The immense advances recently made by a team spearheaded by Maryna Viazovska to crack the sphere-packing problem are simply astounding. This accomplishment was described as the most important breakthrough in mathematics in a century. This mathematical conundrum — to find the best packing of spheres in any number of dimensions — has long fascinated mathematicians. From Viazovska’s work to current advances in AI, we’re witnessing a paradigm shift in the way mathematical proofs are created and verified.
Viazovska came to worldwide attention in 2016 when she generalized the sphere-packing problem to two dimensions. She went on to show that the honeycomb arrangement is indeed the best possible packing of spheres in two dimensions. Later that year, she decided to tackle the eight-dimensional case. She proved that a very special symmetric arrangement known as E8 provides the most efficient packing possible in this dimension. These pathbreaking computations paved the way for more extensive forays into greater dimensions.
In collaboration with her colleagues, Viazovska proved that the Leech lattice represents the optimal sphere packing configuration in 24 dimensions. This proof came without its difficulties. The challenge of the 24-dimensional case meant a lot of background research and code development.
The Role of Artificial Intelligence
Recent developments in artificial intelligence have proven to be the central force behind the formalization of mathematical proofs. Collaborators Sidharth Hariharan and Christina Huygens joined him in providing a template for turning his informal eight-dimensional sphere-packing proof into a formal one. Using this blueprint, a new AI model called Gauss was able to autoformalize Viazovska’s eight-dimensional proof in just five days!
Gauss’s capabilities did not stop there. In an astounding feat of efficiency, it autoformalized Viazovska’s 24-dimensional proof in under two weeks. This tremendous accomplishment required more than 200,000 lines of code to be written. It required an unprecedented understanding of the properties of the Leech lattice.
“We made a research breakthrough sometime mid-January that produced a much stronger version of Gauss,” – Jesse Han
This new frontier of research between AI and human mathematicians has shown us to what extent technology can reinforce and build upon classical mathematical techniques. Commenting in the press release on the technology, Hariharan said he was excited about its potential. He’s convinced it can do more to accomplish great things and to help mathematicians do incredible things.
The Formalising Sphere Packing in Lean Project
The Formalising Sphere Packing in Lean project which started in March 2024. That project became a milestone effort for a new era of ensuring the correctness of complex mathematical proofs. Lean is a cutting edge interactive programming language and proof assistant used to write and verify mathematical proofs. The goal of the Borsuk project is to bring Viazovska’s extraordinary discoveries to life, so that future mathematicians can easily verify and build upon them.
The collaboration between mathematicians and AI has never been more prominent. This combined expertise improves development, speeds up the verification process, and deepens our understanding of complex mathematical concepts. As Jesse Han explained, “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.”
The disruptive potential AI in this context has attracted significant interest from the wider mathematical community. Our friend and colleague Liam Fowl could not have over-stated the significance of this change. He said, “These new results look very, very impressive, and certainly foreshadowed some pretty rapid progress in this direction.”
Acknowledging Human Contributions
Though AI’s potential has certainly turned heads, both Hariharan and Han insist that human beings should get a healthy share of credit for AI’s impressive new capabilities. Their joint work is a testament to merging traditional research with cutting-edge technology.
Harian explained, “They told us that they had done 30 ‘sorrys.’ What that means is that they had successfully established 30 intermediate factual disputes that we were required to establish. This is proof that even with the help of AI, human oversight and guidance are still very much needed to guide through complicated proofs.
While automating their formalization, Gauss identified and corrected a typo in their project. This is a great example of how technology can multiply the good work humans are already doing in mathematics.
“One of them helped us identify a typo in our project, which we then fixed,” – Sidharth Hariharan
The synergy between mathematicians and AI showcases truly remarkable technological breakthroughs. It builds on the even more critical foundational work laid out by researchers prior to these discoveries.
