_We do not know in advance which efforts are going to pay off_. Abstract efforts in topology put us on the road to nuclear energy. Silly number puzzles enabled internet commerce. Non-euclidean geometry gave us synchronized universal GPS.

We should not let our inability to conceive of applications of weird abstract stuff prevent us from making these investments. If our ancestors had fallen in to that trap, we'd be far poorer as a society.

What we can do is ask that people trying new stuff attempt to fail quickly. And that's basically where we are with academia today. Most people who do mathematical work will not have a career in math. They try something new, work for a little while on it, and go do something else when the results turn out to be of modest interest. This leaves behind a messy undigested literature, which is unfortunate. But maybe AI can help us sift that for treasures we missed.

Modern equities and futures markets are highly evolved and rather carefully regulated systems. We've spent centuries learning what the failure modes are and how to guard against them. It's never perfect, it's never going to be perfect -- it's fundamentally a voting system -- but in general, we get liquidity and price discovery at a relatively low cost, while avoiding fraudulent and evil behavior like wash trading and criminal profit laundering.

These new "prediction markets" have been put in place without any of those hard-earned protections. And surprise, they're rife with dirty trick and dirty money.

What LLMs are good at is organizing concepts, filling in detail, and remembering to check corner cases. So their use should help mathematicians to get a better handle on what's terra firma and what's still exploration. Which is great. Proof by it-convinced-other-mathematicians doesn't have a flawless track record. Sometimes major theorems turn out to be wrong or wrong-as-stated. Sometimes they're right, but there's never been a complete or completely correct proof in the literature. The latter case is actually quite common, and formal proof is just what's needed.

Consistent quantum field theories involving chiral fermions (such as the Standard Model) are relatively rare: the charges have to satisfy a set of polynomial relationships with the inspiring name "gauge anomaly cancellation conditions". If these conditions aren't satisfied, the mathematical model will fail pretty spectacularly. It won't be unitary, can't couple consistently to gravity, won't allow high and low energy behavior to decouple,..

For the Standard Model, the anomaly cancellation conditions imply that the sum of electric charges within a generation must vanish, which they do:

3 colors of quark * ( up charge 2/3 - down charge 1/3) + electron charge -1 + neutrino charge 0 = 0.

So, there's something quite special about the charge assignments in the Standard Model. They're nowhere near as arbitrary as they could be a priori.

Historically, this has been taken as a hint that the standard model should come from a simpler "grand unified" model. Particle accelerators and cosmology hace turned up at best circumstantial evidence for these so far. To me, it's one of the great mysteries.

Y'all should read this, and make sure you read to the end. The last paragraph is priceless.

The writers just lacked the courage to do it. They tried to tack a Disney ending onto a tragedy.

Looking this sort of thing up on the fly in lecture is a great use for LLMs. You'll lose track of the lecture if you go off to find the definition in a reference text. And you can check your understanding against the material discussed in the lecture.

But in many cases, it's extra effort for not much reward. The patterns which most mathemematicians are interested in are (generally) independent of the particular foundations used to realize them. Whether one invests the effort into formal verification depends on how hard the argument is and how crucial the theorem.