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I got confirmed from a graduate school starting from next year and I will major algebraic geometry.

Until now, I have never thought that I study little things than others with my age. However, I heard that some of my colleagues already studied Hartshorne at least once and quite a few of them have read Rudin's RCA when they were undergaduates. It's kinda unbelievable to me, but it seems like if they really did study and understood, then they will write absolutely a better Ph.D thesis than mine.

So I'm now very worrying myself. I want to know whether this situation is general. Is it recommenable to study graduate subjects as early as possible? Or are there people here who experienced the same thing too? Was that beneficial?

Between "studying each thing deep and slow" and "skimming many subjects as fast as possible", which one is better?

Rubertos
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  • In the past I used to take a grad course every Spring. This was part of professional development, "strongly recommended" by my employer. Over the winter break I used to obtain the course syllabus, books and chapters for the first month and by the time the course started, I was prepared with the first part of the course. Being prepared is more important than starting grad school "as early as possible" – imranfat Nov 28 '15 at 23:20

2 Answers2

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Your questions in order:

Yes, it is highly advantageous to be exposed to more sophisticated ("graduate") subjects as early as one can tolerate it.

Not clear that one should "study" them.

Yes, some people do "read ahead". I myself found it very helpful.

I would claim that the widely-believed sense of "deep and slow" versus "skimming ..." is a fake comparison, and is not the question anyone should really ask. That is, studying "slowly" cannot possibly be "deep", because "slowly" also entails maintaining naivete and one sort of shallowness for an unfortunately long time.

Likewise, the "skimming... as fast as possible..." is not any sort of "other" alternative. A more sane "other" is "looking around, not getting bogged down in details, trying to see where things are going".

And, my recommendation would be to both read fairly carefully and also look around to see what's going on. Both lower-level details and some idea of the goals and larger enterprise.

Very specifically: much of "comfort" and "facility" consists of familiarity more than anything else. The psychological obstacle of novelty is surprisingly great, while the psychological ease of (even superficial) familiarity is surprising. Simply hearing the words and a bit of a story a year or two (or more) earlier is extremely beneficial, in all my observation (and my own experience).

paul garrett
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  • You are absolutely right. However, unfortunately I think I cannot handle both since I'm studying almost all day. It's matter of a time but I cannot make more extra time to be exposed to next-level mathematics more often. Moreover, if one says that he covered only Rudin's RCA when he was undergraduate, then I may try much harder to cover that too. However, it is Hartshorne's algebraic geometry. As far as I know, there are tons of prerequisites to do that. Isn't it way too fast? – Rubertos Nov 28 '15 at 23:34
  • "Look forward, look back, look down, look all around." Becoming aware of things is not so expensive in terms of time, though perhaps psychologically, when one starts to see how large the world is. The usual notion of "study" is an extreme exaggeration of ideas about school, classes, exams, ... Those things, and their exaggerations, are just an idiosyncratic, distant reflection of the genuine subject of mathematics (and probably most other things, similarly). – paul garrett Nov 28 '15 at 23:40
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    @Rubertos You write "I'm studying almost all day." Are you sure you're not over-exerting yourself? There were definitely points in undergrad where I realized I was burning myself out, and wound up learning more because I studied less (and therefore actually wound up learning what I was studying). Of course, it's very hard to learn where that line is, but my knee-jerk reaction is that you're probably on the working-too-hard side of it, just from the feel of your post and your comment (that said, this sentence is of course basically worthless, since I don't actually know you; oh well :P). – Noah Schweber Nov 28 '15 at 23:41
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    Following up on @NoahSchweber's comment: another psychological aspect is overcoming the quasi-paranoid idea/feeling/concept that you are being pursued by hungry wolves over the tundra... figuratively. That is, it is good to have some sense of urgency, of caring, but not beyond that ill-defined point that one can tolerate without losing one's marbles. A feeling of desperation doesn't seem to usually help anything. Although the journey is important, actually arriving at the destination is a non-negotiable good... – paul garrett Nov 28 '15 at 23:46
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Your question is very broad, and I'm not sure this fully addresses it; but this is too long for a comment, and hopefully you find it useful nonetheless.

I think there's a couple false assumptions here.

First, that there is a "better" way to approach studying mathematics. People vary wildly in how they learn, and ultimately I think it's best to find an approach to math that works well for you. Certainly you shouldn't discount the value of mastering difficult material early (and although rare, it is definitely believable for an undergrad to master such material), but at the same time, just knowing a bunch of stuff doesn't make you a mathematician. Grad school is much more about learning to be a mathematician than it is about learning material.

Second, and far more importantly, your line

It seems like if they really did study and understood, then they will write absolutely a better Ph.D thesis than mine.

This is something I'm still struggling to understand intuitively, so saying this in answer to your question also helps me internalize it: mathematics is not linear. Even ignoring the fact mentioned above that mathematics $\not=$ a bunch of facts, and even ignoring the fact that one's rate of learning changes over time, it is impossible to guess ahead of time how your thesis will compare with someone else's simply because there are so many different facets of mathematics. As you go through grad school, regardless of where you start relative to your peers, you will eventually find yourself an expert in some small area, just as they will in their own small areas. What contributions you make to this small area will surprise you, and it's pointless to try to guess ahead of time whether they will "match up" (however you might measure that) with someone else's.

Your thesis is not predetermined; it will be the product a number of things, including the growth you experience as a mathematician as you go through grad school (as well as a fair amount of chance, let's be honest). Certainly knowing more things at the beginning is an advantage, but it's by no means a dispositive one.

Noah Schweber
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  • To clarify: I am currently finishing up my own time in grad school, so I can't claim any "grand perspective." Still, everything I've written above is stuff I've been told by my advisor, and by other mathematicians young and old. – Noah Schweber Nov 28 '15 at 23:44