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In Matthew Emerton's comment on Terry Tao's blog, he speaks about learning etale cohomology or the theory of Neron models as "black boxes". By this he means that you can learn what the theory is about and how to use it, without going into the detailed proofs of why they can be used.

Which theories (e.g. etale cohomology) can be learned as black boxes?

And where would one go (e.g. find lecture notes) to learn something like that?

Notes on something like this would ideally give you an idea of what is going on, give examples, and most importantly illustrate how they would be used to solve problems. I am mainly interested in arithmetic algebraic geometry and algebraic number theory, so I would especially like to know about "black boxes" in this direction, though "black boxes" in other areas might also be worth knowing about.

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Isn't that answered in the last paragraph of the question? –  José Figueroa-O'Farrill Feb 15 '11 at 20:11
When I was student, Hironaka's theorem on the existence of resolutions of singularities was considered a black box by pretty much everyone. I suspect it is still not far too from the truth today, in spite of the many simplifications. –  Donu Arapura Feb 15 '11 at 20:54
The formalism of Grothendieck's 6 operations (aka Voevsodsky's cross functors) and nearby/vanishing cycles constitutes a huge black box for Betti, étale, de Rham and p-adic and motivic (co)homology. Ayoub's thesis is a very complete SGA(if note EGA)-like reference. Deligne Cisinski's paper on triangulated categories of motives has more concise formulation. This includes the étale cohomology black box the Emerton was talking about: for example proprer base change theorem is the Exchange isomorphism $f'_!g'^* = g^*f_!$, while being Zariski local follows from $g^!f_* = f'_*g^!$ –  YBL Feb 16 '11 at 1:42
One can "learn" anything to some extent as a black box. Learn definitions, statements of theorems, some applications, then try some new applications. But it is not as much fun as knowing how it works. I used Riemann Roch for years as a black box, but after reading Riemann, I understood it - what had been mysterious became clear. But life is short. Black box learning is easier surrounded by experts, absorbing useful knowledge by osmosis. 5 minutes with Hironaka helped more than staring at the paper for much longer. But to make real progress one must study too. –  roy smith Feb 16 '11 at 3:45
I also think this site makes a huge contribution to black box learning. I often read the answers to other peoples questions here for the wonderful succinct expert answers so generously provided by many. –  roy smith Feb 16 '11 at 3:47
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2 Answers

I find Hodge theory pretty scary stuff with its compact inclusions of Sobolev spaces, pseudodifferential operators and parametrixes for elliptic differential operators. However it is very easy to use the results of Hodge theory as emanating from a black box. I remember how exhilarated I was by the argument that a Hopf surface, homeomorphic to $S^1 \times S^3$, could not be Kähler, and much less projective, just because its first Betti number is $b_1=1$. Whereas by Hodge theory a compact Kähler manifold $X$ has betti numbers $b_q(X)$ which are even whenever $q$ is odd.

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it might be simpler to notice that $b_2=0$, so it cannot be symplectic :) –  Pavol S. Feb 16 '11 at 16:32
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I have found Groebner bases to be incredibly useful for testing concrete ideas about varieties, and although I did spend time learning how they work, all I've ever really needed to know is how to interpret the results of a Groebner Basis calculation, and how to choose monomial orders that will produce useful answers. I don't actually know how the most efficient algorithms and their implementations (Fauguère F4 and F5) arrive at an answer---the textbook algorithms are painfully slow for complicated calculations.

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Almost nobody knows how "the most efficient algorithms and their implementations (Fauguère F4 and F5) arrive at an answer" since there are only two really nontoy implementations, and both are closed source (Magma/Steele and Maple/Faugere). I personally think this is a shame. –  William Stein Feb 16 '11 at 3:47
I couldn't agree more. –  known google Feb 16 '11 at 4:11
The main principles behind F4 and F5 have been published (it took a while back then, but now they have been in print for 10+ years). The actual implementations are undoubtedly way more intricate than the basic algorithms suggest, but that's something you could say about any serious implementation of any non-trivial algorithm. That being said, looking at the hundreds of thousands of lines of code might not be the best way to get a feel for the algorithm either. –  Thierry Zell Feb 16 '11 at 5:03
I also think that your answer blends two issues that should be kept very separate: understanding how an algorithm works and understanding its output. Many users only care about being sure that the output is indeed a Groebner basis, and couldn't care less about how the basis was computed. But any user has to learn at some point about various monomial orders, the FGLM algorithm and so on. –  Thierry Zell Feb 16 '11 at 5:08
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