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Superficially I think I understand the definitions of several cohomologies: (1) de Rham cohomology on smooth manifolds (I understand this can be probably extended to algebraic settings, but I haven't read anything about it) (2) Cech cohomology on Riemann surfaces, or schemes (3) Group cohomology in number theory and I have some rough understanding of interpreting cohomology functors as derived functors.

So my question is: what do higher cohomology groups mean concretely?

Some specifics:

For (1): closed forms modulo exact forms, but is there anything more concrete? It does solve some differential equations, but...is there more to it?

For (2): Serre duality implies the 1st cohomology group is dual to linearly independent meromorphic functions satisfying certain conditions wrt a divisor. How about higher cohomologies? My primary source is Forster's book, so the Serre duality treated there might not be the most general possible.

For (3): $H^0(G,A)$ is $G$-invariant elements of $A$, 1st cohomology is the $A$-torsors, 2nd cohomology is extensions of $A$ by $G$. How about higher cohomologies? My primary source is Artin's "Algebraic numbers and algebraic functions", and "Cohomology of number fields". The latter book (p.20, 2nd Edition) states very roughly that (my interpretation, sincere apologies to the authors if I misunderstand anything), higher cohomologies may not have concrete interpretations, but they play significant roles in understanding lower cohomologies and proving results about them.

PS: My background is (in case it's needed), very very rudimentary knowledge in analysis, algebra, algebraic geometry and number theory, but have not seriously learned any algebraic topology (though have seen the proof of Brouwer fixed point theorem via singular homology). I might have a tendency for the analytical and algebraic understanding of things (e.g. my primary impression of cohomology is that it's the obstruction of exactness, the need to extend exact sequences).

A side question: is it advisable to actually seriously learn algebraic topology to get a better idea of cohomology theories?

Thank you very much.

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    There are conceptual pictures available, but they're sort of complicated: http://ncatlab.org/nlab/show/cohomology If you think in terms of extensions, higher group cohomology measures "higher extensions," the simplest example being http://en.wikipedia.org/wiki/Ext_functor#Construction_of_Ext_in_abelian_categories2011-09-13
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    @Qiaochu : I don't think that someone struggling to understand the basics would be enlightened by a discussion of topi and higher categories. In fact, most experts get by without ever studying these things...2011-09-13
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    @Adam: maybe. I personally find it useful to know that a conceptual explanation exists, even if I can't currently understand it.2011-09-13
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    @Qiaochu : That's pretty meta. In any case, I want to remark (largely for the OP) that the people at the nlab have an amazing ability to take things which I use every day and understand at a fairly deep level and write about them in such a way that I have no idea what is going on. It's probably best not to spend too much time there unless you have a really good reason to.2011-09-13
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    Thank you both for your kind help, I will keep your advice in mind.2011-09-13
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    @Qiaochu: Probably the "non-technical" (but still incredibly technical) motivation page might be better: http://ncatlab.org/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks2011-09-13
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    @Adam: If there are things there that you use every day and understand deeply, but you think aren't explained the right way...then add to it! They encourage different explanations and want many ways to view the same thing. Take some initiative. Don't just wait for someone else to write it the way YOU want.2011-09-13
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    @Matt : But I don't agree with their underlying viewpoint (that the best way to understand things is via higher category theory). There are a couple of people who prove good theorems that way, but if you look at the work of most of the exponents of this viewpoint, then you will see a lot of pointless generalization, but few theorems that are interesting to people outside their little world. In fact, if I were in a bad mood, then I might describe things like the ncatlab as damaging to mathematics -- they lead smart but impressionable young people away from the core areas of mathematics.2011-09-13
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    In summary : there are a couple of geniuses (eg Grothendieck or Jacob Lurie) that can prove good theorems via excessive generality. However, a mediocre imitator of Grothendieck is much worse than an average mediocre mathematician.2011-09-13
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    @Adam: So surely you would be benefiting mathematics by writing up your viewpoint...as I said they aren't *really* pushing this viewpoint. They encourage the down-to-earth perspective to be present on all pages. Alright, this isn't the place to get into this argument, I just wanted to let the poster know that the nlab people are really smart and willing to answer any questions you have about anything on the pages including giving you a non-categorical way of thinking about it. So cautioning to not spend much time there isn't great advice in my mind.2011-09-13
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    @Matt : This is straying offtopic, so it will be my last response. Are you honestly claiming that the ncat people are not pushing a point of view on how one should think about mathematics? They certainly claim to be pushing one -- see their page http://ncatlab.org/nlab/show/nPOV. Since I strongly disagree with this point of view, why would I want to contribute to their website? And anyway, a website is a pretty lame place to spend a lot of time writing things up. If I decide to write an expository account of something, I'll write a book.2011-09-14
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    let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/1338/discussion-between-matt-and-adam-smith)2011-09-14

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