Maryna Viazovska, a prominent mathematician, has made significant strides in solving the sphere-packing problem in both 8-dimensional and 24-dimensional spaces. This intricate mathematical problem seeks to know how closely like spheres can be stored inside n-dimensional spaces. Photo Courtesy of Anna Viazovska Viazovska’s revolutionary work has made big waves. It has ignited the imagination of human mathematicians and artificial intelligence working together, most notably with Math, Inc. and their AI system, Gauss.
The sphere-packing problem has long intrigued mathematicians. In two dimensions, the honeycomb structure turns out to be the optimal packing arrangement. This is why, in three dimensions, pyramidal stacking is the best there is. That’s where Viazovska took these ideas further. She used cutting-edge mathematical functions known as quasi-modular forms to prove that the symmetric arrangement we call E8 is the best packing there is in 8 dimensions. In addition, aside from all of this work on sphere packing, she and her collaborators confirmed that the Leech lattice is indeed the optimum packing arrangement in 24 dimensions.
The Role of AI in Formalizing Mathematical Proofs
On February 23, Sidharth Hariharan and the rest of their programming team really leveled up here. They began formalizing Viazovska’s 8-dimensional proof using Lean, a popular programming language and proof assistant that checks the correctness of mathematical proofs. As the specific focus of a larger per-project effort, this initiative seeks to improve the verification process behind sophisticated mathematical proofs.
Hearing about the successful formalization of these proofs, Jesse Han, CEO and cofounder of Math, Inc., expressed excitement about the advancements in AI’s role in mathematics. “We made a research breakthrough sometime mid-January that produced a much stronger version of Gauss,” he stated. Gauss is a reasoning agent that masterfully combines classical natural language reasoning with fully formalized reasoning. This unique combination empowers it to help mathematicians in new, more powerful ways.
The partnership between human mathematicians and Gauss has already paid amazing dividends. Hariharan noted, “When they reached out to us in late January saying that they finished it, to put it very mildly, we were very surprised.” As part of their winning work, the team had to establish 30 important intermediate facts — what Hariharan called the “sorrys” — along the way. “They told us that they had finished 30 ‘sorrys,’ which meant that they proved 30 intermediate facts that we wanted proved,” he added.
Achievements in Sphere-Packing Proofs
Math, Inc. is proud to announce that Gauss autoformalized Viazovska’s proof of optimal sphere packing in 24 dimensions. He did this amazing work of over 200k lines of code in just two weeks! This accomplishment is a tremendous step forward in the ability of AI to aid in discovering math research. Surprisingly, the AI not only duplicated human work, but it spotted and fixed a typo in Viazovska’s original paper that was published.
Han elaborated on Gauss’s dual functionality: “It’s a particular kind of language model called a reasoning agent that’s meant to interleave both traditional natural-language reasoning and fully formalized reasoning.” It is this creative aversion to specialization that empowers Gauss to run rampant across multiple dimensions of mathematical thinking.
Hariharan and Gauss’s collaboration is a glimpse into a new trend in mathematics. In this collaboration, AI is the ultimate partner—amplifying human intelligence, not substituting it. As Liam Fowl, mathematician and one of the collaborators on this project told us, formal verification is the future of mathematical proofs. He stated, “Formal verification of a proof is like a rubber stamp.” This endorsement is a huge step in signifying the growing acceptance of AI-assisted methodologies within our mathematical community.
Future Implications for Mathematics and AI
The impact of these initiatives has generated significant conversations around what the future of AI means for math education. The Formalising Sphere Packing in Lean project Public announcement March 2024 It expands the frontier of mathematical proof, assisted by AI, to the limits.
As this model collaboration continues, researchers are hopeful for more breakthroughs. Fowl remarked on the impressive nature of these results: “These new results seem very, very impressive, and definitely signal some rapid progress in this direction.” This enthusiasm is indicative of a larger optimism in the profession about blending cutting edge technology with long established mathematical techniques.