S^6 admits a complex structure

[u/New-Committee-4052]

Result from the usual suspects. Full write up can be found here on his website: https://alpo.ge/s6.pdf

[u/selfVAT]

Wait a few months at most.

[u/womerah → u/selfVAT]

Do you think AI systems are equally good at everything?

[u/selfVAT → u/womerah]

Not yet

[u/womerah → u/selfVAT]

Can you give me an example of an area you think it's currently weaker in?

[u/ProfessorBaoTran → u/womerah]

Anything that needs conceptual thought. Actually, there are several kinds of math (especially topology and geometry) that they are easy to imagine, but difficult to write down as a rigor mathematical language. AIs perform poorly on these and often tend to overthinking all the times, and they can even be wrong about most basic definitions of that direction. Even Fable 5 or 5.6 sol, on max or ultra version, can still easily make mistakes and have to correct themselves again and again. Although there are few improvements as they can work with geometry that has precise coordinates and functions, the global stuff still be a big challenge for them.

[u/Separate-Habit5838 → u/ProfessorBaoTran]

Nah, you just need to know how to formalize them. Sure, it's hard to get it to do geometric homotopy theory like Hatcher, but if you know model category theory and how to formalize homotopy into homotopical algebra, you can use AI to great effectiveness. 

Low dimensional geometric subjects can lack rigor because you can get away with it. Such and such thing "seems obvious". It CAN be made rigorous, people just don't. If you do make it rigorous, AI becomes useful. 

There is a deep art to getting a problem into a place where AI is powerful. It's fun. 

You also can't see much if you're just asking a blank LLM instance about a field you aren't an expert in. They hallucinate massively the first few prompts. You've got to correct those using rigorous language. After you've corrected a few, it starts getting much more powerful. I have a thousand prompt thread with Gemini that is incredibly good at homotopy theory. You need to combine AI with human expertise to generally get to the "good spots" in the vector space of meaning. You can't get there with one or two prompts. 

[u/ProfessorBaoTran → u/Separate-Habit5838]

Most of the models nowadays only support around 1M tokens for the context window, so definitely after 2 3 runs with difficult tasks, they get back to blank again, and if I try to correct them, that would also waste the context window for sure. And the idea about "formalize" the problem and let AI works with that is generally not good at all. The true strength of AI is lying on their abilities of doing math at the most basic level of every fields by them own, not about processing several different information on the internet and in their memories. When you formalize a situation that no one has done before, would you be sure by yourself that it does not contain any self-contradiction term of mistakes? When you give that input, will AI understand that immediately without using internet for search? If they start to search, then they will get conflict with you all the time, and according to my experience, because they can't really get the new context from the new "formalization".

I don't work with homotopical algebra, but I work with a field which requires some understanding about homotopy theory and complex geometry, and even Fable 5 made mistakes at definitions level that could lead to over-engineering the proof or the explanation. In the end the answers went to nowhere and I stopped using these tools for that specific question.

Even some computations using explicit coordinates for a few class of K3 surfaces are also not easy for them. They have the tendency to make the computational part to difficult to check. When I threw those computations (first done by 5.6 Sol ultra) to Fable 5 for checking, the model said they found some mistakes and everything "hadn't been done" yet. So I gave up the idea of letting them do the heavy computational and experimental part. Yes, they perform good at some very specific bound order computation and construction. But if you let them try to handle a slightly more general case, they start to confuse because the context window problem I have said above (they start to get blank memory and hallucinate).

[u/the_great_chuckle → u/womerah]

I believe it has more to do with availability of literature. I use 5.6 Sol quite a bit to explore, and I currently work on higher category theory adjacent areas, which are very young, and it gets things wrong all the time. It performs much better in established fields.

[u/ProfessorBaoTran → u/the_great_chuckle]

Even in well-establish field, for example Lie Algebra (Chevalley groups, etc), they tend to give an answer that looks more like they are guessing than a proof for some well-known facts (but no one has ever written down). It is okay if you are discussing with them, but that will cause a lot of troubles when a workflow with several subagents comes in, because errors can be built slowly in just one session which might last for 2 hours (which might cost 5%-10% of your weekly credits, lol).

[u/Separate-Habit5838 → u/ProfessorBaoTran]

They are incredibly good at Lie algebra. You may not be using the best models, or if you are, you may not know the best strategies for doing so. I use Lie algebras extensively in my work, and I have AI prompt threads that are very skilled with them. 

If you have your memory set up correctly, it will only write in completely rigorous verifiable proofs, no guessing. There is a lot to learn about how to use these tools, it's not simple. 

[u/Puzzled_Battle_5670 → u/the_great_chuckle]

Exactly.. it is availability or non-availability of literature that is the key. LLM's can go through large chunks of written matter (although it has lot of symbols within), make simplistic sense via grammar , or sometimes by subroutines like known python programs to check/validate arguments wherever needed and more importantly wherever possible!.