October 6th, 2026 was the day that AI doing math really became personal for me. The amenability problem for \(F\) was among the long list of problems to which Open AI announced solutions on that day. This was not the first time that AI had solved a problem I had invested some time in—the question of whether \(\mathfrak{d}=\omega_1\) implies \(\mathfrak{a}=\omega_1\) had been solved by an individual mathematician with the use of AI—but the amenability problem for F was a problem that I had worked on so much over the last 19 years that it's become part of my mathematical identity. My interaction with the problem also had a particularly human dimensions as well, which is part of why I'm writing this today.
My first encounter with the amenability problem for \(F\) was 19 years ago when I arrived at Cornell. I started attending the topology and geometric group theory seminar, in order to branch out a bit. Laurent Bartholdi mentioned the problem at the beginning of a talk he gave—I think it was the first talk in the Cornell topology and geometric group theory seminar which I attended. The talk was about something else, but I became instantly fascinated with and distracted by the problem. Laurent mentioned that Ken Brown had spent time working on the problem. So I came to Ken's office and asked him to tell me about it. I saw it as a way for me to interact with a new colleague, to talk some mathematics together even though our interests where quite removed. Ken was done working on the problem, but he was always eager to listen. The problem resonated a little with some work that had been done a little earlier on the interaction between logic, combinatorics and dynamics, what is now known as the Kechris-Pestov-Todorcevic correspondence. In particular, \(F\) is a subgroup of \(\mathrm{Aut}(\mathbb{Q},\leq)\), which Pestov had shown was extremely amenable using the existence of finite Ramsey numbers. The more I played around with the problem, the more I started seeing how strategies coming from set theory and Ramsey theory might be useful. This began with my demonstration that the Fo̷lner function for \(F\) grew at least as fast as the tower function composed with \(\log(n)\), and in particular faster than any finite iterate of the exponential function. The amenability problem for \(F\) became a side research project of mine and proved to be a great way to learn some group theory and start doing research in that subject. I gradually met people working in group theory who had an interest either in the amenability problem for \(F\) or at least in Thompson's group.
There was a false proof that \(F\) was amenable that received a lot of attention around 2009 and I ended up being one of the people to pinpoint where the problem was. When the dust settled and it was clear the problem remained open, I began investing time in the problem again. In October 2010, I bought a bottle of XO cognac in Casis while at a workshop at CIRM Luminy in Marseille. The idea was that the bottle would be opened when the problem was solved, with any stakeholders invited over to share in the celebration.
In 2012, I thought I solved the problem, proving that \(F\) was amenable. I was very careful to check the proof before making an announcement, doing my best to check the argument line-by-line. I announced the problem on the arxiv and within a few weeks, many people had indicated they felt the proof was correct (it was short--the whole paper was around 12 pages or so, including references, introduction, etc.). It was only by a coincidence of timing that the bottle didn't get opened then. When the error was found (by someone who had a competing claim that \(F\) was nonamenable), it took a toll on me psychologically for a while - with me lacking confidence that I could ever really check my own (or anyone else's) work. Some of the scars from that still linger.
In the window of time when the proof seemed to be correct, I got invited to give talks in a number of seminars. One of those was in Neuchatel. Even though the proof was broken, I had some great conversations with the people there and with Alain Valette in particular. I came home with a problem which would soon grow into Yash Lodha's PhD thesis. When Yash and I traveled to a workshop in Durham, England later that summer, my interest in the problem (and $F$ in general), connected me to Collin Bleak, who became a close friend and collaborator. I've met dozens of people through working on this problem that I would never have crossed paths with. I believe many others only know me through my interest in Thompson's group.
What I hope comes through in what's written above is that this is very much about the people, connecting to them, and sharing mathematics with them. There is a lot more color that I could add, but the above is more than enough to give context.
When Open AI announced a proof that \(F\) was nonamenable, I had a few reactions. First, it did show that the approach I was taking to solving the problem (which would have established F was amenable) was doomed to fail. Second, it was disappointing to me that AI got to it before a human. With a human, I would want to share in their joy and help sort out the status of their proof since I know they would be anxious to have their result confirmed. With Open AI, this is just a PR exercise for them and the solution is just a biproduct of that. In fact this is but one of the vast number of problems whose solutions it announced on October 6, 2026. It was but a pellet of lead in a shotgun blast. I'm not alone in finding that offensive. I am genuinely busy while writing this and probably wouldn't be able to get to reading a human's proof immediately either, but I feel that much less in a rush to read an AI proof.
I am still planning on opening the cognac once a group of peers I trust all agrees that the proof is correct and once we collectively understand the proof. I don't know when that will be but perhaps it will not take long. I will say that I don't feel any desire to invite someone from Open AI to share the cognac. It was meant for stakeholders in the problem. I would have assumed until recently that anyone who solved the problem would fall into this category, but it seems not. I have a feeling that the cognac tasting will feel more like a wake than a celebration.
To be clear, even though my heart wants math to be like an Olympic sport going forward, I know that it's a science. In fact, that was one of the things that drew me to it. When you solve a problem, you are the first person to do it, regardless of how many resources your predecessors had. We work on math because we want to know something. Those of us who talk or write about the humanity of all this (as I just did) aren't wrong. On the other hand, I think we (or at least I) have a hard time defending the Olympic sport view: that this is an artificial human construct to test intellectual strength and assign rewards based on it. Or maybe that we are making art that few people but us can understand or appreciate. That has also been the reality of pure math but one which is easier to ignore if there is some other purpose. AI is separating those two things in an uncomfortable way and making us look in the mirror at ourselves.
What's so hard about this for everyone (especially those in pure math) is that we have to completely rethink what it means to do math and how our skills are important. Many of us are in the denial and anger phases of grief. We have much work to do in the months and years ahead and should be mindful of what that work is.
Justin Moore
Remark The above was written on October 8, 2026 and lightly revised and edited afterwards.