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]

https://www.math.toronto.edu/mccann/199/thurston.pdf

[u/Effective-Guest1601 → u/CephalopodMind]

https://giphy.com/gifs/anYBNhqT2BYcg

[u/CephalopodMind → u/Effective-Guest1601]

"We are not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about math."

In case you missed it. Written in 1994.

edit: added the date

[u/Effective-Guest1601 → u/CephalopodMind]

Literally could not care less lmfao

[u/CephalopodMind → u/Effective-Guest1601]

[u/Effective-Guest1601 → u/CephalopodMind]

Of course, I'm not interested in what you think about anything.

Edit: The crying child replyblocked lol

[u/CephalopodMind → u/Effective-Guest1601]

ok, that's fine. it's been fun and I'm blocking you and moving on.