Thursday, September 7, 2023

Does Alon Amit fully understand Wiles-Taylor proof of Fermat's Last Theorem?

Here's what Alon Amit has to say.

Alon Amit.

Oh goodness. I wish.

When I was a graduate student I took every Number Theory, Algebraic Geometry and Representation Theory course I could find, specifically trying to learn everything Ehud de Shalit was teaching. He's one of very few people mentioned by Wiles in the Annals paper containing the proof, but this wasn't even the main reason: de Shalit is a phenomenal teacher, an outstanding researcher, and this domain of mathematics attracts me like no other.

I took the basic graduate-level courses in commutative algebra and number theory. I took Algebraic Number Theory, two courses on Elliptic Curves, and a course on -adic analysis – all of these with de Shalit. I took reading seminars on these topics. And I believe I'm still years of concentrated study away from being able to comprehend the proof, end to end. I'm not yet capable of reading Ribet's proof of Serre's -conjecture, nor Wiles' earlier papers with Coates, nor the basic papers of Shimura and Eichler on modular forms.

My next project, if and when my life allows me the luxury of being a student again, is to master Diamond and Shurman's "A First Course in Modular Forms". I'm not completely ignorant of the contents of this book; I ostensibly already know much of it. But "knowing" isn't the same as "having it in the palm of your hand", and that's what I think I need to do first.

I want, one day, to do more traveling and be in places I haven't been to yet: New Zealand, Norway, China, South America. In the same spirit, I hope I get to visit this wonderful place called Wiles' proof. It's a selfish goal. I will likely not have an opportunity to meaningfully teach this proof to others, given the enormous amount of prerequisites. Like climbing Mt. Everest, it's something one wants for oneself.

And I really, really want to.

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