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 14:50Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question. 


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. 

