world! Good luck!
August 31, 2010 at 6:41 am
Dear David and Henry:
Thanks for all the advice!
I agree with David, and think two math courses will be a good medium, even though I probably will try to sit in at least on the commutative algebra lectures (or the other course out of the three I mentioned I choose not to take). As far as CS is concerned, I may try to sit in on one of the undergraduate intro to programming courses if I find I still have free time; I am currently undecided about this. Having talked to many graduate students, advanced undergraduates, and professional mathematicians (and having attempted to read research papers), I am fully aware of the limitations of my current background, and I think that taking at least two (and possibly sitting in on another if I have any leftover time) would be a good idea to keep moving forward.
(I think I might have taken three if I could take five courses, but for the freshman fall Harvard does not allow it. The difference between sitting in on a course and doing the problems versus taking it for a grade seems rather academic to me, though.)
My experience in reading papers in foreign languages is that translation is generally not too difficult, since most of the words are cognates. I have only, however, had to read in French, which conveniently (and coincidentally) was the language I studied in middle and high school. As a result, I actually don’t anticipate taking language courses.
Re Math 55: That was my initial plan a few months ago, though a few people (such as my advisor) advised me that it would be more efficient to take other courses, and after a while I decided to do that. I actually would still try to sit in, but right now I don’t anticipate doing so since it conflicts with algebraic geometry.
August 31, 2010 at 11:05 pm
You mentioned that you’re aware of the limitations of your background which I personally believe is a good thing. However, you also mentioned that you tried to read papers and talk with professional mathematicians. If based on this you feel that your background is limited, please don’t! Let me tell you that even for a professional mathematician who spends at least 30 years doing mathematics, there will always be plenty of papers, even in their own field of speciality that is completely alien to them. I’m guessing you’re talking about papers in algebraic geometry in which case this is even more true – the literature is very, very vast – it takes a good deal of time and patience to learn how to get used to it.
I think you’ve done the right thing to continue to expand your knowledge. I’m not sure how long you’ve been doing mathematics or exactly what mathematics you’ve been exposed to but it seems to me that you’ve definitely got a headstart in your mathematics education. On the other hand, I think it’s important for you to keep in mind that, while learning mathematics could be useful in your later research, it could in the end play a much smaller role than you hoped it could play.
For example, I’m no great mathematician by any means, but I have known some mathematicians who have done cutting edge research in their fields, but who knew very, very little mathematics prior to doing so. By this I mean that some of them were exposed to not so good undergraduate math curriculums and had virtually 2 years to learn as much math they could before they really had to spend time on their research. But they were really efficient – they managed to do things early in their career that few people would imagine doing in their lifetime. And once they got tenured, they virtually could do whatever they wanted. (And at that time they broadened their education.) I don’t want to name names here, but I’ve also known some sub-sub-sub … -sub specialists who’ve done tremendous research.
I think it all comes down to the fact that if you’re hard-working enough, you should be able to make really good use of what you have, however little it is. Continue learning math by all means, but I feel (and this is my opinion) that you should assess the situation. E.g., ask yourself how comfortable you feel with doing original research. Try to practice that kind of thing as you learn mathematics if you feel you need to work more on it. I’m not sure how you learn math, but I hope that you’re working out some of the harder problems in your textbooks and really trying to see beyond what your textbook is doing. I think this is an important skill in original research – being able to come up with ways to dodge your limitations – and do significant original research. Getting practice with that kind of thing early would be very useful.
So ultimately what I’m trying to say is that you should have a go at reading some papers and doing original research just to keep up that kind of thinking. Reading and going to lectures, and learning a broad spectrum of math is all well and good, but I’m not sure how much difference it’ll make when it comes to original thinking. It’ll definitely give you more tools at hand which is important, but developing a research mind takes many years and that probably won’t happen by just reading math. I’m not suggesting you haven’t done research before, by the way, I don’t know you in real life, but if you haven’t done much, you might want to consider my ranting above …
September 1, 2010 at 5:47 am
It’s interesting that you mention that–actually, I spent a fair amount of time in high school trying out research projects (such as this one; there’s another (on analysis) that got somewhere but whose writing up has been long delayed), and I went to an REU last summer as well (though didn’t get anywhere). So my impression actually was that I should do a more learning-style summer program this year (Harvard has a program called PRISE that can be like that, I believe, which I was considering). I know the project I did the summer before last (which I had linked to) was something I knew nothing about when I got there, and speaking to my mentors there was one of the things that made me realize that I knew no algebraic geometry!
I always take a crack at the problems in a textbook (unless it’s EGA, of course :)); I’m also intending (if time permits) to do all the problem-sets in the additional math classes that I’ll sit in on. Research has been a fun experience, though, so especially if I can get into one of the really strong REUs, I’ll definitely consider it. (Most of the papers I have been looking as of late have been relatively old ones like Tohoku, FAC, etc.)