No one is asserting that someone taking 100 level calculus is going to self-teach => publishing in algebraic geometry. I agree that your first hump correctly d

No one is asserting that someone taking 100 level calculus is going to self-teach => publishing in algebraic geometry. I agree that your first hump correctly describes the first barrier. I assert that a lateral movement in 'sufficiently different' subsets of mathematics will provide an additional hump within the space of knowledge, quantified by a magnitude of eh maybe > 2/3s minimum. Going from comb. to alg. geo. is amazingly hard; hard enough, I suggest, that it's the equivalent of modern day 'self-teaching', given the large corpus of knowledge one has to attain[1]. I assert it'd take at least ~2 years for one to move from combinatorics to an understanding of say, Sheaf Theory circa 1950 (up to Serre) unless you're one of those rare guys like Tao who can jump from field to field (ugh pun not intended). [1]http://matt.might.net/articles/phd-school-in-pictures/ http://matt.might.net/articles/phd-school-in-pictures/ Getting published in the J. of Top. is being on the arc (perhaps, even deforming the disc in R^2 heh heh). Being able to move laterally s.t. your findings substantial enough they are accepted into the J. of Alg. Geo. requires a requires such an absurdly large lateral movement along the arc that it's a feat analogous in difficulty to self-teaching oneself up until say the 1920s.