I guess that's perhaps why I haven't gotten any further with algebraic topology. I get to the point where I can compute the homology of some spaces, then I don't actually see anything more.
It's like someone showing you a hammer and not letting you smash some nails.
Most students find it very hard to learn from (in my experience at UGA). When I ran summer topology qual prep sessions, I had to provide a fair amount of intuition (and they'd already had the course and the book).
Unfortunately (or fortunately), it tends to fail for homogeneous spaces that are not (locally) symmetric, @MikeM. That's what made my Ph.D. thesis interesting (I suppose).
I just volunteered to "teach" my best AoPS calculus student an informal reading course out of my blue book next year. I wonder how disciplined he'll be. I suggested he try to get as much as possible out of the current course for the rest of the year. :P
There's a lot of serious overlap of differential geometry and 1-dimensional complex analysis. The Schwarz lemma is really about hyperbolic geometry. There are various books, but, as I said, look at Krantz for starters.
To define a subset of projective space as the zeroes of a polynomial, you need the polynomial to be homogeneous. But the polynomial, per se, does not define a function on the projective space.
Oh oh! That's what it was. The thing I was looking at was finding zeros for inflection points :D. Then it started talking about homogeneity and I couldn't get it right in my head.
