Maryna Viazovska’s groundbreaking research on sphere packing has now entered a new level of verification. The incredible crossroads of mathematics and artificial intelligence is a testament to the power of disruptive, creative thinking. In 2016, mathematician Maryna Viazovska solved the sphere packing problem in seven and eight dimensions. She has had an enormous impact on the field of mathematics with her groundbreaking research. This challenge is at the heart of deciding how rich the same circles, spheres, or other like objects can be clustered within n-dimensional space. Her work earned her the prestigious Fields Medal in July 2022, marking her as the second woman to receive this honor in its 86-year history.
The Fields Medal is sometimes called the Nobel Prize of mathematics, awarded for outstanding achievements in the field. Viazovska’s triumphs have added poignancy, having occurred barely months after the invasion of her native Ukraine by Russia began. Her demonstration proved that a symmetric structure known as E8, indeed, achieves the most tightly packed arrangement in eight dimensions. At the same time, she and her collaborators showed that the Leech lattice is indeed the optimal packing way in 24 dimensions.
The Role of AI in Formalizing Proofs
Jesse Han and his crew at the Math, Inc. crew represented a big breakthrough in the mathematical research. They trained an AI reasoning agent called Gauss to autoformalize Viazovska’s proof on packing spheres in 24 dimensions. Remarkably, Gauss managed to accomplish just that in a mere two weeks, generating upwards of 200,000 lines of code. The project, which is titled “Formalising Sphere Packing in Lean,” started in early March 2024. Sidharth Hariharan and many other passionate collaborators provided invaluable inputs to this effort.
Hariharn, who previously worked on formalizing Viazovska’s proofs, noted the efficiency of Gauss in identifying critical intermediate facts during the autoformalization process. In his words, “They told us that they had done 30 ‘sorrys.’ That meant that they had indeed established successfully, 30 intermediate facts that we were asking them to prove.
Fortunately, the complexity of the autoformalization task was non-trivial. Han elaborated on the challenges faced: “It was actually significantly more involved than the 8-dimensional case because there was a lot of missing background material that had to be brought online surrounding many of the properties of the Leech lattice, in particular its uniqueness.”
The Evolution of Reasoning Agents
We believe the development of Gauss is a substantial step in AI’s skills to contend with more structured reasoning. Jesse Han, a member of Gauss’s release team, called Gauss an extraordinary language model. He described it as an advanced reasoning agent, combining some classic natural language reasoning with fully formalized reasoning. This new capability makes it possible for Gauss to do much more than just perform literature searches—not only to put the latest computational tools to use.
Han emphasized the versatility of Gauss: “It’s able to conduct literature searches, call up tools, and use a computer to write down Lean code, take notes, spin up verification tooling, run the Lean compiler, etc.”
The implications of such technology are profound. Hariharan expressed excitement about its potential applications in mathematics: “This is technology that we’re very excited about because it has the capability to do great things and to assist mathematicians in remarkable ways.”
Implications for Future Research
The successful and seamless verification of Viazovska’s proofs through AI indicates a fast-approaching technological singularity at the intersection of math and tech. The promise and achievement of being able to autoformalize complex mathematical proofs creates new opportunities for researchers and mathematicians to explore.
Liam Fowl remarked on the significance of these results: “These new results seem very, very impressive and definitely signal some rapid progress in this direction.” This desire mirrors a rising sense of optimism about the role AI can play in aiding mathematical research.
The more artificial intelligence advances, the more apparent it will be that academia needs to take on a greater role. Counterfactual thinking agents such as Gauss make shortcuts. In doing so, they improve the precision and dependability of mathematical verifications.