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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 in MO. –  lhf May 17 '11 at 18:22
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The rank-nullity theorem isn't deep (in my view). Cayley-Hamilton is a better choice for elementary linear algebra. –  Yuval Filmus May 17 '11 at 18:38
    
The MO question is "first nontrivial theorem" which is related, but hopefully it is clear that I am asking for something a bit different. –  quanta May 17 '11 at 18:40
    
@Yuval, That is a good point and actually I agree but I have not studied a lot of linear algebra recently. –  quanta May 17 '11 at 18:40
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16 Answers

Differential Geometry: the Gauss-Bonnet theorem.

I took a one-semester intro course on differential geometry class and we got to this towards the end of the semester, so I feel that a couple of months is an appropriate time frame for this theorem.

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In Analytic Number Theory, the Prime Number Theorem is a great result.

Riemann's 1859 paper, which outlined a possible approach to proving the PNT, is credited with motivating a large amount of the research done in Complex Analysis in the 19th century.

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It might be too obvious, but Gödel's incompleteness theorems are certainly landmarks, not only because of their historical significance and "popularity" outside mathematics (where usually they are quoted in a wrong way, in a wrong context, in order to claim a wrong claim) but also because it's ideas are actually the roots of Computability theory - and understanding both the theorems and their connections to computability might take some time indeed (but I wouldn't say a couple of months...)

In complexity theory I'd say the Cook-Levin theorem, which leads to the entire theory on NP-completeness, is a good representative. But then again, the theorem itself is mainly technical and not very illuminating - it's the implications that are important and need to be studied.

Yet another main landmark of modern complexity theory is the PCP theorem (actually PCP theorems - there are many versions). This is a very surprising and powerful result.

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A nice example in Algebraic number theory is the solution of the $p=x^2+ny^2$ problem, described in details in Cox's book. Very much like Fermat's last theorem it has its roots in the 17th century (actually, it originated from Fermat...) and is completely solved only using some heavy machinery from number theory (although not as heavy as FLT). I'd say it is a good representative for the power and depth of the field that is still very much accessible.

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In Probability theory I was told by my professor, that that 3 most important theorems are Central Limit Theorem, Law of Large Numbers and Law of the Iterated Logarithm.

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In Finite Group Theory, while the Odd Order Theorem and the Classification are major results, I would put a major landmark at the Sylow Theorems and Hall's Theorem (a generalization of the Sylow Theorems). Especially the former come up all the time, and there are many interesting corollaries that often are not discussed.

Another good possibility is the O'Nan-Scott Theorem for the study of permutation groups.

(Also, I think it would take a lot longer than "a couple of months or so" to really learn and understand the Classification of Finite Simple Groups...)

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On the Sylow theorem there was a thread on MO with illuminating examples why one should care about this result (in case it wasn't clear already). –  t.b. May 17 '11 at 19:04
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For Field Theory, we have The Fundamental Theorem of Galois Theory. It is a basic tool not only for fields, but for algebraic number theory among other fields, and it establishes a beautiful connection between group theory and field theory.

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In subject $X$ the Fundamental Theorem of $X$ is always pretty important.

For example:

  • Fundamental Theorem of Finitely Generated Abelian Groups.
  • Fundamental Theorem of Calculus.
  • Fundamental Theorem of Arithmetic.
  • Fundamental Theorem of Algebra.
  • Fundamental Theorem of Galois Theory
  • Fundamental Theorem of.....
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In Combinatorics you have for example Szemerédi's regularity lemma and all sorts of "related" results, such as the Green-Tao theorem.

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  • In Algebraic Number theory you have the Kronecker-Weber theorem.
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How about the Classification Theorem (of finite simple groups)? –  Yuval Filmus May 17 '11 at 18:37
    
Thank you! –  quanta May 17 '11 at 18:56
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In Complex Analysis the Riemann Mapping Theorem.

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See the recent book Uniformisation des surfaces de Riemann. Sorry, in French only. –  lhf May 17 '11 at 22:19
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  • In Group theory i would say "Structure theorem for finitely generated abelian groups"
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  • In Field theory The Impossibility of Trisecting the Angle and Doubling the Cube
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@lhf: Thanks edited. –  user9413 May 17 '11 at 19:17
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  • Primary decomposition theorem in linear algebra.
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  • Spectral theorem in Functional Analysis.
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