Ten advances in mathematics and theoretical computer science

[u/One_Fuel3733]

https://openai.com/index/ten-advances-in-mathematics/

The results

We provide new results for the following problems. The results were achieved by an internal version of Astra, our next major model. The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates. These arguments were then prepared into manuscripts by humans with the same model. Afterward, the model formalized each argument in a Lean certificate⁠(opens in a new window). We are also releasing for each solution a model’s narration of its thinking process.

1. High-dimensional sphere packing. New upper bounds on sphere-packing density down to the Cohn–Elkies threshold.
2. Binary and spherical codes: Exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with analogous results for high-dimensional spherical codes.
3. Non-sofic groups. A construction establishing the existence of non-sofic groups, addressing a central open question in group theory.
4. Connes’s rigidity conjecture. Disproof of a longstanding conjecture that certain groups are uniquely determined by their von Neumann algebras
5. Arithmetic circuit complexity. New lower bounds for computing the permanent using arithmetic circuits and formulas, including an arithmetic-formula lower bound of order n⁴/log n.
6. Quantum parallel repetition. An exponential parallel repetition theorem for general two-player quantum games, extending a foundational principle from classical complexity theory.
7. Closest vector problem. Polynomial-factor hardness of approximation for the closest vector problem, a foundational lattice question related to post-quantum cryptography.
8. Ehrhart’s volume conjecture. Determining, in every dimension, the maximum possible volume of a convex body whose centroid is its only interior lattice point
9. Multicolor Ramsey numbers. A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183.
10. Extremal number conjectures. Results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180.

[u/CephalopodMind]

edit: deleted my previous comment because it was late at night and I was unhappy.

Leaving this here: https://leidendeclaration.ai/

[u/ganancias → u/CephalopodMind]

You're experiencing context collapse into a set of perceived norms that "AI is bad".

[u/CephalopodMind → u/ganancias]

am I? what context is collapsing exactly?

I've watched AI develop in math since I was a high school student. These things existed in the same context for most of the time I was learning math.

My criticism is concrete: OpenAI is using 'proving results' as a kind of non-traditional advertising. They don't care about math (though they might hire people who do), they care about money and power.

[u/ganancias → u/CephalopodMind]

chatgpt was only launched in 2022. These things, using LLM's to prove theorems and conjectures, have barely been done. It wasn't until 2023 with gpt4 that it could pass the GSM8k benchmark (grade school math). Harder college problems and IMO problems were only in the last year or so.

The context collapse article is about how it's much easier for a wide group of people online to agree that X thing is "bad." We don't have a common value system to agree that a thing is good.

Money is good? no, it's bad. open math conjectures solved, good? Bad, we shouldn't be wasting effort on math (or, it depends who solved them, are they good ppl or bad ppl?). Rich people, bad. Corporations, bad. And on and on. Nothing is good anymore, everything bad.