Maryna Viazovska, one of the most accomplished young mathematicians, made headlines after solving one of the most perplexing sphere-packing problems. She was awarded the prestigious Fields Medal in July 2022, considered the Nobel Prize for mathematics. In 2016, she brought radical new approaches to both flavors of the sphere-packing problem. These innovations have dramatically increased development for uses of artificial intelligence that can now automatically verify mathematical proofs. This exciting new partnership between mathematicians and technology holds great promise for helping students better understand and check their work on increasingly complex mathematical ideas.
The sphere-packing problem, one of mathematic’s best known unsolved problems, challenges researchers to find the densest packing of equal circles or spheres in n-dimensional space. This deep and complex challenge has fascinated mathematicians for centuries, but recent advances have made landmark progress possible in its formal verification. Jesse Han, the CEO and co-founder of Math, Inc., has led a profound culture change within his company. To do this, he created Gauss, an AI that merges human-like reasoning with fully formalized logic.
Breakthroughs in Sphere-Packing Proofs
This year, Gauss has turned out to be a dramatic example of those capabilities. Just two weeks to autoformalize Maryna Viazovska’s proof for 24-dimensional sphere packing, a grand challenge that involves over 200,000 lines of code! The speed and accuracy with which this AI tool has recreated the problem has left many inside the mathematical community stunned.
Han emphasized the complexity of the task, stating, “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 on line surrounding many of the properties of the Leech lattice, in particular its uniqueness.” This is a point issuing from the statement which depicts the level of understanding needed to approach these very high level mathematical ideas.
In the end, Gauss not only formalized proofs, but discovered and fixed a typo in Viazovska’s published paper. This move illuminated the possibility for greater accuracy and precision in mathematical documentation. The ramifications of this technology go far beyond just verification. They seek to change how mathematicians engage with intricate proofs.
The Collaborative Effort
In addition to his research, Sidharth Hariharan, a Ph.D. student at Carnegie Mellon University, has been deeply engaged in the sphere-packing proof verification. He made the experience of formalization a laboratory through which he could pursue the most rigorous and innovative potential understanding of mathematical principles expressed therein. Speaking about the collaborative nature of this project, he noted, “When they reached out to us in late January saying that they finished it, to put it very mildly, we were very surprised.”
A remarkable and deep collaboration developed between these experts to translate their findings into a human-readable, step-by-step “blueprint” for Viazovska’s 8-dimensional proof. This project, more formally packing spheres into higher-dimensional manifolds in Lean, started formally in March 2024. Beyond giving more people the ability to understand complex proofs, the team wanted to push the field of proof verification forward.
“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 added, underscoring the transformative potential of AI in mathematics.
Implications for the Future
This world in which AI can now assist in verifying mathematical proofs marks a new era in the field of mathematics. Within the field, experts agree that this technology will tremendously speed up progress. Liam Fowl, a mathematician observing these developments, stated, “These new results seem very, very impressive and definitely signal some rapid progress in this direction.”
Han and his team hit a key research achievement in about mid-January that resulted in an improved version of Gauss. This new version offers even more efficiency and accuracy when formalizing mathematical proofs.
As researchers by the dozen dive into realizing the full potential of Gauss and similar technologies, the future of mathematical verification appears profoundly bright. There’s transformative potential during this endeavor that aims to apply broad comprehension of AI technologies to uncover fresh, astonishing forms of complex mathematical theories.