for somebody who's out of the loop: what's the fuss over the ABC conjecture?

This is a reference to Inter-Universal Teichmüller Theory. Its Wikipedia article gives a good overview (https://en.wikipedia.org/wiki/Inter-universal_Teichm%C3%BCll...). In maths lasting disagreements over a published "proof" are rare, but IUTT is an example of it. What the article misses: There is a more recent, ongoing effort to formalize the published proof in Lean under the name of "LANA" (e.g. see https://zen.ac.jp/news/zmcpostevent0717e and https://github.com/katobungen/LANA_report_202607/blob/pdf/LA... for a recent update). I guess most mathematicians agree that a successful compile of the proof in Lean would confirm its validity. My personal impression is that the process got stuck at the very point Peter Scholze and Jakob Stix pointed out 8 years ago. Officially LANA has still not reached a conclusion.

https://en.wikipedia.org/wiki/Inter-universal_Teichm%C3%BCll...

Wikipedia is maybe the narrow end of a wedge into this topic but the controversy revolves around a very large and very complex paper that few people are equipped to understand and some of those who are able believe the proof is false.

TDLR for the other two comments: a Japanese mathematician is claiming to have a proof for it, but it is based on an entirely new very complex field of maths which he invented. Getting into it takes years, so other mathematicians are hesitant to invest that much time only to find out that the proof is broken and the field isn't otherwise useful.

It doesn't help that the author is rather withdrawn and not willing to spend any effort in making it more approachable.

Some tried, and said they found gaps in the proof, to which the author responded, but they were not convinced.

And that's essentially the situation since 2018.