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 hurt ((Disclaimer: depending on levels of ability, perseverance and agreement with consensus reality, attempts to solve these problems may well ruin your life)) to try!