Talking to some friends in physics this evening, most of the physics-related results that we could recognize were very mathematical, proving things rigorously where the physics community already had strong expectation. For instance, for a certain model of magnetism (the spin-1 Heisenberg chain), it was strongly expected that there is a finite energy gap between the ground state and the first excited state, but proving this rigorously was quite challenging. So while these are major results in mathematical physics, they probably don't rise to the level of a Millennium problem for the field.

It's interesting to think what a comparable breakthrough in physics might look like, since physics tends to favor things like conceptual understanding and applications over mathematical rigor. Maybe a new quantum algorithm, understanding of high-temperature superconductivity, a precise description of M theory...

From what I can tell, all of the physics results here are quite mathematical. But I am very curious how the internal model they used would perform on more applied problems.

Did you mean to not qualify that? That is a bold statement indeed.

It's funny, because all of the advice he gives in the rest of the video would be amazing to apply to research papers as well. I've had people discourage me from writing research papers pedagogically because it is "unusual," which really meant that it would be looked down upon. It's kind of tragic that the norm is to write research papers that strip away the human details of how a definition is motivated or how something was discovered. This kind of information is usually only shared through in-person discussions or occasionally research presentations, but it is often the core knowledge that enables one to do research in a given field.

It's a great video though. He gives a little checklist of ways to check if your work is clear:

- Do definitions have motivating examples?

- Do proofs [or other results] feel rediscoverable?

- Is there personality?

- Are core ideas illustrated with diagrams?

- Is relative importance highlighted?

...

He also gives a beautiful example of how powerful it is to give a motivating example for an abstraction before introducing it. I really got the sense from watching this of what makes him such a great math educator. Highly recommend his YouTube channel 3Blue1Brown for anyone who hasn't heard of him

*In case there's any doubt, there are three r's in strawberry.

I would rather live in a world where mathematicians get the opportunity to do math with AI, rather than AI companies doing the most important mathematical work and leaving them behind. I hope this for all professions that people want to do.

1. Academic mathematics has traditionally been meritocratic, and this is a good thing. If you work hard and demonstrate results, then you will be given further opportunities to do math research and the resources to do it. We are concerned that AI will reward money and access to frontier models instead.

2. There is a general concern that AI could lead to massive accumulations of wealth (see concerns of massive job displacement). AI companies claim that they want to raise standards of living and benefit humanity, which means that they will need to share their resources and wealth in some way. If they effectively destroy the mathematical community be racing ahead and taking all the major results for themselves, that does not bode well for the rest of us. It seems like a great opportunity to show the world how AI will benefit humanity rather than replace it.

I've seen this sentiment in many discussion here lately, that anyone concerned about these kinds of risks is blindly falling for marketing hype and not thinking for themselves. Please give people the benefit of the doubt and engage with what they are saying. Because if you don't think that bioweapons/nerve agents are a serious concern, I would sleep better if you could convince me if this!

Posting since there was some interest in a related Quanta article here recently https://news.ycombinator.com/item?id=49845998

Comments URL: https://news.ycombinator.com/item?id=49853021

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For instance, figure 3 shows how you can't hide a black hole behind another black hole, which has implications for how a 3D universe can be completely encoded in a 2D region.

https://arxiv.org/html/hep-th/9409089v2

1. First read the abstract and conclusion

2. Then flip through the figures and their captions, and try to understand them as best as you can (though this doesn't work for all papers)

3. Make a deliberate decision about whether it's worth it to dive into the details and read the full text.

LLMs are incredibly useful for parsing dense literature, but I think it's better to ask specific questions about the paper, jargon, equations etc. rather than to have it summarize the paper for you. Summaries are often shallow or even misleading, and they rob you of the opportunity to engage actively with the paper.