The name Maryna Viazovska was all over the news in 2016, the following year. The first case was very difficult, what she accomplished there was very brilliant mathematics. This elaborate and elegant mathematical probe studies how closely like spheres or circles can be packed in n-dimensional space. Viazovska’s revolutionary work ultimately won her the highly coveted Fields Medal. It also established new standards for the field of geomatics.
Her work—they defined the world’s most complex symmetric arrangement—here proved that this symmetric arrangement called E8 provides the densest packing in eight-dimensions. In addition, together with collaborators, she showed that the Leech lattice is the optimal sphere-packing configuration in 24 dimensions. These discoveries set off a new wave of interest from mathematicians and scholars. They are currently in the process of formalizing these proofs to make them more easily understood and verified.
In a stunning development, Gauss — an AI-powered reasoning agent — has autoformalized Viazovska’s 24-dimensional proof. Surprisingly, this masterwork of code—more than 200,000 lines—was done in the span of two weeks. Realizing it took Gauss only five days to formalize the 8-dimensional case would sound pretty ridiculous. He erred additionally a second aphod on the published paper in that interval. This solution is a watershed moment in the nascent field that sits at the crossroads of artificial intelligence and math research.
The Sphere Packing Problem Explained
The dense sphere packing problem, that is arranging non-overlapping spheres to maximally fill space has mystified mathematicians for centuries. That includes how best to pack spheres of various sizes in all dimensions. In two dimensions, the honeycomb structure provides the most efficient packing arrangement. In three dimensions, spheres stacked in a pyramid maximize packing efficiency.
The work by Viazovska filled in the last two exceptional cases, which had stumped theorists for decades. Her work showed that E8 is the most efficient packing method yet discovered in eight-dimensional space. At the same time, her collaboration confirmed that the Leech lattice does indeed serve this purpose in 24-dimensional space. These discoveries did more than push the boundaries of an abstract mathematical field; they unlocked practical new possibilities for application, exploration, and innovation.
“These new results seem very, very impressive, and definitely signal some rapid progress in this direction,” stated Liam Fowl, a researcher who has followed developments in this field closely.
The Role of AI in Formalizing Mathematical Proofs
AI tools’ inclusion in math also led to new ways of formalizing difficult to grasp proofs. Sidharth Hariharan, who met with Viazovska in Lausanne, Switzerland, began exploring formalization as a means to deepen his understanding of mathematical concepts. His conversations with Viazovska spurred a desire, in both of them, to formalize her proofs not out of academic obligation, but intellectual curiosity.
This project – formally “Formalising Sphere Packing in Lean” – started on the 1st of March 2024. Gauss was instrumental in this effort, autoformalizing both the 8-dimensional and 24-dimensional proofs. Hariharan remarked on the effectiveness of the AI: “They told us that they had finished 30 ‘sorrys’, which meant that they proved 30 intermediate facts that we wanted proved.”
This AFM technological leap frees mathematicians to concentrate on even more abstract concepts, confident that the basics have been accurately worked out. According to Jesse Han, CEO and co-founder of Math, Inc., Gauss is what they call a reasoning agent. He emphasized its potential to gracefully integrate human-level natural language reasoning with deep formalized reasoning. “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,” he explained.
Implications for Future Mathematical Research
The developments achieved using Gauss’s extraordinary powers carry promising consequences for future mathematical investigation. Rapidly formalizing wide-ranging proofs accelerates verification efforts. This eases the burden on mathematicians.
Han noted that while the formalization of the 24-dimensional proof was “significantly more involved than the 8-dimensional case,” the project highlights the potential for AI to assist mathematically. “At the end of the day, 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,” Hariharan added.
AI is changing rapidly in maths. Third, it will liberate mathematicians from drudgery, allowing them to focus on more creative pursuits. “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 concluded.