Can you say more about computability of "conformal models" in the Langlands context (beyond vibes, perhaps cites)? In my understanding, "conformal models" are by construction computable..

I'm not sure that's true. My quick searching around is that the first paper proposing strings as a possible description of space and time is Nambu, Nielssen and Susskind in 1969 whereas Langlands first stated his conjectures in his letter to Andre Weil in 1967[1] (ie before string theory had even really kicked off mathematically which didn't happen until Ed Witten got involved in the 1980s). In his letter, Langlands seems to motivate the conjectures entirely from abstract algebra and topology (although this is way above my mathematical pay grade at the moment so I'd be more than happy to accept I misunderstand).

[1] https://publications.ias.edu/rpl/section/21 and https://publications.ias.edu/sites/default/files/letter-to-w... in particular

String theory is still the only self consistent theory of quantum physics.... I'd say that's extremely successful.

No nontrivial (by modern standards) proof has a machine-verifiable version. Math is just too huge.

And (my hot take:) the formal correctness is important, but not really that important in math. Sure, we hope the proofs are correct, but the idea why the proofs are correct is often more important. Humans are not a formal verification machines, and we're often interested in why something is true (and when exactly, i.e. how does it generalise), instead of just asking a binary question. So taking this into account, even if there are holes in the current argument, the important thing is that the experts believe they can be fixed, and the insight we gained along the way.

Edit: I want to back my hot take a little:

* The four color theorem is famous for being the first major theorem to be proven using a computer (using exaustive search on a large search space - too large for human to verify). This was very controversial for the time (maybe still is), because we gained no insight into the problem! Exhaustive search doesn't begin to explain why this is - or is not - the case.

* Even more perversely, It's possible that ABC conjecture was proven in 2012 by Mochizuki. But the proof is very hard to read (very briefly: basically nobody understands it, and Mochizuki was - supposedly - very non-cooperative and refused to answer questions. Some experts claimed holes but their concerns were - quite rudely - dismissed as misunderstanding). Now we live in a split world, where most of the world consider the conjecture unproven (because we can't understand it and learn from it), but some universities consider it proven (because Mochizuki is an expert, and nobody was able to find a mistake and convince the world they're right). In other words, it's a mess.

In both cases, the problem was that we care about understanding the problem, not about just asserting something is true or false.

The article does in fact discuss precisely this:

> The solution for these irreducible representations came to Raskin at a moment when his personal life was filled with chaos. A few weeks after he and Færgeman posted their paper online, Raskin had to rush his pregnant wife to the hospital, then return home to take his son to his first day of kindergarten. Raskin’s wife remained in the hospital until the birth of their second child six weeks later, and during this time Raskin’s life revolved around keeping life normal for his son and driving in endless loops between home, his son’s school and the hospital. “My whole life was the car and taking care of people,” he said.

> He took to calling Gaitsgory on his drives to talk math. By the end of the first of those weeks, Raskin had realized that he could reduce the problem of irreducible representations to proving three facts that were all within reach. “For me it was this amazing period,” he said. His personal life was “filled with anxiety and dread about the future. For me, math is always this very grounding and meditative thing that takes me out of that kind of anxiety.”

By my count at least four of the researchers are employed by American universities and therefore most likely live somewhere in the United States.

And "this stuff" to which you refer is the intended output of their full time jobs*. So presumably, they find time to work on it in the same way a software developer finds time to write code. You just sit down and do it, because you are being paid to do it.

*Did I miss something about how these papers were developed in their spare time?

Karl Schwarzschild found the first exact solutions to Einstein’s gravitational field equations (general theory of relativity) while serving in the trenches in WWI, firing artillery at the Russians.

In the article one of the authors of the proof describes a key breakthrough as happening when he contracted covid and was thereby forced to spend 3 months in bed with nothing to do but think.

As an aside, Europeans have families, lives, distractions etc just like people in the US. Source: have lived in Europe for 30+ years. Have a family and lots of distractions. Have not proved any major mathematical theorems. (yet I suppose- there's still time)

Mathematical research (as far as I know) requires significant amounts of ‘time off’ just pondering and meditating on ideas as much as it requires time sitting at a desk concentrating on a paper or working through things on paper. A lot of people have said their best work was done whilst standing waiting for a bus, in the shower, walking in the woods …and so on.