Maryna Viazovska, one of the world’s leading mathematicians from Ukraine, achieved a milestone in July 2022. She is the first female mathematician to be awarded the prestigious Fields Medal, often referred to as the Nobel Prize for mathematics. Her recognition was an important landmark. In doing so, she became only the second woman in the award’s 86-year history to receive this honor. This prize arrived while her home nation of Ukraine was in the midst of violent conflict, having been invaded by Russia only months prior.
Viazovska’s pioneering work solved what’s called the discrete sphere packing problem in seven and then eight dimensions. This notoriously difficult mathematical problem seeks to determine the optimal arrangement of identical spheres in n-dimensional geometric space. This is the issue she confronted in 2016 and ultimately created a solution for two particular examples. Her work showed that a highly symmetric arrangement known as E8 is the most efficient packing in eight dimensions. She was a steady partner in the collective work of other researchers. Collectively, they showed that Leech lattice is the optimal packing arrangement in 24 dimensions. Her groundbreaking application of these deeply complex mathematical functions, known as quasi-modular forms, proved to be crucial in reaching these proofs.
The Sphere Packing Problem
The sphere packing problem is one of the most alluring math problems, inspiring countless mathematicians. At its core, it poses an intriguing question: how can identical spheres be arranged to occupy the maximum possible volume in a given space? This challenge has resulted in significant research and hundreds of theories throughout the years.
It was progress indeed when, in 2016, Viazovska produced resolutely deterministic solutions to two of the most involved cases. She showed that the E8 lattice is the densest packing configuration in eight-dimensional space. This breakthrough deepened mathematical understanding and laid the groundwork for future research into higher-dimensional packings.
Not long after her E8 breakthrough, Viazovska joined forces with other mathematicians to solve the 24-dimensional case. In combination, they proved that the Leech lattice is indeed the best possible packing in this dimension. Their work has deep implications, not just for mathematics, but to the fields of coding theory and theoretical physics.
A Chance Encounter and AI Breakthroughs
In fact, one of the most important moments in Viazovska’s journey happened on accident—a meeting with Sidharth Hariharan in Lausanne, Switzerland. Her interest in formalizing sphere packing proofs was mostly motivated by her own curiosity on the topic, a desire that was rekindled by this encounter. The partnership resulted in the starting of the Formalising Sphere Packing in Lean project in March 2024.
As a final step of this ambitious endeavor, researchers set out to prove Viazovska’s groundbreaking proofs with the help of advanced AI technology. The project harnessed a reasoning agent named Gauss, which demonstrated remarkable capabilities by autoformalizing Viazovska’s 24-dimensional sphere packing proof within just two weeks.
“They told us that they had finished 30 ‘sorrys’, which meant that they proved 30 intermediate facts that we wanted proved,” – Sidharth Hariharan
Gauss’s journey confirmed Viazovska’s her one true proof. Perhaps like no other conference, it highlighted the new frontier of artificial intelligence, which now can be a powerful partner in mathematical inquiry. This accomplishment is a huge leap forward in the combination of mathematics and technology.
The Impact of Formal Verification
Formal verification of computational proofs has long been considered a key research milestone. According to Liam Fowl, a researcher involved in this work, “Formal verification of a proof is like a rubber stamp.” This cycle makes sure we’re not just correct with our proofs but formatively checked in the most rigorous formal systems to date.
Added to this concern was the difficulty of a 24-dimensional case, which served as another barrier to entry for many researchers. As Jesse Han noted, this situation is infinitely more complicated than the 8-dimensional case. He noted that this complexity is mainly due to the need for a lot of supporting material on the nature and specialness of the Leech 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
Researchers were optimistic about what they have produced so far. Han described a major turning point that took place in mid-January, allowing them to build a more sophisticated version of Gauss.
“We made a research breakthrough sometime mid-January that produced a much stronger version of Gauss,” – Jesse Han
Hariharan echoed this sentiment, stating, “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.”
Looking Ahead
It’s to this last point that Maryna Viazovska’s achievements resound most profoundly, extending well beyond her own personal achievements. She is only the third woman ever to receive the Fields Medal. Her success is a testament to what can be accomplished and serves to inspire generations of mathematicians to come. Her contributions have immeasurably deepened our understanding of high-level mathematics. It further demonstrates how artificial intelligence can be of critical assistance for formalizing intricate proofs.
This partnership between mathematicians and AI systems such as Gauss bodes well for a new era in mathematical research. We all know that technology is changing fast. Get ready for new innovations that will revolutionize the way mathematicians approach the most difficult problems and proof their solutions.