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In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).

For example in natural number theory it is good to know quadratic reciprocity and in linear algebra it's good to know the Cayley-Hamilton theorem (to give two examples).

So, what is one (per post) deep and representative theorem of each subject that one can spend a couple of months or so to learn about? (In Combinatorics, Graph theory, Real Analysis, Logic, Differential Geometry, etc.)

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    Similar to this [question](http://mathoverflow.net/questions/2340/what-is-the-first-interesting-theorem-in-insert-subject-here) in MO.2011-05-17
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    The rank-nullity theorem isn't deep (in my view). Cayley-Hamilton is a better choice for elementary linear algebra.2011-05-17
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    The MO question is "first nontrivial theorem" which is related, but hopefully it is clear that I am asking for something a bit different.2011-05-17
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    @Yuval, That is a good point and actually I agree but I have not studied a lot of linear algebra recently.2011-05-17

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