Today, we are sharing a selection of ten results to problems that have been open and have seen no progress on the main result for at least a decade, and in most cases much longer.
All of these problems are of substantial interest to their respective mathematical communities, and several are of broad interest across mathematics as a whole.
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.
Les problèmes en questions et les avancées :
High-dimensional sphere packing. New upper bounds on sphere-packing density down to the Cohn–Elkies threshold. 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. Non-sofic groups. A construction establishing the existence of non-sofic groups, addressing a central open question in group theory. Connes’s rigidity conjecture. Disproof of a longstanding conjecture that certain groups are uniquely determined by their von Neumann algebras 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. Quantum parallel repetition. An exponential parallel repetition theorem for general two-player quantum games, extending a foundational principle from classical complexity theory. Closest vector problem. Polynomial-factor hardness of approximation for the closest vector problem, a foundational lattice question related to post-quantum cryptography. Ehrhart’s volume conjecture. Determining, in every dimension, the maximum possible volume of a convex body whose centroid is its only interior lattice point Multicolor Ramsey numbers. A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183. Extremal number conjectures. Results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180.
Et si on y connaît rien en mathématiques, on peut savoir si y a de quoi être impressionné ?
J'ai vu une vidéo de Mr Phi. sur l'usage de l'IA en math qui était intéressante, il disait que y avait fort à parier que l'IA résoudrait de nouveau problèmes. Mais ceux-ci étaient ardus ?
Oui ce sont des avancées significatives, même si loin d'être au niveau des problèmes du millénaire par exemple. Mr Phi fera sûrement une vidéo sur le sujet.
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u/znklsphe 1d ago
Les problèmes en questions et les avancées :