This is very different to believing the proof, which would require at least a pass understanding the general approach, seeing that it all actually fits together, then going deeper. At some point you transition to relying on the Lean all hanging together, but as mathematicians we all draw that line somewhere.

But yeah, makes sense. Same thing if you saw news on someone's new database technique to improve performance. If they say the right words, don't say the wrong words, and if you cared enough you'd do spot checks proportional to the claim. If pressed you'd examine the source code, and run independent checks. But if smells roughly right, that's a good first approximation.

But not an expert on this.

While I don't know the specifics, and someone more "in-the-field" than me would recognize all the "named" theorems etc

I am aware that there have been minor issues that have come up with the formalization specifically, and that previous proofs for lower values of n were always needed.

Though it used to be n=5 and lower needed to be checked.

Vaguely. It's describing connections between a number of other mathematics results than can be connected to prove FLT. I assume all the work described is being done to make the proof more presentable, smaller, basically "prettier".

It sounds like they established a minimum and maximum bounds for n in x^n + y^n = z^n, where one proof works for n greater than or equal to 17, and another proof for n < 37 (when prime).

I believe the case (remembering back 40 years here) n is even is very easy, and n is composite and odd slightly less so. Neither really being in the ballpark of what they describe here.

Wiles-Taylor-Wiles was the original proof by Andrew Wiles, and its corrections.

Galois representations is about vectors over Galois extensions, which are essentially adding roots to regular numbers (rationals, integers, etc). That ties into the Langlands program, which is a big area in number theory (that I don’t know much about).

Together with flat deformations and Frey curve, I think they’re talking about a topic in algebraic geometry as applied to number theory.

I also recognize the name Eisenstein from my time as an undergrad, though two decades out and not working in the field I’ve forgotten what his work on ideals implied here. Ideals are a well-known topic though, a sort of structure inside a ring (set with + and *) that is closed under operations — like evens in the integers are the 2Z ideal.

So I’d describe it as “sensible with an undergrad background”.