We have an important and exciting announcement from Math, Inc. Their most recent AI reasoning agent, Gauss, has been able to automatically formalize Maryna Viazovska’s sophisticated proof of how to pack spheres in 24 dimensions. This incredible feat, accomplished in only 2 weeks and comprising over 200,000 lines of coding. For Jesse Han, CEO and co-founder of Math, Inc., the possibilities opened up by Gauss have him excited. This is one of the reasons why he thinks it will radically transform the future of mathematical research.
In 2016, Viazovska created quite the stir when she resolved the sphere packing problem in dimensions 8 and 24. Her most famous work proved that the Leech lattice is the best possible packing of spheres in 24 dimensions. This finding is equally encouraging. That was celebrated when she became the first woman to receive the highly coveted Fields Medal in July 2022, an award sometimes referred to as the Nobel Prize for mathematics.
Just like Gauss itself, the creation of the system was truly a collaborative effort. It kept Hariharan, a first-year PhD student in Carnegie Mellon University, and Viazovska herself fully occupied. The two first crossed paths in Lausanne, Switzerland, where they cemented a friendship over a sprite and a common desire to formalize sphere packing proofs. Prior to Math, Inc.’s advancements, Hariharan and his team had been engaged in similar projects aimed at formalizing Viazovska’s proofs.
The Role of Gauss in Formalizing Proofs
Gauss is a major step forward in the development of powerful reasoning agents intended to serve as assistants to mathematicians. Han described it as “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 technology isn’t merely a tool — it’s changing how mathematicians go about solving complex problems.
This leap brought to Gauss a powerful, role-specific capability that Math, Inc. “We made a research breakthrough sometime mid-January that produced a much stronger version of Gauss,” Han noted. This latter version enabled Gauss to realize outcomes from the Prime Number Theorem (PNT) in just 2-3 days. It thoroughly showcases its effectiveness and speed.
The beginning of the Formalising Sphere Packing in Lean project in March 2024. It seeks to leverage Lean, a rapidly growing programming language and proof assistant, to formalize those mathematical proofs. This new project is in keeping with the recent trend of employing artificial intelligence to deepen mathematical inquiry.
The Significance of Sphere Packing Proofs
The sphere packing problem has historically fascinated mathematicians for its difficulty and its far-reaching consequences in many disciplines. Viazovska’s proof relies on highly specialized properties of the Leech lattice. This mathematical grid will help in proving the perfect sphere packing possible in 24 dimensions. “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,” stated Han.
Besides the Leech lattice, Viazovska proved something quite revolutionary. Her work on quasi-modular forms demonstrated that E8 provides the most efficient packing arrangement in eight dimensions. It is the beauty of this relationship that has inspired a great deal of related research and investigation into higher-dimensional spaces.
Hariharan emphasized the collaborative nature of this endeavor, stating, “They told us that they had finished 30 ‘sorrys’, which meant that they proved 30 intermediate facts that we wanted proved.” This serves as a great example of the art of formalizing intricate proofs, step by step laying the groundwork by building on prior work.
Future Implications for Mathematics
The breakthroughs in the past few years enabled by Gauss would suggest encouraging prospects for those who study mathematics. We recognize that AI is rapidly evolving. Second, it has the potential to liberate mathematicians from drudgery, so they can focus their thoughts on higher-level creative concepts. “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,” Han remarked.
Including AI like Gauss in the process of mathematical research certainly has its complications. With these challenges come unparalleled opportunities to increase efficiency and precision. Liam Fowl commented on this progress, saying, “These new results seem very, very impressive, and definitely signal some rapid progress in this direction.“
Math, Inc. is working diligently to continue to improve Gauss and expand its capabilities. The mathematics community is hopeful about the ways that this technology could revolutionize research methodologies and proof verification processes.