Maryna Viazovska’s name might not ring a bell with you, but she has achieved something considered nearly impossible. Her results have been groundbreaking in both eight-dimensional and twenty-four-dimensional spaces. Her pioneering research has upended foundational concepts in mathematics. In July 2022, she became the second woman ever to win a Fields Medal—the highest honor a mathematician can receive—in the award’s 86-year history. The sphere packing problem is a mathematical question about how closely identical spheres can be packed into n-dimensional space. This curious conundrum has captivated mathematicians for centuries. The surprising math genius of Viazovska’s application of quasi-modular forms proved that the E8 lattice tropically represents the densest packing of spheres in eight dimensions. All this time, she and her co-authors had been proving the Leech lattice accomplishes the same in twenty-four dimensions.
The implications of Viazovska’s findings extend beyond theoretical mathematics. They have prompted renewed interest in formalizing mathematical proofs through programming languages. In March 2024, a pivotal project called Formalising Sphere Packing in Lean emerged when Viazovska met Sidharth Hariharan in Lausanne, Switzerland. Hariharan’s work specifically looked at using the Lean programming language to formalize Viazovska’s proofs. While an impressive undertaking, this effort ignited a broader movement to improve the accuracy and transparency of mathematical verification.
The Sphere Packing Problem Explained
The sphere packing problem is a fundamental question that asks how densely spheres can be arranged in various dimensions without gaps. In three dimensions, for example, the most efficient packing is a pyramid-like lattice where oranges, marbles or any object with some roundness fits perfectly in a pyramid shape. In two dimensions, the honeycomb arrangement turns out to be the best – no surprise there! These different setups are real-life examples of how math in the theory books translates to the extreme applications.
It was this unorthodox path into this very complicated field that led to Viazovska’s extraordinary findings. Just as effective packing arrangement for eight-dimensional spheres, and she had successfully proved this! This accomplishment led not just to furthering mathematical theory, but revealed the complex connections between dimensions and packing efficiency.
In addition to her individual work, Cohn’s collaboration with fellow graduate students led to an exciting discovery about twenty-four-dimensional packing. To prove that the Leech lattice is, in fact, the coolest possible configuration for this dimension, Viazovska and her team set to work. The level of comprehension needed to fully understand these concepts called for a heavy basis of both theory and hands on experience.
“These new results seem very, very impressive, and definitely signal some rapid progress in this direction,” – Liam Fowl
The effort led by Hariharan and Viazovska is an impressive step in the direction of programming computers to help formalize mathematical proofs. Lean, which has been described as a “proof assistant,” lets mathematicians write proofs that a computer can check for total correctness. This process is what makes the outcome of every single step you take in building a proof follow ironclad logical rules.
In March 2024, they set about formalizing Viazovska’s proofs because of their promise to make profound advances in mathematical understanding. Perhaps the greatest of those advances are made. The collaboration aimed to enhance accuracy and rigor within their findings while making them more accessible to the broader mathematical community.
Hariharan noted how the project evolved over time: “We had been building the project’s repository for about two years when we enabled public access in June 2025.” This transition created new avenues for partnership. Mathematicians are more forthcoming with their comments and criticisms, improving the overall product.
Significant advancements were made during this period. Jesse Han, another collaborator, noted: “We made a research breakthrough sometime mid-January that produced a much stronger version of Gauss,” referring to their enhanced proof verification system. That’s because this new version would allow them to reproduce results that once took weeks in only a few days.
“It’s a particular kind of language model called a reasoning agent that’s meant to interleave both traditional natural language reasoning and fully formalized reasoning,” – Jesse Han
Achievements and Impact
Maryna Viazovska’s influence on mathematics goes far beyond her technical results in creating a more effective sphere packing. In this context, her receipt of the Fields Medal serves to quickly underscore her remarkable accomplishment. Beyond that, it encourages the next generation of mathematicians, and especially women who are pursuing STEM fields.
The importance of her work has impressed deep and broad interest from all corners of the academic community. We explore how the intersection of math and computer science has created exciting new areas of opportunity. Now, statements and arguments can be formalized and verified in exciting new ways!
Sidharth Hariharan spoke to the collaborative nature of their work: “They told us that they had finished 30 ‘sorrys’, which meant that they proved 30 intermediate facts that we wanted proved.” This serves as a great example of the careful work involved in formal verification processes, and the power of teamwork to create significant progress.
As research progresses into the sphere packing field, there is still hope for future findings. The underlying theories that relate the eight-dimensional and twenty-four-dimensional cases hold rich opportunities for study and understanding.