Maryna Viazovska is a celebrated mathematician from Ukraine. In July 2022, she shattered ceilings again when she became the first woman to receive the Fields Medal — considered by many to be the Nobel Prize for mathematics. Her selection for this accolade, part of the Academy’s 86-year history, made her the second woman ever to receive this premier award. Her award came at a substantially violent time for her homeland, which was undergoing invasion by the Russian Federation only months before. Viazovska’s groundbreaking work on the sphere packing problem has pushed our mathematical knowledge even further. It has raised awareness of the new possibilities that artificial intelligence opens up to allow us to formalize complicated proofs.
The sphere packing problem is an entirely distinguished mathematical challenge, seeking the densest packing of identical circles or spheres that can fit into n-dimensional space. Mathematician Maryna Viazovska solved one particular rather extraordinary instance of this hard-to-understand problem, making fundamental advances to the field of mathematics. Back in 2016, she had already shown that a symmetric arrangement called E8 gives the most efficient packing in eight dimensions. Along with other researchers, she and her collaborators made a game-changing discovery. In so doing they proved the Leech lattice to be the best possible sphere packing in 24 dimensions. These accomplishments have made her one of the top-tier mathematicians of her generation.
The Sphere Packing Problem: A Mathematical Challenge
The sphere packing problem has long captivated and confounded mathematicians, representing one of the greatest challenges in all dimensions. The packing problem asked how efficiently one could arrange identical spheres. It aspires to ask how such spheres can intervene in space without interrupting, without remnants of a cut. This philosophical inquiry has ramifications, both large and small, across many scientific disciplines, including physics, computer science, and environmental science.
In her work, Viazovska solved the exact same problem in two specific dimensions. She proved that the E8 arrangement is the most efficient possible packing of spheres in eight-dimensional space. Her pioneering research brought to light this incredible discovery. This discovery not only answered an age-old mathematical mystery, but it illustrated Viazovska’s raw analytical genius and unconventional thinking.
In the same vein, her work on the Leech lattice further generalized these results out to 24 dimensions. The Leech lattice, while at first seemingly esoteric, is actually a beautiful and infinitely complicated structure that has baffled mathematicians for 40 years. Viazovska’s proof that it represents the best sphere packing solution in this realm further solidified her status as a leading figure in contemporary mathematics.
The Role of AI in Formalizing Mathematical Proofs
Mathematics is an iterative and ever-evolving process. The convergence of artificial intelligence and mathematical research presents exciting new avenues to explore and discover. In an extraordinary breakthrough, Purdue University’s Sidharth Hariharan aided Viazovska to formalize her proofs. To do this, he leveraged the Lean programming language and interactive theorem prover. This process opened the floodgates to rigorous verification process, which is crucial in the world of mathematics.
On February 23, 2023 the final formalization of Viazovska’s proof for 8-dimensional sphere packing was announced. This mathematical earthquake highlights the importance of precision and clarity in formal mathematical communication. The adoption of formal methods allows proofs to be independently verified, greatly increasing their credibility and robustness.
Math, Inc. celebrated the launch of its proprietary AI system—Gauss. In under two weeks Gauss autoformalized Viazovska’s proof for packing spheres in 24 dimensions. Gauss went through more than 200,000 lines of code to pull off this extraordinary accomplishment. Jesse Han, the CEO and co-founder of Math, Inc. elaborated on Gauss as a robust reasoning agent. In his presentation, he noted its ability to integrate conventional natural language reasoning with fully formalized reasoning. It’s this dual capability that makes the process of confirming mathematical proofs more efficient and accurate.
A New Era of Curiosity and Collaboration
Maryna Viazovska’s motivation for ways to formalize her proofs originally came from the sheer scientific inquisitiveness about the procedural description. She readily adopts new computational methods. This is part of a trend we’re seeing more generally in mathematics, where the combination of human ingenuity and AI have proven to make big breakthroughs possible.
These new, fast-moving AI technologies have already changed how mathematicians solve problems and create solutions. AI systems such as Gauss can automate some of the formalization process. This saves researchers from monotony, allowing them to focus on the more artistic components of their work.
Viazovska’s work with AI represents the beginning of an exciting new chapter in mathematical research. For the first time ever, human intuition and machine precision can truly complement each other. When coupled with other burgeoning technologies, they hold the potential to pave exciting new avenues toward tackling the most difficult challenges that have long puzzled mathematicians.