Frank Calegari has a great post on Terry's blog: I think it aligns most closely to what I've been thinking and trying to say (and I believe communicates it better than I've been able to). Some highlights: "I hope AI can help mathematicians break out of the straitjacket of being constrained by the first natural conjecture they cannot prove, and work on unraveling the deeper mysteries woven throughout our subject." "The natural grooves of our profession have very much been bent toward more narrow technical expertise than a bigger vision. AI allows us an amazing opportunity to broaden ourselves as mathematicians." "There are so many problems in mathematics that are still so fresh and vital but were previously untouchable that the opportunity to approach them is massively exciting. We need to be more ambitious, to create bold research plans that weren’t even conceivable a year ago." "The difference in reaction between the optimists and the pessimists is that, for the pessimists, the game is over. For the optimists, the game has just changed." "[T]here now exists an amazing new tool to study mathematics. To turn away and say that the community of mathematics as it currently exists is more important than mathematics itself would be a terrible mistake." "[I]t takes one look at the arXiv to see many people in our profession rushing out AI-assisted papers in a somewhat unsightly rush to be the “first” to prove things." [Aside: mere days after the qRH result, I had something like a half-dozen emails in my inbox, some from amateurs but some also from tenured faculty, with Lean formalizations of nonvanishing on Re(s)> 7/8-1/10^k for various large values of k. What the hell are you even doing? That kind of thing may have been valuable years ago, when it was difficult and signaled deep understanding and great technique. Today? Who the hell cares?] "To me [this is back to Frank now], someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution." "If you, as a senior mathematician, are stressed about being scooped by AI, then you are not being ambitious enough." [On hiring:] "I am looking for a narrative of where you want mathematics to go, the technical ability to explain why this may be possible, being able to converse about many areas of mathematics, and a demonstrable record of having done all of this in the past." "What I want a graduate student to achieve at the end of their PhD is a view of mathematics that extends well beyond their discipline, and yet with demonstrable technical skills on at least some class of problems." "I advocated above that we need to evaluate work as if AI was involved. But this does not imply that we don’t need to teach students to do deep mathematics without AI just as the existence of computers doesn’t mean we should no longer teach children arithmetic." [I've often said similar things as well...] "All of my friends in the humanities truly believe that having mathematicians in society both as custodians of one of humanity’s oldest intellectual activities and explorers of the frontiers of knowledge is important. I understand that some part of the culture we so deeply care about will disappear, and as humans, an emotional response to that is very natural. But this is a time to explain to everyone that there are so many great new opportunities, possibilities, and reasons to be optimistic. We are on the verge of a new era of mathematics, and I am excited to see what happens." Bravo Frank!

Quasi-RH?!?!???! Are you kidding me? If a human did this, it would be an instant Fields Medal, no questions asked. RH says zeta has no zeros in Re(s)>1/2. The best we had until a second ago was a region that got thinner and thinner the higher up the imaginary axis you go. I thought maybe they’d fatten that up a bit, that’d be a massive breakthrough. But no. They got a zero free strip!!!! Insane

Sept 4 (Friday of Labor Day weekend): FLT formalization announced by Anthropic Sept 7/8: Tristan+Levent -> OpenAI announce Navier-Stokes But somehow amidst all the commotion, we all seemed to have missed the following (!!!). Sept 6: Lech Mazur claims an AI-generated proof that a positive proportion of numbers go back to 1 under Collatz!! And it's Lean-formalized. I went through the Lean (or rather, had my AI do it) and I think it's right. Now I'm trying to make sense of the math (doesn't look like anything is too hard, just a lot of arguments, building on top of Terry's earlier work). Naoufal El Jaouhari, an Applied Maths Engineer based in Paris, wrote to me a few days ago that he'd gotten his AI to formalize the same proof for 3x-1 building on Mazur, which led me to look at Mazur. Now I'm working on a "digestion" of all of this. Weird wild stuff!!

The AMS’s statement is short and sweet: "The news today of progress on resolving the Navier–Stokes problem, one of mathematics' great longstanding challenges concerning the equations that govern the flow of fluids, represents a milestone advance in human knowledge. This story began with Navier, Stokes, Leray, and Ladyzhenskaya and has culminated in the recent breakthroughs of Córdoba and Martínez-Zoroa, then — assisted by new technologies — Alpöge and Buckmaster, with the final steps taken by OpenAI mathematicians. The purpose of mathematics is human understanding, and this achievement, and the process that led to it, will bear fruit for a long time to come."   Ravi Vakil, President of the AMS John Meier, CEO of the AMS

Two summers ago (if anyone still remembers that far back), AlphaProof was one point off of a gold medal on the IMO. Last summer, a half-dozen labs declared gold medal performance. Shortly thereafter, I predicted that the number of systems getting gold this summer would be... Zero Because nobody would care about the IMO anymore, and it would return to being something 17 year old kids do. I think I was largely right? Have you heard anything about the IMO? Basically no; it's just too easy. Prediction: the number of major math problems being solved *by labs* next summer will be... zero. It will be too easy for them, and will return to being something math professors, postdocs, and grad students do... (What do you think?)

There is and always will be (!) a boundary to the frontier of knowledge. AI helps us push that frontier, but mathematics that’s not understood by humans is as valuable as 1000 books that no human will ever read. My advice: start working on harder problems! (Maybe that’s easy for me to say; I keep putting the problems I’m working on into AI, and unfortunately so far, no dice.)

I keep hearing people saying “math is solved”. I claim the opposite: people with a high level of technical skill are even *more* productive, and hence valuable, with new tools. Now is the time to double and triple down on the numbers of math majors and PhD students!