‘Amazing’ Math Bridge Extended Beyond Fermat’s Last Theorem

The proof Wiles finally came up with (helped by Richard Taylor) was something Fermat would never have dreamed up. It tackled the theorem indirectly, by means of an enormous bridge that mathematicians had conjectured should exist between two distant continents, so to speak, in the mathematical world. Wiles’ proof of Fermat’s Last Theorem boiled down to establishing this bridge between just two little plots of land on the two continents. The proof, which was full of deep new ideas, set off a cascade of further results about the two sides of this bridge.

From this perspective, Wiles’ awe-inspiring proof solved just a minuscule piece of a much larger puzzle. His proof was “one of the best things in 20th-century mathematics,” said Toby Gee of Imperial College London. Yet “it was still only a tiny corner” of the conjectured bridge, known as the Langlands correspondence.

The full bridge would offer mathematicians the hope of illuminating vast swaths of mathematics by passing concepts back and forth across it. Many problems, including Fermat’s Last Theorem, seem difficult on one side of the bridge, only to transform into easier problems when shifted to the other side.

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It seems the XKCD server is down due to a hack, but this will do:

https://www.explainxkcd.com/wiki/index.php/435:_Purity

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