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 FigueroaO'Farrill, Andy Putman, Gerald Edgar, Noah Snyder May 12 '10 at 14:50
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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. 


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

