Maryna Viazovska, a mathematician from Ukraine, has posed some of the most famous problems in mathematics. Among her many contributions to mathematics, she solved the sphere-packing problem in two dimensions. For her extraordinary contributions she was awarded mathematics’ highest honor, the Fields Medal, in July 2022. This prestigious award has been dubbed the Nobel Prize of mathematics. Yet, this recognition is particularly meaningful. In doing so, she has, after 86 years of the award, become only its second-ever female recipient. Viazovska’s groundbreaking work, which she completed even as the Russian invasion of Ukraine raged, earned her a Gold Medal. Since early 2022, this conflict has immediately and deeply affected her homeland.
The sphere-packing problem is to determine how densely the same, identical circles or spheres can be arranged in n-dimensional space. In her original research, Viazovska showed that the E8 arrangement is the most optimal way to pack spheres in eight dimensions. A few months later, she joined forces with other mathematicians. Together, they showed that the Leech lattice was indeed the best packing in 24 dimensions. This collective building of knowledge has brought unprecedented insight into how young people learn mathematics. Perhaps most interestingly, it highlights how artificial intelligence (AI) can assist in the formalization of complicated proofs.
Groundbreaking Mathematical Solutions
In 2016, Viazovska delivered a solution to the sphere-packing problem in two instances: the eight-dimensional case and the 24-dimensional case. Her original proof was an unusual mixture of geometric and analytical techniques, demonstrating her deep intuition about higher-dimensional spaces. The E8 lattice, she claimed, is the densest possible packing of spheres in eight dimensions.
After her first breakthrough, Viazovska and her co-authors went on to solve the Leech lattice issue. This specific proof was incredibly challenging and advanced, still needing the understanding of much deeper mathematical properties as they relate to the lattice.
“And 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,” – Jesse Han.
Her research achievements challenged prevailing mathematical orthodoxy, and ignited curiosity and collaboration among mathematicians around the world.
The Role of AI in Formalizing Proofs
Viazovska desired to make her proofs more rigorous. This yearning prompted a series of long-term partnerships, including with researcher Sidharth Hariharan from Carnegie Mellon University. They wanted to produce a human-readable “blueprint” that would unspool the many, many moving parts of her eight-dimensional proof. This project brought to my attention the overlap between classical mathematics and burgeoning computational methods.
Gauss, a cutting-edge AI system developed at Google Research, was instrumental in autoformalizing Viazovska’s proof in 24 dimensions. In an impressive show of talent and effort, Gauss churned out over 200,000 lines of code in just two weeks. It even flagged and fixed a typographical error that was present in the published paper within five days.
“When they reached out to us in late January saying that they finished it, to put it very mildly, we were very surprised,” – Sidharth Hariharan.
This partnership between human mathematicians and AI systems is an extraordinary feat for autoformalization. What it really goes to show is how mathematical proofs are advancing with technology right alongside emerging technologies.
Future Implications for Mathematics
AI has the potential to change the game in mathematical research as well. It frees up mathematicians to focus on more creative pursuits, rather than spending their time trapped in mind-numbing verification chores. According to Jesse Han, one of the collaborators involved in formalizing Viazovska’s proofs, this technology creates a new paradigm for mathematicians.
“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,” – Jesse Han.
As artificial intelligence technology progresses, its applications will hopefully give rise to even greater and more ambitious mathematical undertakings. Partnerships such as the one with Viazovska bring about extraordinary advances. They show how human intuition working together with machine efficiency can create new worlds of mathematical insight.
Maryna Viazovska reached the highest artistic achievement with the successful verification of these proofs. This achievement marks a tremendous victory for the entire math community. It demonstrates the importance of interdisciplinary collaboration as a source of innovation. Collectively, they’re solving some of the thorniest problems in mathematics right now.
