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I'm interested in many things, but algebraic geometry I love.

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comment Philosophy behind Mochizuki's work on the ABC conjecture
David, your comments are precisely the type of answer I'm looking for. It is okay that it's 20 years old.
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comment Philosophy behind Mochizuki's work on the ABC conjecture
You are correct that I was inaccurate on that point, although I did know that it was Weil's idea. As for you argument for patience, I think you have misunderstood my question. I am not asking for a sketch of the methods, but only what those methods aim to achieve. An example of a good answer is David Speyer's comments. So saying "here is the rough argument in the function fields case, and so what we want is a number theoretic analogue of ____" is precisely the answer I was looking for.
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comment Philosophy behind Mochizuki's work on the ABC conjecture
@quid: you're being stubborn. Is it not legitimate to ask questions about mathematics that is available but difficult to read and understand? @Kevin: thanks!
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revised Philosophy behind Mochizuki's work on the ABC conjecture
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comment Philosophy behind Mochizuki's work on the ABC conjecture
@quid: the expositions I've seen (such as kurims.kyoto-u.ac.jp/~motizuki/2010-10-abstract.pdf) are mostly teasers to make people read more. My question is about the sketch underlying the proof of the ABC conjecture, which I don't see evident there. If you have an exposition that you would recommend, I suggest that you write it as an answer.
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comment Philosophy behind Mochizuki's work on the ABC conjecture
Correction: "an enthusiastic report". Sorry, Jordan!
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comment What is the intuition for $\mathbb{Q}^{ab}$ having cohomological dimension $1$?
Carnahan: can you expatiate a little more about your comment?