, specifically the article "Algebraic Geometry", by János Kollár:Finally, if we marry a scheme to an orbifold, the outcome is a stack. The study of stacks is strongly recommended to people who would have been flagellants in earlier times.I feel this way about most of algebraic geometry, but that's only because Penn has a high enough concentration of algebraic geometers that I get tired of hearing people walking around and talking about it all the time.
Also, I find myself screaming at the book quite often. But I do it in a good way; it's often some crucial insight that makes me think "why didn't anybody tell me that before?" (Of course, it's possible they did and I wasn't listening.) And sometimes it's "oh, damn, I thought I came up with that myself". For example, I just read the article on computational number theory, by Carl Pomerance, in which he explains a heuristic reason why Fermat's last theorem is true. I won't give it in full here, but it's basically the following. First, Euler showed it for n = 3. Second, consider all the positive integers which are nth powers for some n ≥ 4. The "probability" that a number m is in this set is about m-3/4. So replace the set of fourth-or-higher powers with a random set S, which contains m with probability m-3/4. Then the probability that a givennumber n can be written as a sum of two such elements of this set S is proportional to n-1/2; independently, it has probability n-3/4 of being in S. So the probability that n can be written as a sum of two elements of S and is also in S itself is proportional to n-5/4, and so we expect finitely many examples. This isn't quite true, because the set of fourth-or-higher powers has some nontrivial structure. But it also took a couple hundred words, instead of a couple hundred pages like the real proof.
But I figured out something like this when I was in college, and I was so proud of myself! So it saddens me to learn I'm not the only one who thought of it. (It also makes me happy, though, because the idea is due to Erdos and Ulam, and there are worse people to be imitating.)
