AI Generates Breakthroughs in Mathematics and Theoretical Computer Science

A team of researchers at OpenAI has made significant strides in mathematics and theoretical computer science using an internal version of their next major model, Astra. This achievement marks a notable milestone in the development of AI systems capable of contributing to mathematical research. The results were achieved by leveraging the power of large language models to tackle some of the most pressing open problems in various fields.

The team has provided new solutions for ten long-standing problems that have seen little progress over the past decade or more. These issues span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics. The results are of substantial interest to their respective mathematical communities and hold broad implications across mathematics as a whole.

The breakthroughs include new upper bounds on sphere-packing density down to the Cohn-Elkies threshold in high-dimensional geometry. Additionally, exponentially improved bounds have been established for binary codes at any prescribed minimum distance, with analogous results for high-dimensional spherical codes. These findings demonstrate the potential of AI systems to accelerate mathematical discovery and provide valuable insights into complex problems.

The team’s work also addresses a central open question in group theory by establishing the existence of non-sofic groups through a construction method. Furthermore, they have disproofed Connes’ rigidity conjecture, which posits that certain groups are uniquely determined by their von Neumann algebras. These results showcase the ability of AI systems to tackle complex and long-standing problems in mathematics.

Another significant contribution is the establishment of new lower bounds for computing the permanent using arithmetic circuits and formulas. This includes an arithmetic-formula lower bound of order n4/log n, which has far-reaching implications for quantum complexity theory. The team’s work also extends a foundational principle from classical complexity theory to general two-player quantum games through an exponential parallel repetition theorem.

The results also include polynomial-factor hardness of approximation for the closest vector problem, a fundamental lattice question related to post-quantum cryptography. Additionally, they have determined the maximum possible volume of a convex body whose centroid is its only interior lattice point in every dimension. These findings demonstrate the potential of AI systems to tackle complex problems and provide valuable insights into mathematical structures.

The team’s work also resolves Erdős problem 183 on multicolor triangle Ramsey numbers with a superexponential lower bound, as well as resolving Erdős problems 146 and 180 through results on compactness and degeneracy conjectures in extremal graph theory. These breakthroughs demonstrate the ability of AI systems to tackle complex and long-standing problems in mathematics.

The emergence of AI systems capable of contributing to mathematical research raises questions about their role in mathematics. The team acknowledges that there are many views as to the impact of AI on mathematics, including concerns raised by signers of the Leiden declaration on AI and Mathematics. They believe that attribution should honestly reflect how a result was produced, claiming human authorship for an entirely AI-generated proof would misrepresent both the system’s contribution and genuine human intellectual work.

The team takes responsibility for the correctness of the mathematical arguments generated by their system while acknowledging the significant role played by Astra in producing these results. They hope that the mathematical community will engage deeply with these findings, place them in context, and bring the ideas behind them to life through new research and discovery.