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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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    What do you mean by "computation"?2011-09-15
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    Is calculation of homotopy/homology groups of spheres/computing fibers of a morphism in algebraic geometry etc computations to you?2011-09-15
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    @Complex123: I think there could be a question here, but in its current state it is very ambiguous. Perhaps you meant to ask something like "do I need a solid understanding of calculus to learn Algebraic Topology?" or "What background undergraduate material is required to study Algebraic Topology?"2011-09-15
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    In computation I would mean like a page full of equations. Like if you look at the Hatcher book most of it is words with sprinkle of equations. I suppose calculation of groups are computations and are needed in both AG and AT. On calculation I mean needing a skill like calculus. See for calculus you need to build up the computational skill. Wondering is that needed in higher level AT and AG. As what I have seen is more ideas. So more of a mechanical question. Do you need strong mechanics to do algebraic geometry? As what I've read of Grothendieck he had weak mechanics.2011-09-15
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    @complex123, To be honest, I don't understand how in calculus you need to build up computational skills. Is it because this is the way it's usually taught? There are many computational stuff in basic AT or AG, such as Mayer-Vietoris sequence or Cech cohomology. These can be used to calculate invariants for many spaces, but the way these are usually taught do not focus on computations/drilling. (perhaps they expect you to be able to do it once you understand it)2011-09-15
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    ...But if you happen to take a class on AT/AG that focuses a lot on computations using these tools, would you consider it a need to build up the computational skill?2011-09-15
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    But solving a calculus problem is just a particular kind of computation, a particular type of mechanics. If I asked you to find the Cohomology ring of the Klein bottle cross the Klein bottle, what would you call that? If you are just learning, (i.e. me) then it is somewhat tricky, and you have to understand how to apply certain theorem. But for someone who knows how to do it, and who has studied the material already, it is "just a computation."2011-09-15
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    @Eric I couldn't figure out a good way of paraphrasing, so I removed the comment. I do think it's important to warn of the dangers of abstraction, but one could get the wrong idea from what I said ("I need to memorize tables of integrals?").2011-09-15
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    Well, in calculus it was just doing a ton of integrals and differentials. I didn't understand anything of it, but it's fine because I needed it to pass exams. But, now I'm reading that you don't need that in higher levels and you don't need to memorize computations. It's not as painful as learning calculus through. However, doing computations over and over again isn't fun.2011-09-15
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    @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.2011-09-15
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    Well, I've been taught analysis and so most of calculus makes sense now. However, even then doing analysis made me feel like all the skills i.e. memorizing integrals and techniques on integrals is pointless. Second year thought calculus was pointless. Third year thought learning metric spaces was pointless as topology is easier. Now at fourth year it seems everything I did was pointless. Like the computations aren't needed. Hence, why I want to forget everything I learn't. Like in away numbers are useless to someone doing number theory. No numbers in number theory.2011-09-15
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    @Complex: Very good, that is right memorizing integrals is not too useful. (It may speed things up, help on competitions or be a nice parlor trick, but otherwise... no.) Analysis is not pointless though, _Ideas_ are never pointless. If you understand the idea and the notion then that is what matters. Usually computations help you understand.2011-09-15
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    I think the question is ill-posed and phrased in a contentious way.2011-09-15
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    I don't understand why people play this psuedo-philosophical game. Of course it might be hard to pin down a precise definition of "calculation" and you could make some argument that all of math is computation or none of math is computation, but it is rather tiresome. Basically everyone that reads the question will know what was intended by "computation" and if you claim otherwise I claim you are being willfully stubborn or just trolling, so I say the question is not ill-posed.2011-09-15
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    To some extent this is an effect of the way things are taught. The main emphasis is on the ideas, as it should be, and the cleanest presentations of the ideas generally does not involve much computation. But if you ever get to apply these ideas to a real-life problem, computation may be necessary. See e.g. Cox, Little and O'Shea, "Ideals, Varieties, and Algorithms" for the computational side of algebraic geometry.2011-09-15
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    In Analytic Number Theory, and in many other branches of pure mathematics, there are very delicate computations.2011-09-15
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    I hate Number theory calculations the most. ANT looks horrible and that is pretty sad as I originally wanted to go do research in Riemann Hypothesis and complex analysis. Yet complex analysis is horrible and just seems like stupid computations.2011-09-15
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    @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.2011-09-15
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    On the m.se front page I read this as "Need for competitions in pure mathematics at the highest level," and I was like, "cool, but then it'd just be a bunch of specialists in their respective mini-competitions." Oh well. :)2011-09-15
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    I think when you've spent over an hour looking for a mistake that turns out to be an undistributed negative sign, then you're definitely doing computation :)2011-09-15
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    I also find this question contentious.2011-09-15

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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.