Maryna Viazovska’s spectacular new solution of the sphere-packing problem has been all the rage. This historical surge in interest has come alongside a breakthrough collaboration between human mathematicians and artificial intelligence (AI). We were able to autoformalize her proofs in 8 and 24 dimensions successfully through this partnership. This accomplishment represents a historic milestone for all of mathematical research. The formalization of her 8-dimensional sphere-packing proof was announced on February 23, showcasing the capabilities of AI in peer collaboration.
The sphere-packing problem asks how closely identical spheres can be packed in n-dimensional space. This breakthrough—that won Viazovska a Fields Medal—solved two different versions of this problem. In her very first example, she showed that a symmetric configuration called E8 realizes the densest packing in eight dimensions. In the other, she and her collaborators proved that the Leech lattice gives the densest sphere packing in 24 dimensions.
Collaborative Achievements in Mathematics
A team of proofreaders worked to check Viazovska’s work. Among them was Sidharth Hariharan, who helped bring the sphere-packing proof to life. During the last week of January, Hariharan’s team released a master plan. Their pioneering work was key in making it possible for Gauss, a new AI tool that can fuzzy autoformalize the 8-dimensional case. Significantly, Gauss was able to find and fix a typo in Viazovska’s published paper after only five days on the market.
And it was appropriately Math, Inc. that broke the news of Gauss’s prodigious accomplishment. In fact, he autoformalized over 200,000 lines of code for Viazovska’s 24-dimensional proof in less than 2 weeks! This rapid progress illustrates not only the efficiency of AI tools but their potential to enhance traditional mathematical practices.
“When they reached out to us in late January saying that they finished it, to put it very mildly, we were very surprised,” – Sidharth Hariharan
In March 2024, the Formalising Sphere Packing in Lean project began. That was made possible by the explosive power of human smarts, combined with the talents of AI. Lean is a cutting-edge programming language and proof assistant. It enables mathematicians to write mathematical proofs that a computer can check for correctness. This milestone marks an important point in the vibrant history of mathematics and technology collaboration.
The Role of AI in Mathematical Research
Jesse Han, CEO and cofounder of Math, Inc., spotlighted the importance of this new collaborative effort. He outlined the important part AI had in automating Viazovska’s proofs. He stressed that hundreds of human contributions led to this achievement. That combination of time-honored mathematical thinking with cutting-edge AI technology is emblematic of a larger pattern unfolding across the research community.
“These new results seem very, very impressive, and definitely signal some rapid progress in this direction,” – Liam Fowl
Han characterized Gauss as a “reasoning agent.” This new specialized language model allows you to do both natural-language reasoning and fully formalized reasoning. This unique dual capability opens the door to a deeper and more nuanced approach to mathematical proofs, increasing their level of validity and reliability.
“Formal verification of a proof is like a rubber stamp,” – Liam Fowl
The potential for AI to help human mathematicians verify evermore complex proofs marks the beginning of a newly revolutionary era in mathematical research. As these technologies mature, they present opportunities to liberate mathematicians from laborious verification tasks. This shift will enable them to focus on what they do best: exploring new mathematical realms.
Future Implications of AI in Mathematics
With the rapid evolution of technology, the demands on mathematicians are likely to change dramatically. Reflecting on the event, Han said he’s hopeful for the ways that AI will help change the future of mathematical discovery. He sees these advances as an enabler for mathematicians and mathematic education. It would empower them to prioritize meaningful creative exploration over being weighed down by mundane minutiae.
“I think the end result of technology like this will be to free mathematicians to do what they do best, which is to dream of new mathematical worlds,” – Jesse Han
This joint enterprise of human ingenuity and AI power is revolutionizing our understanding and verification of complex mathematical proofs. This shift isn’t a temporary development, it’s a permanent change in the profession. As these innovations continue to develop, they hold the potential to push the limit of what we can accomplish in math.