Maryna Viazovska‘s groundbreaking contributions to the field of mathematics have received global recognition, culminating in her receiving the prestigious Fields Medal in July 2022. Viazovska’s work focused on the complicated sphere packing problem. She started out inquiring about ways to pack the same circles, or spheres, as tightly as possible in various dimensions. Together, her work beat both primary whipsawing versions of the problem and accomplished stunningly impressive work. She showed that the E8 lattice realizes the densest packing in eight dimensions and the Leech lattice is the densest packing in 24 dimensions.
The Sphere Packing Problem
It would seem like a simple idea, but the sphere packing problem has baffled mathematicians for centuries. It leads to some deep mathematical questions, for example on the packing of spheres in n-dimensional Euclidean space. In 2016, Viazovska achieved an amazing success by solving the sphere packing problem in the eight and twenty-four dimensions. Her results proved that the E8 lattice provides the densest packing in eight dimensions. In contrast, the Leech lattice does this in 24 dimensions.
The consequences of her research go beyond simple mathematical intrigue. Applications of efficient sphere packing Efficient sphere packing has implications in coding theory, telecommunications, and material science. When you learn how to efficiently pack these spheres, you receive breakthroughs in data communication and storage. Beyond improving processes, this understanding is driving the development of new materials.
In early 2024, the Formalising Sphere Packing in Lean project started with an ambitious project kick off. Its goal was to lay down Viazovska’s proofs in Lean, a programming language uniquely designed for writing computer-verifiable mathematical proofs. We hope that this most recent endeavor serves as a reminder of the incredible synergy at play between mathematics and AI. Tools like Lean free mathematicians to pursue greater levels of rigor and correctness.
Autoformalization and AI Collaboration
A recent milestone is advanced reasoning agent Gauss autonomously autoformalizing Viazovska’s proof of the optimal sphere packing in 24 dimensions. Gauss achieved this monumental task in all of two weeks, producing a codebase of over 200,000 lines of Lean code. This accomplishment marks an important milestone for autoformalization. It also demonstrates the thrilling potential of AI to bolster human collaboration in mathematical research.
Sidharth Hariharan, one of the other collaborators on this project, was just as astonished by Gauss’s capabilities. “When they reached out to us in late January saying that they finished it, to put it very mildly, we were very surprised,” he stated. The efficiency and rigor with which Gauss dealt with the involved formalization process have established new standards in the discipline.
This partnership between AI and human mathematicians has turned out to be beneficial to both. Hariharan soon came to understand that the formalization process was becoming a way for them to automate everything. All the while, though, he was immersing himself in the practice of mathematical understanding. “They told us that they had finished 30 ‘sorrys’, which meant that they proved 30 intermediate facts that we wanted proved,” he explained.
All that innovative work didn’t come without challenges. Jesse Han pointed out that “it was actually significantly more involved than the 8-dimensional case,” due to the extensive background material needed regarding the properties of the Leech lattice. All of this complexity highlights the importance of pairing human subject matter expertise with AI capacity to help guide AI through these complex, mathematical terrains.
The Future of Mathematical Research
The recent advancements represent the leading edge of a radical change in how mathematical proofs can be conceived and checked. AI is changing the landscape of formal verification. Now mathematicians all over the world dream of a future where they can express their creativity, unencumbered by the need for tedious verification work. “A programmer used to be someone who punched holes into cards, but then the act of programming became separated from whatever material substrate was used for recording programs,” Han remarked. “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.”
Together with her 8-dimensional proof, the successful formal verification of Viazovska’s 24-dimensional proof was announced on February 23. Accomplishments like this illustrate just how quickly AI is advancing the field of mathematical research. Liam Fowl emphasized the significance of these developments, stating, “Formal verification of a proof is like a rubber stamp.” This rigorous procedure of the sequent calculus guarantees complete precision, which increases trust in mathematical results.
As artificial intelligence becomes more integral to our lives, it will play an increasingly greater role in mathematics. These new tools can help researchers not just in double-checking existing proofs but in pushing mathematics into previously unknown frontiers. Systems that combine human intuition and machine precision can lead to unprecedented new discoveries.