Maryna Viazovska, a prominent mathematician from Ukraine, made headlines in July 2022 when she received the prestigious Fields Medal, often referred to as the Nobel Prize for mathematics. This honor was a historic achievement in her career. In doing so, she became only the second woman to ever receive the Fields Medal since its inception 86 years ago. Her accomplishment stands out in stark relief to her home country, Ukraine, being attacked by Russian forces. This lends her remarkable journey a particularly poignant touch.
Viazovska is perhaps best known for her groundbreaking work on the sphere packing problem. This very complicated mathematical problem is about how many of the same size spheres can you pack together in n-dimensional space. In 2016, she set her sights on a two-dimensional challenge. She proved that a symmetric arrangement known as E8 provides the densest packing possible in eight-dimensional space. That same year, she worked with other mathematicians to prove that the Leech lattice is the most efficient packing of 24-dimensional spheres.
The Sphere Packing Problem
The sphere packing problem has historically fascinated mathematicians, offering both frustration and discovery to the contour of those impassioned, geometric minds. It’s a question of how close the same shape—say, a bunch of oranges—can be packed together in a defined area while maintaining close contact. This is a question that is more than theoretical. It’s not just theoretical—PDF has practical applications in areas such as coding theory, crystallography, and data science!
Maryna Viazovska’s contributions to this problem are nothing short of monumental. Her early efforts in 2016 proved as much, setting the stage by showing that E8 is the optimal packing configuration for eight dimensions. This remarkable breakthrough was not just an example of her mathematical genius, it branched out into uncharted territory in research about the packing configurations of higher dimensions.
Her more recent collaboration with Sidharth Hariharan turned out to be just as fruitful. Together, they tackled the challenge of formalizing proofs related to the Leech lattice, which was considerably more complex due to its unique properties and nuances.
Collaboration and Innovation
The close partnership between Viazovska and Sidharth Hariharan pushed their research on sphere packing forward in leaps and bounds. They first met in Lausanne, Switzerland, where Hariharan’s burgeoning skill in formalizing proofs rekindled Viazovska’s interest in this area of mathematics. With an emphasis on the importance of rigorous proof verification, they started the Formalising Sphere Packing in Lean project in March 2024.
This collaboration would prove remarkably fruitful as they dove headfirst into the Leech lattice’s many complexities. Hariharan noted that this process was “significantly more involved than the 8-dimensional case,” as it required him to bring together background material surrounding the properties of the lattice itself.
This collaborative effort led to substantial breakthroughs. “We made a research breakthrough sometime mid-January that produced a much stronger version of Gauss,” said Jesse Han, who was part of the team working alongside them. Though their actual results were modest, their efforts showed a new path forward for formal verification in mathematics.
The Role of AI in Mathematical Proofs
However, the field of mathematics has recently experienced fundamental change due to the emergence of artificial intelligence and its effect on proof verification. Against this backdrop, Math, Inc. created Gauss, a state-of-the-art reasoning agent that can automatically formalize mathematical proofs. Amazingly, Gauss autoformalized Viazovska’s 24-dimensional sphere packing proof in only two weeks of wall-clock time, checking more than 200k lines of code.
“Formal verification of a proof is like a rubber stamp.” This development marks the beginning of an exciting new era of fast, efficient mathematical verification techniques. Aesthetic intuition together with rigorous proof systems gives mathematicians superpowers. This helps them focus their attention and resources on the most innovative and exciting parts of their work.
As events unfolded, Jesse Han reflected on the possibility of what such technology could do in the future. Technology of this type will help mathematicians to take control of their field. It will give them more time to do what they do best—envisioning new mathematical universes. This mindset change embodies the increasing understanding that AI is meant to complement human creativity, not take its place.
The future of collaboration and interaction between human mathematicians and AI systems like Gauss looks bright based on these initial encouraging results. Hariharan recounted a moment when the system identified a typo in their project: “One of them helped us identify a typo in our project, which we then fixed.” These examples show how AI can support what humans do best in complex and demanding areas that call for exactness and accuracy.