Tuesday, March 22, 2016

[zmzqakfa] Errors in estimating the difficulty of the modularity conjecture

Before Wiles's proof of Fermat's Last Theorem, mathematicians estimated that the Taniyama-Shimura conjecture to be so difficult that it would never be proven.  How did they come up with that estimate?  Where was their error in estimation?

Was some step in Wiles's proof easier than estimated?  Did mathematicians underestimate the amount of work that one mathemetician could do?  (This seems hard to believe, given so many famous examples of other prodigious mathematicians.  Did Wiles do more work in producing his proof than any mathematician previously in any other field?)  Did Wiles get extremely lucky, relatively quickly finding a proof technique that worked amidst a huge number of others that would not have worked but there was no way he could not have known?  Did an unexpectedly large number of proof ideas get usefully eliminated (after the estimation of difficulty) by other mathematicians preceding Wiles?

Perhaps the estimate that the conjecture would never be proven was never a serious estimate, but one touted after the fact in the hagiography of Wiles.  It might have been that no one (or very few) before him had seriously tried, so no one actually knew how difficult it was.

Relatedly, a nice collection of papers about the modularity theorem, some liberated from behind paywalls.

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