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abc: the story so far

You should take some time to read this very well-written piece about Shin Mochizuki’s claimed proof of the abc conjecture: “The Paradox of the Proof”, by Caroline Chen.

It covers the story from all angles: a biog of Mochizuki, a clear, non-nonsense description of the conjecture, the tale of the mathematical community’s attempts to understand it, and some insightful rumination on the nature of proof.

via Marcus du Sautoy on Twitter, among others

(TL;DR – still nobody knows whether the proof is correct or not)

ABC, as easy as pp1-40

Here’s something that slipped to the bottom of our news queue: Shin Mochizuki has uploaded a 40-page overview of his “Inter-universal Teichmüller theory” papers – the ones which he claims prove the abc conjecture.

Don’t expect to understand any of it, but maybe someone else will.

PDF: A Panoramic Overview of Inter-universal Teichmüller Theory by Shinichi Mochizuki

Previously: Proof News

via Jordan Ellenberg on Twitter.

Proof News

Here’s a little catch-up with the status of the claimed proofs of some big statements that were announced recently.

At the end of August, Shin Mochizuki released what he claims is a proof of the abc conjecture (link goes to a PDF). Barring someone spotting a huge error, it’s going to take a long time to verify. It’s mainly quiet at the moment, apart from a claimed set of counterexamples to one of Mochizuki’s intermediate theorems posted by Vesselin Dimitrov on MathOverflow, which was quickly shut down because the community there didn’t approve of MO being used to debate the validity of the proof. No doubt there are other niggles being worked out in private as well.

At the start of September, Justin Moore uploaded to the arXiv what he claimed was a proof that Thompson’s group F is amenable. Like Mochizuki’s abc proof, experts thought Moore’s proof was highly credible. We were waiting for my chum Nathan to write about it, since his PhD was all about Thompson’s groups F and V, but it turns out we don’t need to: at the start of this week, Justin retracted his paper because of an error which “appears to be both serious and irreparable”. The amenability of Thompson’s group F has been proven and disproven many times, so I still want Nathan to tell me (and you) all about it.

In lighter news, via Richard Green on Google+, recent uploads to the arXiv show that Goldbach’s conjecture and the Riemann hypothesis are true. I’d love to know how it feels to upload a six-page paper which you know proves something like the Riemann hypothesis. It must be a lovely state of mind. Certainly much better than what people like Moore and Mochizuki must go through, waiting for the first email to arrive telling them they’ve made a terrible mistake and their work is not yet complete.

If I’ve inspired you to have a go yourself, look at Wikipedia’s list of unsolved problems in mathematics and take a crack at one this weekend. Can’t hurt1 to try!

  1. Disclaimer: depending on levels of ability, perseverance and agreement with consensus reality, attempts to solve these problems may well ruin your life []