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I'm fourth year undergrad student and I've noticed the skills that I've built up to do computation isn't actually being used.

A good example is algebraic topology, I've never really used calculus in it or PDEs technique. It just seems everything that has been developed is useless to algebraic topology. Only thing I use is group theory and then most of it like common sense reasoning with pictures and heavy use of category theory.

So the soft question is, what computation do you need in algebraic topology/algebraic geometry? As it seems you need none apart from group theory and commutative algebra in AG. Algebraic topology seems to be more understanding as opposed to calculation.

Edited:

What should really asked is this. What mechanical skills do you need in high end Algebraic topology and geometry. As I've read that Grothendieck didn't know that 57 wasn't a prime and that Bourbaki was saying that you don't need heavy calculations. So was wondering is it worth it to revise all of analysis and skills like solving PDEs, relearning Linear algebra e.t.c., when it seems the skills are useless.

Because I really don't want to relearn computations and certainly don't want to relearn analysis and complex analysis. Plus, I've been reading you don't need it. I suppose the big problem is that undergraduate algebraic geometry looks nothing like graduate text books in algebraic geometry. So what computational skills do you need for graduate level AG.

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    $\begingroup$ What do you mean by "computation"? $\endgroup$ – Eric Naslund Sep 15 '11 at 16:49
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    $\begingroup$ Is calculation of homotopy/homology groups of spheres/computing fibers of a morphism in algebraic geometry etc computations to you? $\endgroup$ – Soarer Sep 15 '11 at 16:50
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    $\begingroup$ @Complex: Computations are very important as they help by adding to your understanding of what is going on. However, it seems that your complaint is based on the terrible way mathematics is taught at the lower levels; only computations and no understanding. I don't think you should associate calculus with just pages of computations, but rather how you in particular were taught calculus at that time. $\endgroup$ – Eric Naslund Sep 15 '11 at 17:23
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    $\begingroup$ I think the question is ill-posed and phrased in a contentious way. $\endgroup$ – Ryan Budney Sep 15 '11 at 18:07
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    $\begingroup$ @complex123: I do not understand what point there is in your telling what areas of math you believe are «horrible» and «stupid». You do not seem to be that well versed in them, so we (and you yourself...) should take your judgements of them with a bit of care. Indeed, your argument is more or less based on your not knowing what (what you call) computation is for; do you know what kind of computations are carried out by algebraic number theorists?. If my vote were not final, I would vote to close. $\endgroup$ – Mariano Suárez-Álvarez Sep 15 '11 at 20:43
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Some of the comments above reflect that the idea of 'computation' is unclear in the question, and I agree. It is. In particular, calculating homotopy equivalents, homology, fundamental groups, morphisms, etc. are all calculations. And some of them are very hard. So in that sense, even in the algebraic fields, there are often many calculations. So 'skills for computation' or 'calculation' won't suddenly become useless.

Perhaps you meant, "Do I need calculus or the skills of analysis to learn algebraic geometry or algebraic topology?" Often, that answer is no. Not to begin. But the deeper the result, the more it will intersect with other fields (usually). For example, suppose that you ended up going towards the Riemann Hurwitz Formula or Mapping Class Groups, both of which are heavily entrenched in algebraic geometry and algebraic topology. Firstly, just a glance at the linked pages will show a small amount of the sheer amount of computation involved in manipulating objects with these ideas. But for a little background: Riemann-Hurwitz has grand cross-overs with complex analysis, which is really cool. For that matter, there are many things in complex analysis that can be viewed with an algebraic flavour (and many that perhaps don't).

Although I don't do much algebraic geometry myself, I know that there are several crossovers into the realm of complex analysis other than the one I mentioned above. And if you have any intention of applying anything you do, then some of the more computational sides of number theory and analysis may become very useful. Some of the people with whom I research happen to be very good ad algebraic geometry, and can sometimes use it to shed light on something we're looking at. And that's sort of the idea, right?

Really, much of the useful things from the analyses are tools, just like the tools we use from group theory. You call them computations - I call them the application of tools. And in theory, you already know them, right? Will it be so bad to relearn these ideas while they're still relatively fresh?

But in the end, it comes down to what you want to do. It is possible to be an algebraic geometer and perform relatively little computation compared to, say, an analytic number theorist or something. It is possible also to not know that 57 is divisible by 3, as you mention in your comments. But there will be come computation involved, and it is silly and restrictive to be afraid of these things.

I finish with one more thing 'in the end:' if you study math in grad school, then in all likelihood you will not have the opportunity to ignore the analyses. Almost every reputable grad school has qualifying exams, or some other form of examination, on those subjects because they are so often useful. And one day, when you figure out what you really want to do in whatever subject you pursue, you can then truly begin the process of specialization.

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