In July 2022, Maryna Viazovska became the first woman to receive the prestigious Fields Medal for her groundbreaking research on sphere packing. Considered by many to be the Nobel Prize for mathematics, this award honors outstanding achievements in pure and applied mathematics. Viazovska’s work addressed two surprisingly fundamental variants of the sphere packing problem. This research problem aims to determine how closely we can pack the same-sized circles (or spheres) in n-dimensional space. Her research culminated in significant findings: she proved that a symmetric arrangement known as E8 represents the best packing in eight dimensions and shortly thereafter demonstrated that the Leech lattice is optimal for 24 dimensions.
The Sphere Packing Problem Explained
The sphere packing problem has mystified mathematicians for ages. At its heart, it wants to know how efficiently the same identical spheres can be packed into a defined space without leaving any holes. This challenge stretches across many dimensions, which is what makes it a rich and in many ways, mysterious area of study.
It was precisely in the special circumstances of these eight and twenty-four dimensions, where the intuition of the proof would fail, where Viazovska’s work shined. Her discoveries pushed the frontiers of mathematics to a deeper understanding. They gave their findings concrete applications to coding theory and cryptography.
To prove their optimality, Viazovska used very complex mathematical structures to win out the E8 and Leech lattice packings. The E8 lattice shows optimum packing in the eight-dimensional space, realized by a highly symmetrical configuration. The Leech lattice, conversely, is known for its exceptional packing properties in 24 dimensions.
The Role of AI in Proof Verification
The formal verification of Viazovska’s 24-dimensional proof was a truly collaborative process that fed the experience of the human mathematician into the latest AI technology. The project, called Formalising Sphere Packing in Lean, started in March 2024. Its aim was not to repeat existing ways of learning, but rather deepen comprehension through rigorous proofs.
Sidharth Hariharan, a major contributor to this initiative who identified the educational benefits of formalizing proofs, described how formalization is an educational tool. He noted that they had previously worked on formalizing the eight-dimensional sphere packing proof before transitioning to the more complex 24-dimensional case.
Jesse Han, one of the team members behind the project, shared that this language model is a special type referred to as a reasoning agent. It actually straddles the line between classical natural language reasoning and fully formalized reasoning. This cutting-edge AI model allowed researchers to cut down the detailed verification process considerably.
With the magic of Gauss on their side, the team accomplished exceptional efficiency. The AI produced answers that previously took weeks in only a few days. We spent two years exclusively constructing the repository for the project. In June 2025, we celebrated our proudest accomplishment—opening access to the public to foster transparency and collaboration. Hariharan underscored.
Impacts and Future Prospects
The impact of this technological partnership has echoed across the mathematical community since. In fact, a number of experts have recently raised their voices in enthusiasm, sharing their excitement over AI’s prospects to help mathematicians better solve complex problems.
The proof is accepted, but formal verification of a proof acts like a rubber stamp, Liam Fowl explained. He pointed to the key importance of AI in providing complete validation. During this 45-minute discussion, he lavished praise on the remarkable outcomes generated by these great collaborative efforts. He added, “These new results appear extremely, extremely impressive and obviously portend some very rapid progress in this realm.”
Matthew Han seconded this sentiment about how much more complicated the process of formalizing the 24-dimensional proof is versus its eight-dimensional one. He noted that the job was much more complicated. Even from the start, there were many background things that needed to be developed, especially about different aspects of the Leech lattice.
As researchers and instructors alike refine these cutting-edge methods, the role of AI in mathematics is only set to grow. Hariharan wrapped up the discussion on a hopeful note. As he put it, “At the end of the day, this technology is really exciting! It can do wonderful things and help mathematicians do amazing things.”