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I was asked this question by a friend's daughter a few days back, and I was interested to see what people here think. I realize of course that this is quite subjective, but still, I get the impression that certain fields, like algebraic geometry, require much more "initial outlay' in terms of absorbing difficult concepts than others, before one can contribute. So, what are your experiences ? Is there any consensus among mathematicians about which areas are especially "hard" or "difficult to get into" ?

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closed as off topic by S. Carnahan, José Figueroa-O'Farrill, Andy Putman, Gerald Edgar, Noah Snyder May 12 '10 at 14:50

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This should be community wiki. It also runs the risk of being purely subjective. Are we just taking an opinion poll, or is there more to it than that? – Pete L. Clark May 12 '10 at 13:46
Sorry, I'm voting to close. If there is a consensus about hardest area, I haven't heard it. – S. Carnahan May 12 '10 at 14:02
I'm also voting to close. The "hardest area" will depend strongly on the student. I don't think there is any kind of consensus. – Andy Putman May 12 '10 at 14:10
Perhaps could we put this question in another form than ordering ? === I hope that Mathoverflow could prove in the long run to be a resource to store useful mathematical folklore before it finds its way in books and treatises. In the meantime we could say: in which area of mathematics do you think good textbooks are sorely needed in english ? – ogerard May 12 '10 at 14:51

For our undergraduate math students, the reputation is that the hardest courses are the ones where you first have to write proofs... abstract algebra and elementary analysis.

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Learning 'almost etale extensions' from Faltings' original paper is very tough.

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