auntienomen

The QCD and Yang-Mills results don't seem to go beyond what the experts know/expect. Basically, they're saying there is a path integral measure, but shrugging when asked what functions are integrable with respect to it. The latter question is key for mass gap and confinement, and existence of the measure only means that the question of whether a mass gap exists makes sense.

It's probably just a terminology mismatch. But practicing scientists take a lot of care to honestly say what they're saying, so it doesn't land well as rhetoric.

I think it's the "can we honestly say" phrasing that got you flamed here. It reads as "those stupid scientists don't know what they're doing".

Dark matter is what we'd get if we had a lot of stable particles that didn't interact via electromagnetism, but only by gravity and short range forces.

The standard model has lots of this kind of thing. Leptons don't feel the strong force. Neutrinos are electrically neutral. The weak force only couples to left-handed particles. It seems completely natural to expect some dark matter. The only question is how much.

This is horseshit. Turn off all learning and conversation sharing, then feed chatgpt the same sequence of prompts. Tell a story with a surprise ending. It will learn the surprise.

It's impressive but long expected and nowhere near as hard.

Agents require training data for RL. This data is rare in comparison to what we feed foundation models, and Google is sitting on a dragon's hoard of sensor and tracking data. I'm not counting them out yet.

Minor quibbles: Hawking radiation has a blackbody spectrum, so we get a variety of wavelengths, with a peak determined by the size of the hole. The outgoing particles should eventually include any kind of particle that can be created in the Standard Model, not just photons and gravitons.

Strangely, it is _not_ the instrinsic curvature of spacetime that produces Hawking radiation. There's a variation on Hawking radiation -- called Unruh radiation -- experienced by accelerated observers in flat spacetime. Any two observers will agree on the value of a field (e.g. the electric field) at a point, but an accelerated observer will experience empty spacetime as a _thermal bath of particles_, with a temperature proportional to the acceleration. Relatively accelerated observers don't agree on what the vacuum is.

I don't think the distinction between holography and string theory is as clean as you're making it. The vague idea of holography predates string theory, but most of the working examples we have of holography come from string theory. That's what AdS/CFT is: It's an explicitly holographic description of superstring theory in an anti-Desitter space. It's frankly bizarre that the author of this article doesn't mention that fact. I suspect we are seeing a popular science rebranding of string theory without the word strings.

It will also reduce your heating bill.

Most research isn't useful. Research on average is extremely useful. Great ideas are rare, and we don't know in advance where they are.

If you do know, let me know and I'll lead a round for your VC fund. I for one would never have predicted that abstract topology would lead to nuclear weapons within a few generations.

And conceptually novel approaches to outstanding problems are the sort of thing that a retrain should pick up on, because they would be hard to compress into what it already knows.

It'd be really nice to have bolt-on after-market automation. I'm not sure if the car companies will want to accommodate this, but I'd really love to make my old Honda self-driving.

I remember a company called Ghost Locomation that was working on this.

Those notations are used when writing down the models, because they make clear the intrinsic geometry, the basic symmetries, etc. But they're not used so much in the study of solutions to the equations. Solutions tend to have peculiar features, tend to break underlying symmetries, etc. and there only needs to be one nasty particular solution to prove the NS conjecture wrong.

Frenkel does a nice job explaining the Langlands program in general. But Buzzard's complaint about Langlands, I believe, refers specifically to the proof of a version of the Geometric Langlands Conjecture by Gaitsgory et al. The proo f is of order thousand pages of mathematical text and builds off of thousands of pages of higher-categorical algebraic geometry by Lurie & others. It's a ripe target for formalization because it's terrifically complicated, not well understood or thoroughly digested yet, and relatively important. A formal proof would be reassuring to mathematicians, whereas Fermat's Last Theorem is relatively unique in that so many mathematicians have examined the proof that it's not very likely to be wrong.

Do we have any examples of an current day AI system introducing a novel concept or perspective. We've got plenty of counterexamples discovered and some theorems proven, but afaik nothing analogous to a new definition.

This comment misrepresents what the cancelled program does.

There are days when I suspect Google does not believe in doing user research.

Yeah. In math at least it's clear that you don't know what's going on. In other fields, it's very easy to think you understand without knowing how much you're missing.

Glad to see that Cython got a shout out here. It's not that widely used, but it's a lovely experience. It uses the CPython ABI to drastically reduce the friction of switching between C & Python.

More generally, don't trust BigCo AI agents with _any_ IP. They have no incentive not to use your IP as training data.

A crucial caveat: basic research investment runs on the same logic as venture capital investment. We know that most mathematical efforts will be worthless. Our experience has lead us to expect that a very small number of such efforts -- some of them very far removed from applications -- will have payoffs so large that they change the shape of our society.

_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.

I'd argue that civilization is the story of people finding ways to overcome what they previously had to tolerate.

The problem with prediction markets is fundamentally that they're unregulated.

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.

ITPs are far older than LLMs in general, sure, but that's a pedantic distraction. What everyone is talking about here (both the comments, and the article) are ITPs enriched with LLMs to make the "smart" proof assistants. The LLMs used in ITPs are not vastly different from the usual chatbots and coding assistants. Just a different reinforcement learning problem, no fundamental change in their architecture.

Exactly this. LLMs really aren't built for discovering new mathematics, especially _interesting_ new mathematics. They're built to try the most obvious patterns. When that works, it's pretty much by definition not interesting.

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.

It's going to have to be a mixture of approaches. Stratospheric injection buys time for more holistic solutions.

There's always some risk of confusing the model with the reality, but yeah, if you have chiral fermions interacting through gauge fields and gravity, the charges have to say satisfy all of the anomaly cancellation conditions (there's about half a dozen) or the model will be inconsistent.

There does appear to be a deeper reason, but it's really not well understood.

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.